# Derived Satake *Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).* For a connected group, equivariance on a point leaves only vector spaces in the perverse heart. Its derived category still remembers the cohomology of the classifying space. This is already enough to show why deriving the abelian Satake equivalence does not give derived geometric Satake. We calculate that difference, prove the torus model, and specify the additional categories in the general theorem. ## 1. Which derived category? Work with complex algebraic groups and their classical topology. Cohomological grading means \(H^i(K[s])=H^{i+s}(K)\); in particular a vector space \(V[-2]\) has its vectors in degree \(2\). Write \(\Lambda\) for a characteristic-zero field. The commutative polynomial-simplex model in §5 applies to every such field. Smooth complex forms are used in §3 to prove restriction of classifying-space cohomology; the rational descent there proves its general coefficient-field statement. There are two constructions to distinguish: - \(D^b(\mathrm{Sat}_G)\), the bounded derived category of the abelian perverse category; - \(D^b_{L^+G,c}(\mathrm{Gr}_G)\), the Borel equivariant constructible category, with bounded underlying complexes and finite Schubert support. In the second category, equivariance includes the compatible derived data on all levels of the action nerve. Equivalently one uses finite-dimensional sufficiently acyclic free-action models, in the range of degrees under consideration. Equivariant perverse sheaves and perverse sheaves on stacks, Theorem B.9 constructs those bounded models and their comparisons. Its §4 calculates the first extra class for \(\mathbf G_m\). An isomorphism on the first action product, with a cocycle on the second, describes heart equivariance; it does not replace the full derived construction. On each finite model, bounded-below injective complexes and their Hom complexes give the enhancement used for derived sections. Sheaves of modules and their derived categories, Theorem 6.1 and Lemmas 6.2–6.3 prove resolution existence and the quasi-isomorphism comparison of those Hom complexes. Their differential is \(d(f)=d_Jf-(-1)^{|f|}fd_I\), and composition satisfies the graded product rule by expansion. The bounded model comparisons are taken in the degree range under consideration. The explicit compatible torus construction is given in §5; no general enhanced-descent theorem is inferred from triangulated full faithfulness alone. Mapping objects retain these complexes and composition; cones are homotopy cones. Thus \(R\operatorname{Hom}(A,B)\), and not just its cohomology groups, is part of the construction. A **perfect** module over a differential graded algebra is an object obtained from finite sums of shifts of the free module by finitely many cones and direct summands. Equivalently it belongs to the smallest idempotent-complete stable subcategory containing that free module. This definition does not require the algebra to be concentrated in degree zero. ## 2. The unit sees a classifying space The base point \(e=t^0\) is fixed by \(L^+G\). Its unit sheaf is \(\mathbf1=i_*\Lambda_e\). Closed extension is fully faithful also in the equivariant derived category: in every free-action model it is closed extension along the corresponding closed subspace, and the ordinary identity \(i^*i_*=1\) identifies the unit and counit. It is compatible with the transition models. Hence \[ R\operatorname{Hom}_{L^+G}(\mathbf1,\mathbf1) \simeq R\Gamma(BL^+G,\Lambda). \tag{2.1} \] Indeed on a model of the point quotient the object is the constant sheaf. Its derived endomorphisms are the sections of its coefficient-resolution complex. Products are cup products, because composition of constant-sheaf endomorphisms is the product in that resolution. This proves (2.1) as an endomorphism algebra, rather than only an equality of dimensions. Evaluation at \(t=0\) induces \[ L^+G\longrightarrow G, \tag{2.2} \] with constant loops as its section. Its kernel is contractible in the coefficient topology of formal jets. Explicitly \(g(t)\mapsto g(st)\), \(0\leq s\leq1\), fixes constant loops and at \(s=0\) is \(g(0)\). Substitution preserves the defining polynomial equations of \(G\) and the group product. On each jet quotient it is polynomial in the jet coordinates and \(s\); the maps are compatible under truncation. On the inverse-limit topology it is therefore continuous. The kernel is contracted to its identity. This induces the classifying-space comparison at each bounded model and in each fixed cohomological degree. One can also check it on the bar nerve: the maps on its degree-\(r\) spaces are the corresponding maps on \((L^+G)^r\) and \(G^r\), and the contraction respects faces and degeneracies because it is a group homomorphism at each parameter. Realizing the homotopy gives the same comparison. Consequently \[ \boxed{\ \operatorname{Ext}^*_{D^b_{L^+G,c}(\mathrm{Gr}_G)} (\mathbf1,\mathbf1)=H^*(BG,\Lambda).\ } \tag{2.3} \] The jet construction of The Satake category, §2 supplies the finite algebraic quotients used here; their successive kernels are vector groups in the classical topology. Equation (2.3) is a topological Borel computation. It is not a claim that the algebraic loop quotient functor is unchanged on nonreduced rings. ## 3. Two classifying-space calculations ### 3.1. A circle and a degree-two generator The group \(\mathbf C^\times\) retracts to its unit circle by \(z\mapsto z|z|^{-s}\). A model of the circle's universal bundle is \[ S^\infty\longrightarrow\mathbf {CP}^\infty. \tag{3.1} \] Here the unit vectors have finitely many nonzero complex coordinates, with the weak topology from their finite spheres. This space is contractible. If \(S\) inserts a zero first coordinate, the normalized vector \((1-s)v+sS(v)\) never vanishes: for \(s>0\), use the last nonzero coordinate of \(v\) in its new shifted position. This deforms \(v\) to \(S(v)\). The latter is orthogonal to the first coordinate vector, so the path \(\cos(\pi s/2)S(v)+\sin(\pi s/2)e_0\) contracts it to \(e_0\). The homotopies are continuous at every finite stage and on products with the compact parameter interval. The compact-exhaustion product proof is in Grassmannians and classifying maps, Lemma 2.1. Coordinate charts of projective space give local trivializations of (3.1). The integral computation \[ H^*(\mathbf {CP}^n,\mathbf Z) =\mathbf Z[c]/(c^{n+1}),\qquad |c|=2, \tag{3.2} \] is proved in Gysin sequence and projective splitting, Theorem 3.2. Briefly, the unit-circle bundle of the tautological line has total space \(S^{2n+1}\). Its Gysin sequence says that multiplication by its Euler class is an isomorphism between consecutive even cohomology groups below the top degree. The cell decomposition has one cell in degrees \(0,2,\ldots,2n\) and none elsewhere; thus there are no further groups, and the powers of the class generate. The class \(c=-c_1(\gamma)\) evaluates to \(+1\) on the complex-oriented projective line. Compatibility of the inclusions and stabilization in every fixed degree give \[ H^*(B\mathbf G_m,\Lambda)=\Lambda[c],\qquad |c|=2. \tag{3.3} \] There is one group in every nonnegative even degree and no odd group. The universal-coefficient passage has no torsion term, since the integral cell homology is free. For a split torus of rank \(r\), take products of these bundle models. The chain product comparison and the same free even cell groups give \[ H^*(BT,\Lambda)=\Lambda[c_1,\ldots,c_r],\qquad |c_i|=2. \tag{3.4} \] ### 3.2. \(SL_2\) and a degree-four generator The maximal compact subgroup of \(SL_2(\mathbf C)\) is \(SU_2\), which is the group of unit quaternions. Polar decomposition gives a deformation to it: write \(g=up\), with \(u\) unitary of determinant one and \(p\) positive Hermitian of determinant one, then use \(up^{1-s}\). For the right quotient, \(gSU_2\mapsto gg^*\) identifies the quotient with the positive Hermitian determinant-one matrices; its inverse is the coset of the positive square root. The matrix logarithm identifies that space with the vector space of traceless Hermitian matrices, so it is contractible. The locally trivial associated bundle \(EG/SU_2\to EG/G\) has that fibre. Its homotopy sequence and the weak-equivalence cohomology comparison identify the classifying-space cohomology rings. Those two results, including their proofs, are Homotopy fibres and the Serre spectral sequence, Theorems C.2 and B.4. A universal unit-quaternion bundle is \[ S^\infty_{\mathbf H}\longrightarrow\mathbf {HP}^\infty. \tag{3.5} \] The same shift and normalization contraction proves that its total space is contractible. The quaternionic projective charts are the charts of one-dimensional right quaternionic subspaces, and their unit frames give local trivializations. For the finite space \(\mathbf {HP}^n\), normalize the last nonzero homogeneous coordinate to one. Its strata are affine quaternionic spaces of dimensions \(0,1,\ldots,n\). The closed filtration by \(\mathbf {HP}^j\) therefore attaches one real cell in every degree \(4j\). The characteristic map is the projectivization of \((z,\sqrt{1-\|z\|^2})\) on the closed quaternionic ball; on the boundary it lies in the preceding projective space. Compactness and the projective chart inverse show that the attachment quotient is the stated space. Thus its additive integral cohomology is \(\mathbf Z\) in degrees \(0,4,\ldots,4n\), with no other groups. Let \(Q\) be the tautological quaternionic line, viewed as an oriented real rank-four bundle. Its sphere bundle is \(S^{4n+3}\). The rank-four Gysin sequence, proved in Gysin sequence and projective splitting, Theorem 1.1, now makes multiplication by \(e(Q)\) an isomorphism from degree \(4j\) to degree \(4j+4\) for \(0\leq j0. \tag{3.12} \] Here \(\eta\) is a closed curvature representative of the global Chern class. Its fibre integral is the same Chern number on every fibre by local trivialization and naturality. A positive power of the ample line embeds \(G/B\) in projective space. Divide its restricted positive Fubini--Study form by that power; its positive top volume proves \(s>0\). The Chern and orientation comparisons identify it with the integral Chern number. These comparisons and positivity are proved in The Satake category, Theorem 5.7 and Connections, curvature and characteristic forms, Parts A and D. Equation (3.12) proves that pullback along \(q\) is injective over \(\mathbf C\). It is also injective over \(\mathbf Q\). For any space, split its rational singular-chain complex into its homology and two-term contractible summands by choosing complements to boundaries in cycles and to cycles in chains. Its cohomology with values in a field \(K\supset\mathbf Q\) is therefore \(\operatorname{Hom}_{\mathbf Q}(H_j(-;\mathbf Q),K)\). The coefficient map from \(K=\mathbf Q\) to \(K=\mathbf C\) is injective, without a finite-dimensionality assumption. A rational class killed by flag pullback is killed after extension to \(\mathbf C\), where the preceding paragraph proves injectivity. It is thus zero already over \(\mathbf Q\). The map \(V/T\to V/B\) has fibre \(B/T\), an affine space by the ordered root coordinates of AG-RG-04 §5. Its torsor product charts make it a locally trivial contractible-fibre bundle. GL-PERV Lemma B.5 proves its adjunction unit is an isomorphism, without a nonproper fibre-base-change assumption. We have therefore proved, initially for \(K=\mathbf Q,\mathbf C\), \[ H^*(BG;K)\hookrightarrow H^*(BT;K) \tag{3.13} \] in each prescribed range, and hence in every degree by increasing the frame. At the same finite frame the right action of \(N_G(T)\) on \(V/T\) preserves its map to \(V/G\). It factors through \(W\), and acts on the character Chern classes by the ordinary Weyl action. Thus the image of (3.13) lies in \(H^*(BT;K)^W\). This invariance does not require an unproved statement about arbitrary group actions on classifying spaces. ### 3.5. Invariant polynomials and characteristic classes The centre of the enveloping algebra and Harish-Chandra’s theorem, Theorem 3.1 proves Chevalley restriction for a complex semisimple Lie algebra, with an actual graded polynomial proof: \[ \mathbf C[\mathfrak g]^{\mathfrak g} \xrightarrow{\sim}\mathbf C[\mathfrak t]^W. \tag{3.14} \] For reductive \(\mathfrak g=\mathfrak z\oplus[\mathfrak g,\mathfrak g]\), tensor that proof with the polynomial algebra on the centre. The Weyl group fixes the centre, so the same statement follows. Infinitesimal invariants equal \(G\)-invariants here. Root-vector infinitesimal invariance integrates by its finite polynomial exponential; torus infinitesimal invariance means weight zero; the torus and root groups generate the connected group by the proved Bruhat coordinates of AG-RG-04. Thus the notation in (3.14) really gives the algebraic adjoint invariants. We need the characteristic-class map for this general \(G\), not merely the matrix-group version in DG-CHAR. It has the following short proof from the existing principal-connection foundations. A smooth principal \(G\)-bundle has a connection by Connections and parallel transport, Theorem B.1. That theorem patches local product connections with a partition of unity and works for every finite-dimensional Lie group. For a complex group regard its connection as a form valued in its complex Lie algebra on the real base. Principal curvature, transformation, and Bianchi are proved in Curvature and holonomy groups, Theorems A.4, A.6–A.7. Let \(f\) be an invariant homogeneous polynomial of degree \(a\), and \(p\) its symmetric polarization. If \(\omega\) is the connection and \(F=d\omega+\tfrac12[\omega,\omega]\), put \[ \mathrm{CW}_f=p\bigl(iF/(2\pi),\ldots,iF/(2\pi)\bigr). \tag{3.15} \] Invariance makes this horizontal and invariant, hence a base form. To verify closedness and independence explicitly, for homogeneous valued forms \(\beta_j\) of degrees \(b_j\), differentiate the polarization identity \[ \sum_j p(X_1,\ldots,[Z,X_j],\ldots,X_a)=0. \] Move the connection one-form to the front in each term. Its Koszul sign cancels the differentiation sign, so \[ d\,p(\beta_1,\ldots,\beta_a) =\sum_j(-1)^{b_1+\cdots+b_{j-1}} p(\beta_1,\ldots,D\beta_j,\ldots,\beta_a). \tag{3.16} \] Since \(DF=0\), (3.15) is closed. For two connections take their affine path, with difference \(\alpha\) and curvature \(F_t\). The local bracket formula gives \(\dot F_t=D_t\alpha\). Equation (3.16) then gives \[ \mathrm{CW}_f(F_1)-\mathrm{CW}_f(F_0) =d\left[ a\left(\frac{i}{2\pi}\right)^a \int_0^1p(\alpha,F_t,\ldots,F_t)\,dt\right]. \tag{3.17} \] The difference is horizontal and equivariant; hence its displayed primitive is global. Pullback of the connection proves naturality. Sums and products of polynomials give sums and products of their curvature forms because two-form coefficients commute. This proves the required general Chern–Weil map. It is the invariant-contraction proof of Connections, curvature and characteristic forms, Theorem B.2 with its matrix-group restriction removed by the actual principal-group bracket calculation (3.16). Apply it to the finite principal bundle \(V\to V/G\). Restricting to \(V/T\) gives the principal \(T\)-reduction; connection independence permits a connection induced from that reduction. Thus restriction of (3.15) is the torus characteristic polynomial of \(f|_{\mathfrak t}\). For a character \(\chi\), Connections, curvature and characteristic forms, Theorem D.1 in the line-bundle case identifies its normalized curvature with \(c_1(\mathcal L_\chi)\). Therefore \[ \operatorname{res}(\mathrm{CW}_f) =f|_{\mathfrak t}\bigl(c_1(\mathcal L_{\chi_1}),\ldots, c_1(\mathcal L_{\chi_r})\bigr), \tag{3.18} \] when expressed in the linear coordinates \(d\chi_j\). These character classes generate \(H^*(BT;\mathbf C)\). If the associated standard-character line is taken to be the tautological line, its Chern class is the negative of the positive generator \(c_j\) of Lesson 13. Equation (3.18) retains that sign: identify \(d\chi_j\) with \(c_1(\mathcal L_{\chi_j})\), not silently with its negative. Choosing the dual character line gives the positive generator. Either convention gives the same polynomial-ring assertion. Every Weyl-invariant polynomial in these degree-two classes lifts through (3.14) to some \(f\). Equation (3.18) proves it lies in the image of (3.13). We have proved \[ H^*(BG;\mathbf C)\xrightarrow{\sim}H^*(BT;\mathbf C)^W. \tag{3.19} \] All constructions were performed on genuine finite smooth manifolds in the relevant range. No infinite-dimensional differential-form manifold or imported Borel theorem was used. ### 3.6. Rational descent and all characteristic-zero coefficients The rational injection (3.13) first gives finiteness in each degree over \(\mathbf Q\), because the torus groups are finite dimensional in each degree. On a sufficiently large finite frame, that torus calculation is the projective model calculation by the same common-product comparison used in GL-SAT-06, equation (5.26). Here is the coefficient passage without a finite-CW assumption. For the singular chain complex over \(\mathbf Q\), choose degreewise complements to boundaries in cycles and to cycles in chains. It splits as its homology with zero differential plus two-term contractible summands. Applying \(\operatorname{Hom}_{\mathbf Q}(-,\Lambda)\) therefore gives \[ H^j(Y;\Lambda)=\operatorname{Hom}_{\mathbf Q} (H_j(Y;\mathbf Q),\Lambda). \tag{3.20} \] If \(H^j(Y;\mathbf Q)=\operatorname{Hom}(H_j(Y;\mathbf Q),\mathbf Q)\) is finite dimensional, then \(H_j(Y;\mathbf Q)\) is finite dimensional: any arbitrarily large independent finite family in the latter has arbitrarily large independent dual functionals, extended from its span by a vector-space complement. In that case (3.20) is the canonical scalar-extension isomorphism \[ H^j(Y;\mathbf Q)\otimes_{\mathbf Q}\Lambda \xrightarrow{\sim}H^j(Y;\Lambda). \tag{3.21} \] Apply this degreewise to the sufficiently large finite Borel models. Finiteness follows from the trace injection and the bounded torus comparison, so (3.21) applies. The maps are natural and preserve products. Thus it also applies to the stable \(BG\) groups. Finite-group invariants commute with field extension by the averaging projector \(|W|^{-1}\sum_w w\). The map \[ H^*(BG;\mathbf Q)\longrightarrow H^*(BT;\mathbf Q)^W \] becomes (3.19) after extension to \(\mathbf C\), by (3.21). Faithful extension reflects kernel and cokernel degreewise, so it is already an isomorphism over \(\mathbf Q\), and then over every \(\Lambda\): \[ H^*(BG;\Lambda)\xrightarrow{\sim}H^*(BT;\Lambda)^W. \tag{3.22} \] Likewise (3.14) descends to the rational split model and extends to \(\Lambda\). In each polynomial degree, infinitesimal invariants are the simultaneous kernels of finitely many rational linear matrices, one for each element of a rational Lie basis. Kernels commute with field extension. The root-group and torus argument of §7 identifies them with the algebraic group invariants over every characteristic-zero field. The restriction map is rational, becomes an isomorphism over \(\mathbf C\), and therefore is an isomorphism over \(\mathbf Q\) and \(\Lambda\). Combining it with the character Chern-class identification of the torus and (3.22) proves (3.11). This also proves odd cohomology vanishing for \(BG\), but the stronger invariant-polynomial identification is what closes Lesson 13's general outline assertion. It is a cohomology-ring theorem; it does not claim commutative differential graded formality of \(BG\) or the general derived Satake equivalence. ## 4. Why the abelian equivalence cannot simply be derived For a split torus, The Satake category and Consequences and examples, §7 prove that the heart is the category of finite supported graded vector spaces. Every short exact sequence splits weight by weight, by choosing a complement in each vector space. A bounded complex in this heart splits into its cohomology and contractible two-term complexes: choose complements to boundaries inside cycles and to cycles inside each term. Therefore \[ \operatorname{Hom}_{D^b(\mathrm{Sat}_T)} (\mathbf1,\mathbf1[j])=0\quad(j\ne0). \tag{4.1} \] For \(T=\mathbf G_m\), equation (2.3) instead gives a nonzero degree-two morphism in the Borel equivariant category. The two categories cannot be equivalent through a functor sending their units to one another. In degree one, both calculations give zero, so the difference is already compatible with a semisimple heart. For \(SL_2\), the unconditional rank-one semisimplicity proof of Identifying the dual group, §3.3 gives the analogous vanishing in the derived heart, while (3.8) gives the extra degree-four Borel class. One must specify the group when asserting a nonzero higher self-Ext: its degree is two for a circle and four for \(SL_2\). ## 5. The torus category as perfect differential graded modules Let \(\mathcal C_T\) be the bounded Borel equivariant category supported on one reduced point of \(\mathrm{Gr}_T\). In the classical coefficient models, it is generated by its constant object \(\Lambda\). To see this, \(BT\) is simply connected: the local trivial bundle with contractible total space has \(\pi_1(BT)=\pi_0(T)=0\). Thus each finite-dimensional local system is constant. This also follows in every sufficiently acyclic finite projective model, whose fundamental group is zero. The cohomology sheaves of an equivariant object on the point are such local systems on the model; its underlying bounded finite-dimensional complex gives only finitely many of them. Ordinary truncation triangles express it as finitely many extensions of shifts of finite sums of \(\Lambda\). Direct summands are retained. Consequently \(\mathcal C_T=\operatorname{thick}(\Lambda)\). Put \(A_T=R\operatorname{Hom}(\Lambda,\Lambda)=R\Gamma(BT,\Lambda)\), retaining its differential graded algebra. The functor \[ F=R\operatorname{Hom}(\Lambda,-): \mathcal C_T\longrightarrow\operatorname{Perf}(A_T) \tag{5.1} \] is an equivalence. Here is the generator proof. It sends \(\Lambda\) to the free module \(A_T\), and the comparison of mapping complexes is an isomorphism on that generator by the definition of \(A_T\). Fixing one argument, the objects for which the comparison is an isomorphism form a stable subcategory closed under direct summands, because both mapping functors turn triangles into triangles. First extend in the second argument, then in the first. This proves full faithfulness on the thick subcategory. Its image contains the free module and is closed under shifts, cones and summands; full faithfulness lifts an idempotent and its splitting. It is therefore all of \(\operatorname{Perf}(A_T)\). This is differential graded Morita equivalence on the generated category, proved here. For every characteristic-zero field, we now construct a commutative model \[ A_T\simeq P_\Lambda=\Lambda[c_1,\ldots,c_r], \qquad |c_i|=2,\quad d=0. \tag{5.2} \] The generators retain the projective normalization \(c_i=-c_1(\text{tautological line})\). ### 5.1. Polynomial simplex forms For the affine standard simplex define the commutative differential graded algebra \[ \Omega_{\mathrm{pol}}(\Delta^p;\mathbf Q) = \mathbf Q[t_0,\ldots,t_p,dt_0,\ldots,dt_p]/ (\textstyle\sum t_i-1,\sum dt_i). \tag{5.5} \] The \(t_i\) have degree zero, the \(dt_i\) degree one, the latter anticommute, and \(d(t_i)=dt_i\). Its extension to \(\Lambda\) is defined by the same formula. Every affine map taking vertices to vertices induces a pullback of commutative differential graded algebras. **Lemma 5.1.** Compatible polynomial forms on any union of faces extend to the simplex, degree by degree. Polynomial forms on a simplex have cohomology \(\mathbf Q\) in degree zero and zero otherwise. Polynomial forms vanishing on its boundary have cohomology \(\mathbf Q\) in degree \(p\), with oriented integration giving the isomorphism. **Proof.** The ideal of restriction to face \(i\) in the differential-form algebra is \((t_i,dt_i)\). For any proper subset of the vertices, its \(t_i\)'s can be included in affine coordinates: eliminate the \(t_j\) of one remaining vertex. In these coordinates the face ideals are coordinate ideals. The monomial basis, including exterior monomials in the differentials, proves the following distributive property: the intersection of a list of face ideals is spanned by the monomials containing, for every listed index \(i\), either \(t_i\) or \(dt_i\). Successively correcting a lift on one face by a term vanishing on the previous faces therefore extends every compatible family for a proper subset of the faces. For the complete boundary, localize the coordinate ring where \(t_j\) is invertible. Face \(j\) is absent there, and the other faces have precisely the coordinate description just given. Thus the boundary restriction is locally surjective on the affine cover \(D(t_j)\). It is a map of modules over the affine polynomial coordinate ring, so its cokernel is zero when all these localizations vanish. Concretely, denominators in the finitely many local lifts can be cleared by powers \(t_j^m\); the ideal generated by these powers is the unit ideal. Indeed, expand \((\sum t_j)^{(p+1)(m-1)+1}=1\): every monomial contains one \(t_j^m\). Combining the cleared lifts with this identity gives a global polynomial lift. This proves the extension claim without a smooth partition of unity. Contract \(\Delta^p\) affinely to one vertex. Pull a form back along this polynomial homotopy, contract with the parameter vector, and integrate the polynomial parameter from zero to one. A monomial integral is a rational number \(1/(a+1)\). The product rule and the fundamental theorem of polynomial calculus give \[ dh+hd=\mathrm{id}-\mathrm{ev}_{\mathrm{vertex}}. \tag{5.6} \] This proves the absolute cohomology calculation over \(\mathbf Q\), and after every characteristic-zero extension. Integration on an oriented \(p\)-simplex is rational: repeated polynomial integration gives rational monomial integrals. Stokes follows by repeated one-variable polynomial calculus, with the alternating face signs. Hence integration is a map to simplicial cochains. The boundary-extension assertion gives short exact sequences when a finite simplicial complex is built by adding its simplices in increasing dimension. In dimension zero integration is the identity. Induct on the dimension. For \(\partial\Delta^p\), all its simplices have smaller dimension, so integration is already a cohomology isomorphism. Compare the short exact sequences for forms and simplicial cochains of \((\Delta^p,\partial\Delta^p)\). The absolute calculation (5.6) and the boundary induction imply the relative assertion. Attaching \(p\)-simplices now proves the comparison for every finite complex. In particular the relative cohomology is one line in degree \(p\), and the oriented integration of a normalized top form is \(+1\). No multiplicativity of simplex integration is asserted. \(\square\) The proof also works with a coefficient vector space \(V\): tensor the finite-dimensional-degree diagram over the field with \(V\), or apply the formulas pointwise to locally constant \(V\)-valued coefficients. ### 5.2. A commutative sheaf resolution Let \(X\) be a compact smooth manifold. Choose a finite cover \(\mathcal U=(U_i)\) by strongly convex neighborhoods. The required convex neighborhoods are proved in Riemannian connections and convex neighbourhoods, Theorem B.5. Each nonempty finite intersection is contractible: the unique globally minimizing geodesic between two of its points belongs to every member, and contraction to a fixed point is smooth. Intersecting such an intersection with any sufficiently small strongly convex neighborhood has the same property. For \(J=\{i_0<\cdots0). \tag{5.11} \] Each degree term of (5.7) is a finite extension of these acyclic sheaves, by the degreewise version of the filtration (5.10). It is therefore acyclic. A bounded acyclic resolution computes derived global sections, by the earlier bounded-below sheaf-resolution argument. This proves (5.9). Finally, the product is the actual derived product: (5.8) is a map of sheaf differential graded algebras, and multiplication on its acyclic resolution represents the product of the constant sheaf. Integration was used only to prove a quasi-isomorphism of underlying complexes. It was not used as a purported multiplicative map to singular cochains. \(\square\) Because every \(U_J\) is connected, (5.9) is just the algebra of compatible polynomial forms on the finite nerve of the cover. This gives an explicit rational commutative replacement for the smooth real/complex forms used in current Lesson 13. **Naturality and coherence.** For \(f:Y\to X\), take a finite strongly convex cover \(\mathcal V\) of compact \(Y\) refining \(f^{-1}\mathcal U\). Choose an index assignment \(a\) with \(V_j\subset f^{-1}U_{a(j)}\). The induced affine vertex maps of nerves pull polynomial forms back and give a unital commutative differential graded map \[ A_{\mathcal U}\longrightarrow A_{\mathcal V}. \tag{5.12} \] Its sheaf augmentation identifies its cohomology map with the actual pullback. For refinements of the same \(X\), it is a quasi-isomorphism. If two index assignments are chosen, their combined images on a nonempty intersection \(V_J\) lie in one simplex of the target nerve: that intersection lies in all the indicated inverse images. Linear interpolation between the two affine vertex maps gives a polynomial homotopy with parameter algebra \(\mathbf Q[s,ds]\). With several choices, barycentric interpolation gives the corresponding simplex of homotopies. It restricts to the prescribed homotopies on every face. Common strongly convex refinements therefore give explicit compatible comparisons, including higher composition comparisons. They preserve multiplication throughout. ### 5.3. Compatible classifying-torus models Use \(X_n=(\mathbf {CP}^n)^r\). Lemma 5.2 supplies a rational commutative model \(A_n\) from a finite strongly convex cover. For the inclusion \(X_n\to X_{n+1}\), choose a common good refinement on \(X_n\) of its own cover and the pulled-back cover of \(X_{n+1}\). Its model \(B_n\) gives maps \[ q_n:A_n\longrightarrow B_n,\qquad r_n:A_{n+1}\longrightarrow B_n, \tag{5.13} \] where \(q_n\) is a quasi-isomorphism and \(r_n\) represents restriction. This avoids assuming that arbitrarily chosen good covers are nested. Define the following commutative differential graded algebra: \[ D_{\mathbf Q}= \left\{(a_n,p_n): a_n\in A_n,\ p_n\in B_n[s,ds],\ p_n(0)=q_n(a_n),\ p_n(1)=r_n(a_{n+1})\right\}. \tag{5.14} \] Products and differentials are componentwise. This is the explicit path model of the inverse system (5.13), not an unspecified commutative strictification. Projection to the \(a_n\)'s is surjective in every graded degree: interpolate prescribed endpoint values linearly in \(s\). Its kernel is \[ \prod_n B_n\otimes\Omega_{\mathrm{pol}}([0,1],\{0,1\}). \] Relative polynomial integration identifies the cohomology of this kernel in degree \(j\) with \(\prod_n H^{j-1}(B_n)\). The resulting long exact sequence is the kernel/cokernel sequence for \[ (x_n)\longmapsto(q_nx_n-r_nx_{n+1}) \tag{5.15} \] on the cohomology products. This is checked by differentiating the endpoint interpolation; the overall sign in each degree depends on the tensor-order convention and does not change the kernel or cokernel. By the integral projective-space calculation of Lesson 13, §3.1, its product comparison and coefficient extension, \[ H^*(X_n;\mathbf Q)= \mathbf Q[c_1,\ldots,c_r]/(c_1^{n+1},\ldots,c_r^{n+1}). \tag{5.16} \] In every degree the restriction system eventually stabilizes. The map (5.15) is consequently surjective on each cohomology product: solve on the stable tail by successive lifting, then solve the finitely many preceding coordinates backwards. Products of vector-space complexes preserve cohomology, since primitives can be chosen in every coordinate. The long exact sequence therefore gives, multiplicatively, \[ H^*(D_{\mathbf Q})=\mathbf Q[c_1,\ldots,c_r]. \tag{5.17} \] Choose a closed representative \(z_{i,n}\in A_n^2\) of each integral Chern class's rational image. The two classes in \(B_n\) agree. Hence choose \(b_{i,n}\in B_n^1\) with \[ db_{i,n}=r_n(z_{i,n+1})-q_n(z_{i,n}). \] The path \[ p_{i,n}=q_n(z_{i,n})+d(s b_{i,n}) =q_n(z_{i,n})+s\,db_{i,n}+ds\,b_{i,n} \tag{5.18} \] is closed and has exactly the two required endpoints. Thus the families \((z_{i,n},p_{i,n})\) are actual closed degree-two elements \(z_i\in D_{\mathbf Q}\). Freeness of the polynomial algebra gives a commutative differential graded map \[ \mathbf Q[c_1,\ldots,c_r]\longrightarrow D_{\mathbf Q}, \qquad c_i\longmapsto z_i. \tag{5.19} \] It is a quasi-isomorphism by (5.17), not merely a ring isomorphism on an unidentified endomorphism algebra. Repeat (5.14) with \(\Lambda\)-valued polynomial forms. The exact same relative-interval and stable-cohomology proof gives \[ H^*(D_\Lambda)=\Lambda[c_1,\ldots,c_r]. \] Extend the rational cycles (5.18) into \(D_\Lambda\). They give \[ P_\Lambda\xrightarrow{\ \sim\ }D_\Lambda \tag{5.20} \] as commutative differential graded algebras. No embedding of \(\Lambda\) into \(\mathbf C\) is needed, and no claim that scalar extension commutes with unrestricted products is needed. Equivalently \(D_{\mathbf Q}\otimes\Lambda\to D_\Lambda\) is a quasi-isomorphism, because the preceding explicit calculation identifies its induced cohomology map. The sheaf resolutions of §5.2, their pullbacks, and their polynomial comparison homotopies show that (5.14) models the actual stable derived sections of the projective Borel models. In each prescribed finite morphism range the projection to a sufficiently large \(A_n\) is the corresponding comparison. The frame/product comparisons in Equivariant perverse sheaves and perverse sheaves on stacks, Lemma B.7, Proposition B.8 and Theorem B.9 identify these projective models with the bounded Borel category. We check the enhancement directly. On the finite free unit generators, the sheaf mapping complexes are the derived section complexes just resolved by \(A_n\). Their comparisons preserve composition because multiplication of the commutative sheaf resolution represents composition of the unit endomorphisms. Fixing one argument and then the other, mapping complexes of finite cones are cones of these complexes, so the quasi-isomorphism extends through every finite cone. A retract takes the image of the corresponding commuting projectors and retains the comparison. The path model records the actual refinement homotopies, and its projection to \(A_n\) is a quasi-isomorphism in every fixed range once \(n\) is large enough. Thus these compatible comparisons identify the stable mapping complexes with \(D_\Lambda\). Tensor products are supplied by the same strictly commutative sheaf multiplication, whose associativity, unit and symmetry identities commute with every refinement and polynomial homotopy. This proves the enhanced comparison used here, in addition to the earlier triangulated model construction. Start the system with \(X_0=\operatorname{pt}\) and its one-member cover, so \(A_0=\Lambda\). Projection from (5.14) to this component is an augmentation \(D_\Lambda\to\Lambda\). It represents restriction to the chosen base point of the projective Borel models, and hence forgetting equivariance on the point. Since its target has no degree-two elements, (5.20) intertwines it with the usual polynomial augmentation \(c_i\mapsto0\). This also fixes the underlying-point functor; it is not deduced from an abstract unaugmented Morita equivalence. For additional concreteness, a finite-cell \(P_\Lambda\)-module \(M\) is realized on \(X_n\) by \[ \mathcal A_n\otimes_{P_\Lambda}M, \tag{5.21} \] using (5.18). On free generators its derived mapping complex is \(A_n\); finite cones and retracts extend this comparison. Since (5.20) is an isomorphism in every stable range, take \(n\) beyond the finitely many shifts and degrees used in any such mapping comparison. These are precisely the common-model comparisons of the bounded equivariant construction. Equivalently, use the existing associative generator equivalence with the actual unit endomorphism algebra, and replace that algebra by the explicitly identified model \(D_\Lambda\). This constructs the object by finite cones of actual equivariant unit morphisms. Its quotient component is (5.21), because that identity holds on the unit and its mapping algebra, then on finite cones and retracts. The augmented comparison just established identifies the underlying point complex as \(\Lambda\otimes_{P_\Lambda}^{\mathbf L}M\), which is bounded and finite dimensional. One can check the required resolution comparison directly: the free space \((\mathbf C^{n+1}\setminus0)^r\) has zero \(H^2\) for \(n\ge1\), so the pulled-back Chern cycles have primitives. It has a finite good cover obtained from a finite strongly convex cover of \((S^{2n+1})^r\) times its radial \(\mathbf R^r\) factor. Hence §5.2's polynomial model applies there as well. Formula (5.18), with endpoint zero, compares the polynomial action to its augmentation and gives the actual point-resolution isomorphism. The finite cones of unit maps, rather than an unsupported assertion about uniqueness of all algebra nullhomotopies, provide the compatible derived triples. ### 5.4. The symmetric tensor comparison The generation argument above applies to every characteristic-zero field: at a point all equivariant cohomology objects are finite constant vector spaces, and finite ordinary truncations generate the category from its unit. Its associative generator argument therefore identifies the category at one label with perfect modules over its actual derived endomorphism algebra. The quasi-isomorphism (5.20) has the following symmetric tensor realization. On the finite model the multiplication map is \[ (\mathcal A_n\otimes_P M)\otimes_\Lambda (\mathcal A_n\otimes_P N) \longrightarrow \mathcal A_n\otimes_P(M\otimes_P N). \tag{5.22} \] Use finite-cell representatives, so the displayed relative tensors already compute derived tensors. On two free generators (5.22) is multiplication \(\mathcal A_n\otimes_\Lambda\mathcal A_n\to\mathcal A_n\). It is a quasi-isomorphism: the augmentation \(\Lambda\to\mathcal A_n\) is a stalkwise quasi-isomorphism, and tensor over a field preserves it. Fixing one argument, the class of modules for which (5.22) is a quasi-isomorphism is closed under finite cones and retracts; then fix the other. It follows for all perfect modules. Associativity, the unit, and graded symmetry of these maps are the corresponding identities of the commutative sheaf differential graded algebra. Refinement and path comparisons preserve those identities. They therefore give a coherent symmetric tensor equivalence, rather than only an associative Morita equivalence or a cohomology-algebra calculation. Distinct reduced points of \(\operatorname{Gr}_T\) have no morphisms, bounded support is finite, and convolution adds labels. This proves (5.4) below. The character projectors of the dual torus, proved in Lesson 12, equation (7.6), identify those labels with its weight decomposition. Thus (5.4) is also the perfect dual-torus-equivariant polynomial model over every characteristic-zero field. In the generated category the resulting monoidal comparison is \[ F(K)\otimes_{A_T}^{\mathbf L}F(L) \xrightarrow{\sim}F(K\otimes L). \tag{5.3} \] The sheaf multiplication just proved gives this map on the unit, and finite cones and commuting projectors extend it in both variables. Its associativity, unit and graded symmetry are those of that multiplication. The reduced support of \(\mathrm{Gr}_T\) is indexed by \(L=X_*(T)\). Different supported points have no morphisms between them, as closed restriction detects; point convolution adds the labels. The full bounded finite-support torus statement is therefore \[ D^b_{L^+T,c}(\mathrm{Gr}_T) \simeq\bigoplus_{\lambda\in L}^{\mathrm{finite}} \operatorname{Perf}(A_T), \qquad (M*N)_\gamma=\bigoplus_{\lambda+\mu=\gamma} M_\lambda\otimes_{A_T}^{\mathbf L}N_\mu. \tag{5.4} \] The dual torus has character lattice \(L\). Its adjoint action on its Lie algebra is trivial, and its representation coactions split modules into weights, by the coefficient-projector proof in Lesson 12, equation (7.6), degree by degree. Finite cell modules and their summands have finitely many weights. For every characteristic-zero coefficient field, (5.2) therefore makes (5.4) the torus instance of perfect dual-torus-equivariant modules over the symmetric algebra with its degree-two generators. The unit at label zero corresponds to the **free module**, whose endomorphism algebra is \(A_T\). This is not a category of ordinary graded modules with all differentials discarded. Cones, derived tensor products and mapping complexes are essential. For example the cone of multiplication by \(c\), with the suitable shift, gives the perfect module \(\Lambda=A_{\mathbf G_m}/(c)\), while the unit remains \(A_{\mathbf G_m}\). The two objects have different endomorphism algebras. ## 6. The elementary Koszul dual pair Let \(V\) be finite-dimensional, and put \[ A=\operatorname{Sym}(V[-2]),\qquad d=0, \qquad k=\Lambda=A/(V). \tag{6.1} \] Choose a basis \(v_1,\ldots,v_r\), all in degree two. The differential graded module \[ P=A\otimes\bigwedge(e_1,\ldots,e_r), \qquad |e_i|=1, \qquad d(e_i)=v_i \tag{6.2} \] has the augmentation \(P\to k\). It is a finite semifree resolution. In one variable its differential maps \(a e\) to \(a v\). Multiplication by \(v\) in \(\Lambda[v]\) is injective, and its cokernel is \(\Lambda\); hence its cohomology is exactly the augmentation line. Tensor the one-variable resolutions over the field. Splitting complexes into cohomology and contractible summands, as in Lesson 12, Lemma 1.1, shows that their tensor has the same augmentation cohomology. Filtering by the number of exterior factors makes (6.2) semifree, because its differential lowers that number. Thus it computes derived Hom. **Proposition 6.1.** As a differential graded algebra, \[ R\operatorname{Hom}_A(k,k) \simeq B=\bigwedge(V^*[1]), \qquad |\epsilon_i|=-1, \qquad d=0. \tag{6.3} \] **Proof.** The complex \(\operatorname{Hom}_A(P,k)\) has zero differential, since every \(v_i\) acts by zero on \(k\). Its basis is dual to the exterior basis of the \(e_i\); dualizing a degree-one generator gives degree \(-1\). To verify multiplication, let \(\iota_i\) be contraction with \(e_i\) on \(P\). It has degree \(-1\); \(\iota_i^2=0\), and \(\iota_i\iota_j=-\iota_j\iota_i\). Since \(d=\sum_i v_i\iota_i\), the graded commutator of \(d\) with each \(\iota_j\) is zero. These operators give a differential graded algebra map \[ B\longrightarrow\operatorname{End}_A(P). \] Postcomposition with the augmentation is a quasi-isomorphism from \(\operatorname{End}_A(P)\) to \(\operatorname{Hom}_A(P,k)\): the finite semifree filtration makes \(\operatorname{Hom}_A(P,-)\) preserve quasi-isomorphisms. The images of the products of contractions are the dual exterior basis, with their exterior signs. Thus the algebra map induces an isomorphism in cohomology and proves (6.3). Changes of basis identify contractions with \(V^*\), so the result is independent of the chosen basis. \(\square\) The negative degree in (6.3) is essential. For one variable the resolution has free generators in degrees zero and one. An \(A\)-linear map from its degree-one generator to the augmentation line has degree \(-1\), not degree \(+1\). The reverse computation is equally explicit in characteristic zero. Resolve the augmentation line as a \(B\)-module by \[ Q=B\otimes\Lambda[f_1,\ldots,f_r], \qquad |f_i|=-2, \qquad d(f_i)=\epsilon_i. \tag{6.4} \] In one variable \(d(f^n)=n\epsilon f^{n-1}\). For \(n>0\) its coefficient is invertible, so all the negative-degree pairs contract and only the constant augmentation remains. Tensoring proves the assertion for every \(r\). Polynomial-degree filtration makes \(Q\) semifree. It is infinite, but still computes derived Hom: a nullhomotopy against an acyclic target is constructed successively on that filtration. On each new free generator, the already prescribed differential makes the obstruction a cycle; acyclicity supplies a primitive. The successive choices define a homotopy on the union. This proves that \(\operatorname{Hom}_B(Q,-)\) preserves quasi-isomorphisms. The commuting operators \(\partial/\partial f_i\) are closed endomorphisms of degree two. Their products, followed by augmentation, evaluate on the polynomial monomials by the nonzero factorial coefficients. They therefore identify the polynomial algebra on these operators with the cohomology of \(\operatorname{Hom}_B(Q,k)\). As in the preceding proof, postcomposition with augmentation compares it with \(\operatorname{End}_B(Q)\). Consequently \[ R\operatorname{Hom}_B(k,k)\simeq A. \tag{6.5} \] There is no completed power series algebra in this calculation: in each fixed cohomological degree only finitely many monomials of the \(f_i\) occur, and their duals have that same finite degree. Characteristic zero was used in the factorial coefficients; this proof does not assert an identical divided-power statement in positive characteristic. Applying the generator proof of §5 to these two augmentation objects gives the precise category statements \[ \operatorname{thick}_A(k)\simeq\operatorname{Perf}(B), \qquad \operatorname{thick}_B(k)\simeq\operatorname{Perf}(A). \tag{6.6} \] The second equivalence is \(R\operatorname{Hom}_B(k,-)\); the first is its corresponding \(A\)-module construction. These are statements about the specified generated subcategories. They do not identify all differential graded modules over \(A\) with all modules over \(B\). Since \(B\) is nonpositive with \(H^0(B)=k\), a \(B\)-module with bounded finite-dimensional cohomology belongs to \(\operatorname{thick}_B(k)\). Its ordinary truncation triangles have cohomology objects that are finite sums of shifts of the augmentation module; the negative-degree elements act trivially on a single cohomology object. There are finitely many such triangles. Conversely a finite construction from \(k\) has bounded finite-dimensional cohomology. Thus the second category on the left of (6.6) is exactly the bounded coherent module category of this exterior algebra. If \(\mathfrak h\) is a vector space placed in degree zero, the coordinate algebra of the derived self-intersection of the origin in \(\mathfrak h\) is \[ \mathcal O(\operatorname{pt}\times_{\mathfrak h}^{\mathbf R} \operatorname{pt}) =\Lambda\otimes^{\mathbf L}_{\operatorname{Sym}(\mathfrak h^*)}\Lambda \simeq\bigwedge(\mathfrak h^*[1]). \tag{6.7} \] To prove it, resolve the origin over the degree-zero polynomial algebra by its ordinary Koszul complex, with free exterior generators in degree \(-1\) and differential equal to the coordinate functions. Tensoring with the origin kills that differential. Taking \(\mathfrak h=V\), (6.7) is precisely \(B\). Equations (6.5)–(6.6) explain why perfect modules with degree-two polynomial generators can also be expressed by coherent modules on this derived self-intersection. For a torus, apply this independently at each weight label in (5.4), using the coefficient hypothesis of (5.2) for the polynomial presentation. Its spectral description can therefore use either perfect modules over \(\operatorname{Sym}(\widehat{\mathfrak t}[-2])\) or bounded coherent modules over \(\bigwedge(\widehat{\mathfrak t}^*[1])\), with the dual torus's weight decomposition. The latter algebra is the coordinate algebra of \(\operatorname{pt}\times^{\mathbf R}_{\widehat{\mathfrak t}}\operatorname{pt}\). Under the second equivalence in (6.6), its augmentation module corresponds to the free polynomial module, the Satake unit. Its product is the product transported from (5.4); this assertion does not replace that product by ordinary tensor over the exterior algebra. ## 7. What renormalization changes For the exterior algebra \(B=\Lambda[\epsilon]\), \(|\epsilon|=-1\), the augmentation module \(k\) is not compact in the category of all differential graded \(B\)-modules. This can be checked from (6.4), rather than from terminology. Take the direct sum \(M=\bigoplus_{n\geq0}k[2n]\), all with trivial \(\epsilon\)-action. In degree zero a map from its resolution to \(M\) may assign the generator \(f^n\) any scalar in the \(n\)-th summand. Those choices are independent; the Hom differential is zero because \(\epsilon\) acts by zero. Thus \[ \operatorname{Hom}_{D(B)}(k,M)=\prod_{n\geq0}\Lambda, \qquad \bigoplus_{n\geq0}\operatorname{Hom}_{D(B)}(k,k[2n]) =\bigoplus_{n\geq0}\Lambda. \tag{7.1} \] The natural direct-sum comparison is not surjective. This is exactly failure of compactness. On the other hand \(k\) is a generator of the bounded coherent category \(\operatorname{thick}_B(k)\). If one **defines** its renormalized presentable category by \[ \operatorname{Ind}(\operatorname{thick}_B(k)), \tag{7.2} \] then that object is compact there: by the construction of Ind-completion, maps from an original object commute with filtered colimits. The equivalence (6.6) extends to \[ \operatorname{Ind}(\operatorname{thick}_B(k)) \simeq\operatorname{Ind}(\operatorname{Perf}(A)). \tag{7.3} \] Its compact objects are exactly the original idempotent-complete perfect category. To see the last assertion, write an object as a filtered colimit of finite constructions; if the object is compact, its identity factors through one stage, making it a retract of that stage. These formulas give a concrete reason for specifying the completion in a derived Satake theorem. The full module category over the exterior algebra and (7.2) have different compactness behavior despite containing the same bounded coherent subcategory. ## 8. The general theorems and their categories The bounded nonrotating equivalence is proved in [Theorem C.81.5.1](#c815-the-bounded-nonrotating-derived-equivalence), for connected complex reductive groups and every characteristic-zero coefficient field. The loop-rotation, renormalized, singular-support and factorization equivalences below are stated with their separate categories and hypotheses. Their proofs require additional arguments and none is used as a premise for the results proved here. ### 8.1. The bounded equivariant theorem Let \(G\) be a complex simply connected semisimple group and \(\widehat G\) its dual over an algebraically closed characteristic-zero coefficient field. [Bezrukavnikov–Finkelberg, Theorem 5, §2.7](https://arxiv.org/pdf/0707.3799v4#page=7) gives monoidal triangulated equivalences \[ D^b_{G(O),c}(\mathrm{Gr}_G) \simeq\operatorname{Perf}^{\widehat G} (\operatorname{Sym}(\widehat{\mathfrak g}[-2])), \qquad D^b_{G(O)\rtimes\mathbf G_m,c}(\mathrm{Gr}_G) \simeq\operatorname{Perf}^{\widehat G}(U_\hbar^{[\ ]}). \tag{8.1} \] On the right, equivariant perfect modules are generated by \(A\otimes V\), for finite-dimensional representations \(V\), under finite cones and summands. Both displayed algebras have differential zero. In the second, Lie generators and \(\hbar\) have degree two, and the relation is \(xy-yx=\hbar[x,y]\). The paper simplifies to simply connected \(G\) after §2.4; the reductive categorical formulation is given next. The first equivalence is proved in [C.81.5](#c815-the-bounded-nonrotating-derived-equivalence), with the broader connected reductive and characteristic-zero coefficient hypotheses there. The second is the loop-rotation statement. [Appendix C.82](#c82-quantum-free-generators-and-rotation-coefficients) describes its free bimodule tensor generators and the actual rotation mapping coefficients. [Appendix C.83](#c83-the-quantum-kostantwhittaker-generator-comparison) proves the actual quantum Kostant–Whittaker free-generator mapping and balanced tensor comparison, together with the geometric torus parameter calibration. The general rotation-cohomology-to-Whittaker module comparison and its full enhanced geometric tensor comparison are additional requirements. Equation (8.1) is a monoidal theorem, without an asserted factorization upgrade. ### 8.2. Ordinary and renormalized spherical categories For a reductive group in characteristic zero, put \[ \mathcal H_{\widehat G} =B\widehat G\times^{\mathbf R}_{\widehat{\mathfrak g}/\widehat G} B\widehat G =(\operatorname{pt}\times^{\mathbf R}_{\widehat{\mathfrak g}} \operatorname{pt})/\widehat G. \tag{8.2} \] Write \(\mathrm{Sph}=D\text{-mod}(\mathrm{Gr}_G)^{G(O)}\), with its ordinary cocompletion. Let \(\mathrm{Sph}^{loc.c}\) consist of objects compact after forgetting equivariance, and define \(\mathrm{Sph}^{ren}=\operatorname{Ind}(\mathrm{Sph}^{loc.c})\). [Arinkin–Gaitsgory, §§12.3–12.5](https://arxiv.org/pdf/1201.6343v4#page=109) distinguish the following monoidal equivalences: | Geometric category | Spectral category | Exact theorem | | --- | --- | --- | | \(\mathrm{Sph}^{loc.c}\) | \(\operatorname{Coh}(\mathcal H_{\widehat G})\) | Theorem 12.3.3 | | \(\mathrm{Sph}^{ren}\) | \(\operatorname{IndCoh}(\mathcal H_{\widehat G})\) | Corollary 12.3.4 | | \(\mathrm{Sph}\) | \(\operatorname{IndCoh}_{\mathrm{Nilp}}(\mathcal H_{\widehat G})\) | Theorem 12.5.3, Corollary 12.5.5 | | \(\mathrm{Sph}^{c}\) | \(\operatorname{Coh}_{\mathrm{Nilp}}(\mathcal H_{\widehat G})\) | Corollary 12.5.5 | Here Nilp constrains singular support in \(\widehat{\mathfrak g}^*/\widehat G\), rather than ordinary support in the underlying point quotient. Proposition 12.4.2 and Corollary 12.4.5 give the polynomial Koszul-dual presentation of all IndCoh. Remark 12.4.3 excludes compatibility of that presentation with factorization. The functors of §12.2.4 are \(\Xi:\mathrm{Sph}\to\mathrm{Sph}^{ren}\) and \(\Psi:\mathrm{Sph}^{ren}\to\mathrm{Sph}\), with \(\Xi\dashv\Psi\) and \(\Xi\) fully faithful. Renormalized compactness refers to \(\mathrm{Sph}^{loc.c}\). ### 8.3. Factorization is additional structure For reductive \(G\) over an algebraically closed characteristic-zero field, [Campbell–Raskin, Theorem 6.6.1](https://arxiv.org/pdf/2310.19734v2#page=59) proves an equivalence \[ \mathrm{Sat}_G:\mathrm{Sph}_G \xrightarrow{\sim}\mathrm{Sph}^{spec}_{\widehat G} \quad\text{in }\operatorname{AssocAlg}(\operatorname{FactCat}). \tag{8.3} \] Both categories are renormalized. Proposition 6.3.2 specifies geometric renormalization; §5.7 specifies spectral renormalization using integrable factorization modules. Theorem 8.10.1 and Corollary 8.10.1.1 identify its one-point algebraic models. The QCoh formal-completion presentation there precedes passage to integrable objects and renormalization. Proposition 10.2.1 identifies QCoh with the tempered subcategory, rather than with all ordinary spherical objects. The spherical theorem (8.3) is unconditional; its proof is not supplied here. The half-density convention is specified in [Gaitsgory–Raskin, §§1.5–1.7](https://arxiv.org/pdf/2405.03648v3#page=23): \[ \mathrm{Sph}^{non\text{-}ren}_G =D\text{-mod}_{1/2}(L^+G\backslash LG/L^+G). \tag{8.4} \] Renormalization ind-completes the objects compact after either forgetful functor to the Grassmannian. Theorem 1.7.2 gives the unique monoidal factorization equivalence with all IndCoh on the spectral Hecke category, compatible with the Casselman–Shalika action. Its one-point stack is the derived zero self-intersection over \(\widehat{\widehat{\mathfrak g}}_0/\widehat G\); that self-intersection agrees with (8.2). Appendix E constructs the Ran category. Example 1.7.4 sends \(\mathrm{IC}_\lambda\) to \(\mathrm{nv}(V^{-w_0\lambda})\), preserving its dual-weight convention. Remark 1.7.5 makes the half-density twist essential for factorization. ## 9. Reading the spectral expression The notation in (8.2) can be understood without assuming the general theorem. For a vector space \(\mathfrak h\), its derived zero self-intersection has exterior coordinate algebra (6.7). Its underlying ordinary point does not describe all its functions: the negative cohomological degrees record the intersection's excess tangent directions. For an acting group, equivariance additionally records compatible action data. In the torus case the action on \(\mathfrak h=\widehat{\mathfrak t}\) is trivial, and coefficient projectors give exactly the weight labels in (5.4). In this finite exterior-algebra example, coherent means bounded finite-dimensional cohomology, as proved in §6. IndCoh is its Ind-completion as in (7.2). Equation (7.1) explains why this differs from simply taking all modules over its coordinate algebra. In the general statement, singular support requires additional operators on derived morphisms. It is part of the external theorem in §8; no assertion that ordinary support or a point's set of coordinates determines it is needed for our calculations. Likewise, factorization specifies compatible categories for moving finite collections of points, their products when the points are disjoint, and their behavior under collision. A one-point equivalence supplies only one part of that data. Fusion, §6 proves the classical collision product of perverse sheaves. It does not construct the derived spectral category on Ran. Thus (5.4) is a proved fixed-point torus tensor equivalence; (8.3) includes the further moving-point structure by its external theorem. The relevance to geometric Langlands is a precise preview. Hecke correspondences act on sheaves on \(\operatorname{Bun}_G\), as described in Consequences and examples, §8. A general derived Satake equivalence expresses the local category of kernels for that action through \(\widehat G\). Constructing the global spectral action also requires those global correspondences and their derived descent. This lesson's torus calculation neither assumes nor proves that further action. ## 10. Exercises with complete solutions **Exercise 13.1 (easy).** Compute \(H^*_{\mathbf G_m}(\operatorname{pt},\Lambda)\), including its product and odd degrees. **Solution.** By definition this is \(H^*(B\mathbf G_m,\Lambda)\). The contraction and projective bundle model in §3.1 give \(B\mathbf G_m\simeq\mathbf {CP}^{\infty}\). There is one even cell in each nonnegative even degree. The finite Gysin calculation (3.2) proves that the positive generator \(c\) in degree two has powers generating all those groups. Their restrictions stabilize, and all odd groups are zero. Hence the answer is the polynomial algebra \(\Lambda[c]\), with \(|c|=2\), including its multiplication. **Exercise 13.2 (easy).** State and verify derived Satake for a rank-\(r\) torus over any characteristic-zero coefficient field. Identify the unit and convolution. **Solution.** Put \(L=X_*(T)\) and \(A=\Lambda[c_1,\ldots,c_r]\), with zero differential and each generator of degree two. At one support point, truncation triangles and constant local systems give \(\operatorname{thick}(\Lambda)\). The mapping-complex generator argument of (5.1), together with the commutative polynomial-simplex model (5.2), identifies it with \(\operatorname{Perf}(A)\). Distinct support points have no morphisms; finite support therefore gives the finite sum over \(L\). The tensor comparison (5.3) proves that convolution adds labels and tensors over \(A\), exactly (5.4). Coefficient projectors identify labels with characters of \(\widehat T\). The object at label zero is the free module \(A\). Every ingredient is proved in §5, so this is the fixed-point differential graded tensor equivalence, with no claim about Ran factorization. **Exercise 13.3 (medium).** For \(G=\mathbf G_m\), show that \(\operatorname{Ext}^1\) between simple objects in the Satake heart vanishes, while the unit has nonzero \(\operatorname{Ext}^2\) in the Borel equivariant derived category. Explain the corresponding degree for \(SL_2\). **Solution.** Simple torus objects are one-dimensional vector spaces at single labels. A short exact sequence splits by the weight projectors and an ordinary vector-space complement. This proves vanishing of every heart \(\operatorname{Ext}^1\), including self-extensions. In the Borel category, closed extension at the base point and loop contraction prove (2.3); (3.3) identifies its degree-two self-Ext with \(\Lambda c\ne0\). The bounded derived heart has zero morphisms in that degree by the complex-splitting proof of (4.1). For \(SL_2\), (3.8) instead says that degree two is zero and the first positive class is \(c_2\) in degree four. The group cannot be omitted from the assertion. **Exercise 13.4 (medium).** For \(G=SL_2\), compare \(H^*(BG,\Lambda)\) with \(\operatorname{Sym}(\mathfrak g^*[-2])^G\). Do not assume the general characteristic-class theorem. **Solution.** Polar decomposition and the quaternionic universal bundle reduce the first ring to \(H^*(\mathbf {HP}^{\infty},\Lambda)\). Its rank-four Gysin sequence gives the polynomial ring \(\Lambda[c_2]\) with generator of degree four, by (3.6)–(3.7). For the second ring, write \(p=z^2+xy\). Restriction of an invariant to diagonal trace-zero matrices is even by the Weyl interchange. Subtract its corresponding polynomial in \(p\); the difference vanishes on the dense distinct-eigenvalue locus and is zero. Thus (3.9) is \(\Lambda[p]\), with \(p\) of degree four when the linear coordinates have degree two. Sending \(p\) to \(c_2\) defines the graded algebra isomorphism. This computes the rings and a chosen isomorphism; a universal normalization in arbitrary rank is not being inferred. **Exercise 13.5 (hard).** For \(A=\Lambda[c]\), \(|c|=2\), and \(B=\bigwedge(\epsilon)\), \(|\epsilon|=-1\), prove both derived endomorphism calculations and the corresponding generated-category equivalences. Why does this not equate all \(A\)-modules with all \(B\)-modules? **Solution.** Write \(k\) for the augmentation object. Resolve \(k_A\) by \(P=A\otimes\bigwedge(e)\), \(|e|=1\), \(de=c\). Multiplication by \(c\) is injective, so the augmentation is a quasi-isomorphism. The closed contraction \(\iota_e\) has degree \(-1\), square zero, and its two classes give all of \(\operatorname{Hom}_A(P,k)\). The finite free comparison therefore proves \(R\operatorname{End}_A(k)\simeq B\). Resolve \(k_B\) by \(Q=B[f]\), \(|f|=-2\), \(df=\epsilon\). Its positive pairs cancel since \(d(f^{n+1})=(n+1)\epsilon f^n\); the augmentation is a quasi-isomorphism. The increasing polynomial-degree filtration makes it semi-free, and the homotopy-lifting argument in §6 shows it computes derived Hom. The closed operator \(\partial_f\) has degree two. Its powers, followed by augmentation, give the dual basis to \(f^n\), with nonzero coefficients \(n!\). Thus \(\Lambda[c]\to\operatorname{End}_B(Q)\), \(c\mapsto\partial_f\), is a quasi-isomorphism. There is one basis element in each nonnegative even degree, without a completion. The two generator mapping-complex arguments give \(\operatorname{thick}_A(k)\simeq\operatorname{Perf}(B)\) and \(\operatorname{thick}_B(k)\simeq\operatorname{Perf}(A)\). The second takes \(k_B\) to \(A\), while the free \(B\)-module goes to a different object. These equivalences specify their generated subcategories. In all \(B\)-modules, the same \(k_B\) fails compactness by the product-versus-sum calculation (7.1). It is compact in its own Ind-completion (7.2). There is also a direct failure of the first functor on all \(A\)-modules: the nonzero module \(A[c^{-1}]\) is sent to zero by \(R\operatorname{Hom}_A(k,-)\). Its Hom complex from \(P\) has differential multiplication by \(c\), now an isomorphism, and is contractible. Thus that functor is not faithful on all modules. Passing to larger categories without specifying the completion would invalidate the assertion. ## Appendix A. Arithmetic Ext purity and the formal generator category This appendix proves the purity and formality steps for the actual spherical generators. It does not identify their full graded Ext category with the dual Lie-algebra model; that further comparison is needed for (8.1). We work over \(\overline{\mathbf F}_q\), with finite \(E/\mathbf Q_\ell\), \(\ell\nmid q\), and a chosen \(a\in E\) with \(a^2=q\). The coefficient \(E(1/2)\) has geometric Frobenius \(a^{-1}\). Write \(A_\lambda=I_\lambda(d_\lambda/2)\) for the normalized geometric IC with its specified split Weil structure. The finite rational coefficient operations and trace orientations are proved in Semi-infinite orbits and weight functors, Appendices O–Q, with the supported section comparisons in The Satake category, Lemmas A.1.1.1–A.1.2.1. The split Tate stalk and costalk operators are proved in Consequences and examples, Theorem C.3.1.1. The free [Equivariant Satake category and Kostant–Whittaker reduction, §§6.5–6.6](https://arxiv.org/abs/0707.3799v4) supplies the formality context; the algebraic and geometric arguments used below are proved here. ### A.1. Arithmetic classifying-space cohomology We use actual finite free-frame models in every prescribed degree range. Their algebraic construction is the one in Equivariant perverse sheaves and perverse sheaves on stacks, Propositions B.6–B.8. Here is the constant-adic acyclicity calculation needed to use the same frames in this characteristic. For \(V_{N,r}\) consisting of \(r\) independent columns in \(k^N\), successive column-forgetting maps have, on minor charts, fibre \(\mathbf A^{r-1}\times(\mathbf A^{N-r+1}\setminus0)\). Relative zero-section localization and the supported vector trace give \[ H^j(\mathbf A^d\setminus0,E)= \begin{cases}E,&j=0,\\ E(-d),&j=2d-1,\\0,&\text{otherwise}.\end{cases} \tag{A.1.1} \] Indeed the zero section has supported coefficient \(E(-d)[-2d]\), obtained by composing the \(d\) coordinate section comparisons. The ordinary coefficient of affine space is \(E\), by the relative affine-line calculation iterated \(d\) times. The localization triangle then gives exactly (A.1.1). Its maps are the zero-section class and the trace, so are Frobenius compatible. This calculation is relative on the trivializing charts. For a pulled-back admissible coefficient complex the same localization triangle tensors with that complex, by the actual section comparisons and proper projection maps of Lesson 5 Q; it gives the same unit range. The units for successive column projections are therefore isomorphisms through degree \(2(N-r)\); their cones start above that range, and left t-exact ordinary direct image preserves that bound under composition. This proves the needed stable-range comparison. It does not compare arbitrary sheaves merely by their pointwise fibres. All quotient and flag maps below are algebraic maps on these free frames. Quotients are obtained by the finite homogeneous-space and torsor charts in the cited construction; only its topological acyclicity calculation has been replaced by (A.1.1). Increasing \(N\) increases the proved coefficient range. Equivalently their mapping and cohomology comparisons can be computed on the whole action nerve; the frame units identify those comparisons in each fixed range. **Theorem A.1.1.1.** If \(L/\mathbf F_q\) is split connected reductive, then \(H^{2j+1}(BL,E)=0\), every \(H^{2j}(BL,E)\) is finite dimensional, and its actual Frobenius is \(q^j\operatorname{id}\). The same holds for a split group \(H=U\rtimes L\) whose \(U\) has the affine ordered coordinates of a split unipotent group. **Proof.** For a split torus \(T\), use the product of the \(r\) projective-space frame models. The split affine cells of each projective space and their oriented cycle classes give \[ H^{2j}(BT,E)=E(-j)^{b_j},\qquad H^{2j+1}(BT,E)=0. \tag{A.1.2} \] This is the actual scalar operator, by Lesson 12 C.1.1.1, rather than an inference from a trace. For \(U\), every level \(U^s\) of the group nerve is an affine space. Its ordinary constant cohomology is \(E\) in degree zero. The augmented nerve of these constant complexes contracts by the degeneracy maps: insertion of the identity gives the contracting homotopy in positive simplicial degrees. Consequently \(R\Gamma(BU,E)=E\). The same relative calculation on torsor charts identifies the unit \(E\to R(BH\to BL)_*E\), so \(BH\) has the same groups and operators as \(BL\). Take a split Borel \(B\subset L\). The map \(V/B\to V/L\) of a sufficiently large common frame is smooth proper with fibre \(L/B\), of dimension \(m\). There is an ample \(L\)-linearized flag line, as proved in AG-RG-03 Lemma 5.1. Let \(\eta\) be its descended Kummer first Chern class. Structural trace and projection give \[ \pi_*(\eta^m\cup\pi^*x)=s x,\qquad s=\int_{L/B}c_1(\mathcal L)^m\in\mathbf Q_{>0}. \tag{A.1.3} \] We justify that scalar in the adic setting. A positive power \(\mathcal L^d\) gives a projective embedding. Choose \(m\) hyperplanes meeting its smooth image transversally in finitely many reduced points after a finite separable field extension. Such hyperplanes exist: the incidence of a point and a hyperplane containing its projective tangent space has dimension smaller than the hyperplane parameter space; proper projection makes its bad image closed. Apply this argument successively to the smooth intersections. The final intersection is nonempty and finite, by the projective dimension theorem used in the flag construction. The divisor section comparisons identify the product of their Kummer classes with the supported class of those points. Each rational point has trace one. Thus the top trace is their positive number divided by \(d^m\). The supported products and trace commute with that field extension; this is the displayed nonzero rational scalar already over the original field. Smooth proper base change identifies the same fibre scalar on every geometric fibre of the finite frame. Hence \(\pi_*(\eta^m)\) is \(s\) times its unit section. Projection proves (A.1.3). Its classes lie in \(H^{2m}(-,E(m))\), and trace is correspondingly twisted, so this left inverse commutes with Frobenius. Since \(s\ne0\) in the characteristic-zero coefficient field, flag pullback is injective. The successive affine root coordinates of \(B/T\) identify \(H^*(BB,E)=H^*(BT,E)\), using the same ordinary affine-line unit. In each stable range we have therefore obtained \[ H^*(BL,E)\hookrightarrow H^*(BT,E). \tag{A.1.4} \] The right side is finite dimensional in each degree, zero in odd degree, and has exactly the scalar in (A.1.2). Its Frobenius-stable subspace has that same scalar. This proves the theorem, including the unipotent extension case. ∎ ### A.2. Pure equivariant Ext from actual restrictions **Theorem A.2.1.1.** For any two normalized spherical IC generators, every finite-dimensional group \[ \operatorname{Ext}^n_{L^+G}(A_\lambda,A_\mu) \tag{A.2.1} \] has Frobenius with sole eigenvalue \(a^n\). A Jordan part is allowed in this assertion. The proof applies to all original split connected reductive groups, using their actual sufficiently large jet action. **Proof.** Fix a finite closed union containing both supports, and a sufficiently large jet level. Its orbit stabilizers are connected and have split reductive Levi quotient with split unipotent radical: the constant parabolic and the remaining root and torus coefficients are the actual stabilizer coordinates of Lessons 4 §2 and 6 §§1–2. The classifying-space groups of every stabilizer consequently have the actual scalar \(a^p\) in degree \(p\), by A.1. On an orbit \(S\) of dimension \(d\), put \(K=i_S^*A_\lambda\), \(L=i_S^!A_\mu\). The ordinary cohomology fibres of \(K\) in degree \(u\) and \(L\) in degree \(v\) have actual scalar operators \(a^u,a^v\), by Lesson 12 C.3.1.1 and its dual costalk comparison. For the exceptional restriction this includes the stratum normalization: the point's additional exceptional pullback contributes \([-2d](-d)\), so dividing its scalar by \(q^d=a^{2d}\) gives precisely \(a^v\) on the orbit fibre. Equivariance transports these fibres over the whole orbit; connected stabilizers leave their cohomology systems constant, by Lesson 6 B.1. The action-nerve mapping complex on this orbit has its ordinary hypercohomology spectral sequence. At a point, complexes of finite-dimensional vector spaces split into their cohomology and contractible pairs. Thus its degree-\(t\) fibre Hom is \[ \bigoplus_u\operatorname{Hom}_E(H^uK,H^{u+t}L) \tag{A.2.2} \] with sole scalar \(a^t\). No splitting of the full equivariant objects is being asserted. The cohomology local systems of this fibre Hom are constant on the classifying space of the connected stabilizer. The nerve spectral sequence consequently has terms \(H^p(BH,E)\) tensored with (A.2.2). Each term of total degree \(n=p+t\) has scalar \(a^n\). A differential has total degree one and commutes with Frobenius. It is zero, since its source and target have distinct sole eigenvalues \(a^n,a^{n+1}\). More explicitly, an intertwiner between finite generalized eigenspaces with distinct eigenvalues is zero: apply the relatively prime annihilating polynomials and their Bézout identity. The orbit Ext groups thus have a finite filtration whose degree-\(n\) quotients all have eigenvalue \(a^n\). They are finite dimensional. The bounded fibre degree range and the finiteness of \(H^p(BH)\) give the required convergence. Finally order the finitely many orbits so that the successive unions are closed. Source localization triangles give a finite filtration of the global mapping complex with terms \(R\operatorname{Hom}_S(i_S^*A_\lambda,i_S^!A_\mu)\). All triangles and maps are the actual equivariant coefficient maps and commute with Frobenius. Its spectral sequence has the same degree and eigenvalue property, so again every differential is zero. A finite extension of generalized \(a^n\)-eigenspaces is still a generalized \(a^n\)-eigenspace: a product of their annihilating polynomials annihilates the extension. This proves (A.2.1). It does not turn such an extension into a split scalar Frobenius module. ∎ ### A.3. Minimal models and a nonresonant automorphism We spell out the algebra used to pass from these groups to mapping complexes. A minimal \(A_\infty\) category has graded Hom spaces and operations \(m_r\) of degree \(2-r\), with \(m_1=0\). Suspend the Hom spaces by an operator \(s\) of degree \(-1\). On their tensor coalgebra of composable words, these operations are the components of a degree-one coderivation \(b\), with \(b^2=0\). All signs below are the tensor Koszul signs. A coalgebra map is determined by its components on single-output words. It is invertible whenever its first component is invertible, by induction on word length. **Lemma A.3.1.1.** A differential graded category over a field has a minimal \(A_\infty\) model on its cohomology Hom spaces, with its ordinary binary composition. A quasi-equivalence or an equivalence of enhanced objects induces an \(A_\infty\) equivalence of these models. **Proof.** On each Hom complex choose representatives for cohomology, complements to boundaries in cycles, and complements to cycles. This gives maps \(i,p,h\) with \[ pi=1,\qquad dh+hd=1-ip,\qquad h^2=hi=ph=0. \tag{A.3.1} \] On words the tensor maps \(I,P\) give a contraction for the tensor differential \(d_0\). Its homotopy is the signed sum \[ H_r=\sum_{j=1}^r(ip)^{\otimes(j-1)} \otimes h\otimes1^{\otimes(r-j)}. \tag{A.3.2} \] Expanding \(d_0H_r+H_rd_0\) telescopes to \(1-(ip)^{\otimes r}\). The same side conditions give \(H^2=HI=PH=0\). Its coproduct identity is \(\Delta H=(H\otimes1+IP\otimes H)\Delta\), verified by dividing the sum at each cut of a word. Let \(\delta\) be the coderivation for the original binary product. It lowers word length by one. Define \[ I_\infty=(1+H\delta)^{-1}I,\qquad P_\infty=P(1+\delta H)^{-1},\qquad b_H=P\delta(1+H\delta)^{-1}I. \tag{A.3.3} \] The inverses here are finite geometric sums on every fixed word, not analytic limits. The recursion \(I_\infty=I-H\delta I_\infty\), its counterpart for \(P_\infty\), and (A.3.1) give \((d_0+\delta)I_\infty=I_\infty b_H\), \(P_\infty(d_0+\delta)=b_HP_\infty\), and \(P_\infty I_\infty=1\). To check these identities, substitute the recursions, use \(d_0H+Hd_0=1-IP\), and collect terms containing \((d_0+\delta)^2=0\). At word length \(r\) the leftover terms have smaller word length and vanish inductively, starting with \(r=1\). The same coproduct identity for \(H\), applied to those recursions and to the coderivation identity for \(\delta\), proves that \(I_\infty,P_\infty\) are coalgebra maps and \(b_H\) is a coderivation. Applying \(P_\infty\) and \(I_\infty\) to the square-zero identity gives \(b_H^2=0\). Its first component is zero and its second is \(p\,m_2(i\otimes i)\), the cohomology product. For a dg functor \(f\), the map \(P_\infty f I_\infty\) is an \(A_\infty\) functor with first component \(H(f)\). If that component is invertible, the word-length recursion constructs its coalgebra inverse, which also commutes with the coderivations. Changing the choices uses the same construction on the identity of the original category, so gives an equivalence between the models. Closed homotopy equivalences of enhanced objects give the same conclusion after adjoining those objects to one dg category. Units are preserved on cohomology; equivalently the construction can be normalized at identity arguments using the side conditions in (A.3.1). This is sufficient for the unital finite-cone and retract envelopes. ∎ **Theorem A.3.1.2.** Let \(\mathcal C\) be a minimal \(A_\infty\) category with degreewise finite-dimensional Hom spaces. Suppose it has an \(A_\infty\) automorphism \(F\) fixing its objects, such that the first component on every degree-\(n\) Hom space has sole eigenvalue \(a^n\), where \(a\) is not a root of unity. Then an \(A_\infty\) change of coordinates with first component the identity sends \(\mathcal C\) to its ordinary graded cohomology category, with all \(m_r=0\) for \(r>2\). Jordan parts of \(F_1\) are permitted. **Proof.** First linearize \(F\). Suppose its components from arity two through arity \(r-1\) have already been removed. Conjugate by a coalgebra change of coordinates whose only new component is \(g_r\), of unsuspended degree \(1-r\). At arity \(r\) its new component is \[ F_r+g_rF_1^{\otimes r}-F_1g_r. \tag{A.3.4} \] For a tuple of input degrees \(n_1,\ldots,n_r\), the output of \(g_r\) has degree \(\sum n_i+1-r\). The operator on that finite-dimensional Hom block \[ g\longmapsto gF_1^{\otimes r}-F_1g \tag{A.3.5} \] is invertible. Indeed its two commuting multiplication operators have respective sole eigenvalues \(a^{\sum n_i}\) and \(a^{\sum n_i+1-r}\); their difference is nonzero for \(r>1\). Subtract those scalars to obtain commuting nilpotent operators. Their difference is nilpotent, by its binomial expansion. The inverse of the nonzero scalar plus that nilpotent operator is a finite geometric polynomial. Solve (A.3.5) uniquely to make (A.3.4) zero. There are possibly infinitely many degree tuples, but a graded multilinear cochain is their product of finite Hom blocks. The inverse just described is applied componentwise. It requires no uniform bound on the Jordan sizes. Iterating in \(r\) gives a change of coordinates: every component on a fixed word has stabilized after finitely many steps. The new automorphism has only \(F_1\). It commutes with the transformed coderivation. For \(r>2\), the transformed \(m_r\) has output degree \(\sum n_i+2-r\). Thus the equation \[ F_1m_r=m_rF_1^{\otimes r} \tag{A.3.6} \] intertwines generalized eigenspaces with distinct sole eigenvalues \(a^{\sum n_i+2-r}\) and \(a^{\sum n_i}\). The same invertibility calculation forces \(m_r=0\). The binary product is unchanged by a coordinate change with linear term the identity, and its two eigenvalues agree. This proves the assertion. The construction did not assume a weight decomposition on the original cochain spaces, or semisimplicity of their Frobenius action. ∎ ### A.4. The actual arithmetic generator envelope Let \(\mathcal E_G\) be the graded category with objects \(A_\lambda\) and actual morphisms \(\operatorname{Ext}^*_{L^+G}(A_\lambda,A_\mu)\), with their genuine Yoneda composition and zero differential. Let \(\operatorname{Perf}(\mathcal E_G)\) denote the idempotent-complete finite-cone envelope of its represented modules. **Theorem A.4.1.1.** The geometric bounded finite-support spherical derived category, with the actual rational-adic coefficient enhancement, is equivalent to \(\operatorname{Perf}(\mathcal E_G)\). It is the category of its actual graded Ext generators; no dual Lie-algebra identification or monoidal upgrade is asserted by this theorem. **Proof.** On finite jet and free-frame diagrams take the compatible coefficient resolutions used in the rational-adic enhancements of Lesson 5 P–Q and Lesson 12 N. Their Hom complexes and composition give the enhanced generator category. At each coefficient level the underlying injective-resolution existence and comparison are proved in Sheaves of modules and their derived categories, Theorem 6.1 and Lemmas 6.2–6.3. Compatible cartesian action-nerve diagrams retain the equivariant data. Increasing the frame compares these complexes in each required degree range by the pulled-coefficient unit calculation of A.1; none of the action levels is discarded. Increasing a sufficient jet level has the same property: its kernel is split unipotent with \(R\Gamma(BU,E)=E\), so the corresponding nerve unit identifies mapping complexes. The finite model thus defines the same generator enhancement at every sufficient level. Arithmetic Frobenius pullback is an exact equivalence of these coefficient sites, and hence gives a dg functor on the resolution enhancement. Each chosen Weil structure identifies its generator with its pullback by a closed enhanced homotopy equivalence. Adjoin the pullback representatives when necessary, apply A.3.1.1, and identify them with the original labels. This gives an \(A_\infty\) automorphism of the minimal generator category whose first component is precisely the actual Frobenius on Ext. By A.2 its sole eigenvalue in degree \(n\) is \(a^n\), and \(a\) is not a root of unity. A.3.1.2 therefore makes this category equivalent to \(\mathcal E_G\). For completeness, the generators and their finite envelope really give the bounded category. Bounded perverse truncations express any bounded finite-support object by finitely many cones of its perverse cohomology objects. The geometric spherical heart is semisimple with simples \(A_\lambda\), proved in Lessons 6 B.1 and 11 C–D. Each perverse cohomology object is a finite sum of those simples. Retracts remain bounded and finite supported, as their ordinary and perverse cohomology are retracts of the corresponding finite objects. Enhanced Yoneda is fully faithful on the generators: evaluation of a represented module at its representing object gives its mapping complex, with exactly the original composition. Its comparison extends in the first argument and then the second through shifts and finite cones, because Hom from or into a cone is the corresponding mapping cone. It extends through retracts by taking the images of their commuting projectors. Its essential image is therefore exactly the finite-cone and retract envelope of the represented modules. A quasi-equivalence of the generator categories gives the same comparisons on every such construction, again in both variables. Applying the preceding formality equivalence proves the theorem. The remaining step toward (8.1) is an actual composition-preserving identification of \(\mathcal E_G\) with the free dual-group polynomial generators. Compatibility with convolution and change to complex coefficients also need their own enhanced comparisons. This theorem supplies neither merely from a list of Ext dimensions. ∎ ![Orbit restriction scalars give actual Ext purity; enhanced Frobenius gives the formal generator category without discarding Jordan parts](assets/arithmetic-ext-to-formal-generators.png) Theorems A.1.1.1–A.2.1.1 and A.3.1.2–A.4.1.1 give every group, map, degree and nonresonance condition displayed. The last box is the actual graded Ext category; its further dual Lie-algebra identification is a separate requirement. Editable SVG source. ### A.5. Why a one-sided weight inequality is insufficient **Proposition A.5.1.1.** The condition \(H^n_j(D)=0\) for \(jj\). This is a subalgebra: equality in the product degree bound forces both factors to be closed diagonal elements. Its map to diagonal cohomology is multiplicative. Full diagonal purity makes both the inclusion and that projection quasi-isomorphisms, weight by weight. In (A.5.1) the projection kills the nonzero classes \(ac,bc,abc\), so is not a quasi-isomorphism. This identifies exactly the missing hypothesis. ∎ ![The explicit exterior algebra satisfies the one-sided weight bound but has a nonzero Massey product and off-diagonal cohomology](assets/one-sided-weight-massey-obstruction.png) Proposition A.5.1.1 and Exercises A.6.1–A.6.3 compute the full differential, cohomology, weight obstruction and enhanced-automorphism condition. A degree bound on cohomology alone does not create the required chain comparison. Editable SVG source. ### A.6. Exercises **Exercise A.6.1 (medium).** In the algebra (A.5.1), compute every differential on its eight exterior basis elements. Verify both the one-sided weight inequality and its failure to imply diagonal formality. **Solution.** On \(1,a,b\) the differential is zero; on \(c\) it is \(ab\). The product rule gives \(d(ab)=0\), \(d(ac)=-aab=0\), \(d(bc)=-bab=0\), and \(d(abc)=ab\,ab=0\). Thus the only nonzero differential is \(c\mapsto ab\). The classes and weights are exactly (A.5.2); every weight is at least the degree, while \(ac,bc,abc\) are off the diagonal. The triple product from A.5 is nonzero and cannot be imported from the square-zero diagonal algebra. This calculation tests a chain-level obstruction rather than a parity count. **Exercise A.6.2 (medium).** For an arity-\(r\) coordinate change \(g_r\) and an arity-\(r\) operation \(m_r\), compute the two nonresonance exponents. Allow nontrivial Jordan blocks and give the inverse used in the first equation. **Solution.** The output degrees are \(\sum n_i+1-r\) and \(\sum n_i+2-r\). Their Frobenius eigenvalue ratios to the input product are therefore \(a^{1-r}\) and \(a^{2-r}\). The first is different from one for every \(r>1\), and the second for every \(r>2\). On a fixed finite Hom block (A.3.5) is \(sI+N\), where \(s=a^{\sum n_i}-a^{\sum n_i+1-r}\ne0\) and \(N\) is nilpotent, after identifying the commuting input and output multiplications. Its inverse is \(s^{-1}\sum_{j=0}^{M-1}(-N/s)^j\) for any \(M\) with \(N^M=0\). The same argument forces the higher operation to vanish. No diagonalization of the Jordan blocks was used. **Exercise A.6.3 (advanced).** Let \(H=k1\oplus kx\oplus ky\), with \(|x|=1,|y|=2\), all products of positive-degree elements zero, and \(m_3(x,x,x)=y\) as the only higher operation on such arguments. Verify this defines a minimal strictly unital \(A_\infty\) algebra. Show that the graded automorphism \(F_1(x)=ax,F_1(y)=a^2y\), with \(a\ne0,1\), cannot lift to an \(A_\infty\) automorphism. **Solution.** In the reduced bar coalgebra the suspended positive generators have degrees zero and one. Its only nonzero coderivation component replaces three consecutive \(sx\)'s by \(sy\). Two overlapping replacements give zero because a replacement inserts \(sy\); two disjoint replacements cancel in the square, since the coderivation has odd degree. Hence \(b^2=0\). Extending the ordinary unit with all higher operations containing it zero gives the unital identities: a unit in a binary vertex removes that vertex, and the two adjacent removals have opposite Koszul signs. On positive inputs the binary products are zero, so there are no remaining mixed binary terms. This proves the stated identities. For an automorphism's arity-three equation on \(x,x,x\), the terms involving \(F_2\) and binary products are zero: \(F_2(x,x)\) has degree one, hence is a multiple of \(x\), whose product with \(x\) is zero. Thus the equation is \(a^2y=a^3y\), impossible for \(a\ne0,1\). A graded cohomology automorphism alone is therefore insufficient for formality. The enhanced Frobenius lift in A.4 is an essential input. This appendix leaves the dual Lie-algebra Ext computation, its convolution compatibility, and the renormalized, singular-support and factorization enhancements as actual further proof requirements. ## Appendix B. The polynomial generators and their full Kostant restriction Let \(R\) be the prescribed split connected reductive coefficient group over a characteristic-zero field \(E\), with its actual torus, character lattice and centre. Eventually \(R=\widehat G\); none of the constructions below replaces it by its adjoint quotient. We prove the algebraic generator comparison needed for (8.1). The remaining geometric comparison must identify these modules and their composition with actual equivariant cohomology. Write \(\mathfrak r=\operatorname{Lie}R\), \(T\subset B\), \(\mathfrak t=\operatorname{Lie}T\), \(r=\dim T\), and \(N=|\Phi^+|\). Choose a nondegenerate invariant form to identify \(\mathfrak r\) with \(\mathfrak r^*\); the derivative formulation below also describes the resulting action directly on the coadjoint space. Every linear coordinate of \(A=E[\mathfrak r^*]=\operatorname{Sym}(\mathfrak r)\) has degree two. Nilpotent means nilpotent in a faithful algebraic module; the toral centre's Jordan component is semisimple and its nilpotent component is zero. Geometric orbit arguments may be checked after algebraic closure. The polynomial maps and the smooth descent diagrams constructed below are defined over \(E\), and their verified identities descend faithfully flatly to it. The root and torus groups and their full schemes are proved in Root groups and the rank-one subgroups, §§2–5. We use Complete reducibility, Theorems 4.1 and 5.1, [Representations of \(sl_2\)](../RT-LIE/RT-LIE-05.html), and Chevalley restriction, Theorem 3.1, with the coefficient descent of §3 above. The classical highest-weight and character statements are also available from Identifying the dual group, Theorem B.4.1.1. The free [Equivariant Satake category and Kostant–Whittaker reduction, §§2.6 and 4.1](https://arxiv.org/abs/0707.3799v4) gives the restriction problem; we prove its polynomial, regular-locus and morphism assertions here. ### B.1. Polynomial invariants and the sum of their degrees **Lemma B.1.1.1.** The algebra \(E[\mathfrak t]^W\) is polynomial on \(r\) homogeneous generators of degrees \(d_1,\ldots,d_r>0\), and \[ \sum_i(d_i-1)=N. \tag{B.1.1} \] **Proof.** First take the semisimple part and its auxiliary simply connected torus \(T_{\rm sc}\), whose character lattice is the full weight lattice. This auxiliary torus is only used to prove a statement about the linear Weyl action. It does not replace \(R\) in any representation or centralizer comparison. The invariant Laurent algebra \(E[T_{\rm sc}]^W\) is polynomial in the fundamental characters \(\chi_{\omega_i}\). Indeed its orbit sums have one basis element for each dominant weight. The product \(\prod_i\chi_{\omega_i}^{n_i}\) has highest weight \(\sum_i n_i\omega_i\) with coefficient one and only lower weights. Induction on the positive root height expresses every orbit sum as a polynomial in these characters. The induction terminates: a strictly lower dominant weight has strictly smaller nonnegative height. Algebraic independence follows by taking a maximal dominant weight among the finitely many monomials in a proposed relation; its coefficient cannot be supplied by a lower monomial. These are exactly the finite highest-weight and tensor arguments already proved in Lesson 11. Complete the torus quotient at the image of the identity. Its invariant algebra completes to a formal power-series ring in \(\chi_{\omega_i}-\dim V_{\omega_i}\). Its fibre over that point has only the identity as a geometric point: a finite-group invariant separates two distinct finite orbits, by prescribing values on their finite point sets and averaging. The Laurent algebra is finite over its invariants, because each coordinate and its inverse satisfy their monic orbit polynomials. Thus completing at that invariant ideal is the same as completing the torus algebra at the identity; the two ideals have cofinal powers. Invariants commute with this completion. To verify this, let \(I\) be an ideal of the invariant algebra. The averaging projector is linear over that algebra, so \(I^nE[T_{\rm sc}]\cap E[T_{\rm sc}]^W=I^n\). It also lifts an invariant class modulo \(I^n\) by averaging any lift. Taking the inverse limit proves the claim. Formal logarithms of a character basis identify the completed torus with the formal vector space \(\mathfrak t_{\rm sc}\), equivariantly for \(W\): the characteristic-zero series \(\log(q_i)\) has inverse \(\exp(z_i)\), and logarithms turn multiplication of characters into addition. Hence \(E[[\mathfrak t_{\rm sc}]]^W\) is a regular power-series algebra on \(r_{\rm ss}\) variables. For clarity this implies polynomiality of the graded invariant algebra, rather than merely a formal statement. Let \(C=E[\mathfrak t_{\rm sc}]^W\), with positive ideal \(\mathfrak m\). It is finitely generated: the coefficients of the orbit polynomials of a linear coordinate basis generate a Noetherian algebra over which the polynomial ring is finite; its invariant submodule is therefore finite over that algebra. The degree filtration and the \(\mathfrak m\)-adic filtration are cofinal, since the degrees of a finite homogeneous generating set are positive and bounded. Its completion is consequently the invariant formal series just calculated. Thus \(\dim_E\mathfrak m/\mathfrak m^2=r_{\rm ss}\). Choose a homogeneous basis and representatives \(f_i\). Induction on degree generates \(C\) from them: a positive homogeneous element modulo their linear span is in \(\mathfrak m^2\), whose factors have smaller positive degrees. The \(f_i\) are algebraically independent, because the fraction field of \(C\) has transcendence degree \(r_{\rm ss}\), by the finite integral extension to the polynomial ring. A relation among \(r_{\rm ss}\) generators would lower that transcendence degree. Adjoining the centre's linear coordinates proves polynomiality for all \(R\). Finally averaging the trace of \(W\) on symmetric powers gives \[ \prod_i(1-t^{d_i})^{-1} =\frac1{|W|}\sum_{w\in W}\det(1-tw)^{-1}. \tag{B.1.2} \] This identity follows by diagonalizing each finite-order \(w\) and multiplying its geometric series; it is an identity of formal power series. Put \(u=1-t\). On the left the first two Laurent coefficients are \((\prod d_i)^{-1}u^{-r}(1+\frac12\sum(d_i-1)u+\cdots)\). On the right the identity contributes \(|W|^{-1}u^{-r}\), and each reflection contributes \((2|W|)^{-1}u^{1-r}\). All other elements contribute lower pole orders. There are precisely \(N\) reflections: the root hyperplanes are the chamber walls and all their translates; a reflection's generic fixed point must lie on one of them, since a point off them has trivial chamber stabilizer. There is one reflection per positive root. Comparing these two coefficients proves \(\prod d_i=|W|\) and (B.1.1). For a torus \(W=1\), all degrees are one and the same assertion is immediate. ∎ ### B.2. A polynomial Kostant slice with its positive grading Choose pinned root vectors \(e_i,f_i\) with \([e_i,f_i]=\alpha_i^\vee\). Let \[ h=2\rho^\vee=\sum_i c_i\alpha_i^\vee,\qquad e=\sum_i e_i,\qquad f=\sum_i c_i f_i. \tag{B.2.1} \] The \(c_i\) are positive integers, because \(2\rho^\vee\) is the sum of the positive coroots. The actual cocharacter \(2\rho^\vee\) belongs to the original coroot lattice. **Theorem B.2.1.1.** The affine slice \(\Sigma=e+\mathfrak r^f\) is mapped isomorphically by the adjoint quotient \[ \chi:\mathfrak r^*\longrightarrow \mathfrak b=\operatorname{Spec}E[\mathfrak r^*]^R. \tag{B.2.2} \] This is a polynomial isomorphism with a polynomial inverse on every parameter algebra. Under the slice action \[ t\cdot x=t^2\operatorname{Ad}_{h(t^{-1})}x, \tag{B.2.3} \] its coordinate weights are \(2d_i\). Every slice point is regular. **Proof.** The simple-root relations give \([e,f]=h,[h,e]=2e,[h,f]=-2f\). On the adjoint module, complete reducibility gives \(sl_2\) summands of highest weights \(2m_i\): all \(h\)-weights are even root heights. Each summand has one weight-zero line, and \(\mathfrak r_0=\mathfrak t\). There are therefore \(r\) summands, allowing \(m_i=0\) on the centre. Since \(\dim\mathfrak r=2N+r\), \[ \sum_i m_i=N. \tag{B.2.4} \] The lowest lines form \(\mathfrak r^f\), with slice weights \(2(m_i+1)>0\). We check an invertible differential at \(x_*=e+f\). The principal triple acts through an actual \(SL_2\) homomorphism into \(R\): in a faithful module its rank-one representation integrates by the polynomial highest-weight formulas, Theorem 2.1 and Proposition 2.2, and the two root-group images are the polynomial exponentials of \(e\) and \(f\), contained in the original positive and negative unipotent groups. These root groups generate \(SL_2\). Its determinant-one matrix \(\left(\begin{smallmatrix}1&-1/2\\1&1/2\end{smallmatrix}\right)\) conjugates the diagonal generator to \(e+f\). Hence \(x_*\) is conjugate to \(h\) and is regular semisimple. The kernel of \(d\chi\) at a regular semisimple point is its orbit tangent space. Indeed the differential of \(R\times\mathfrak t_{\rm reg}\to\mathfrak r\) is \([\mathfrak r,x]+\mathfrak t=\mathfrak r\), and the finite Weyl quotient is étale off its reflecting hyperplanes. Here are the required checks. If \(w\) fixes a regular vector over any characteristic-zero field, its rational fixed subspace has no root identically zero, since that vector witnesses this. A rational real point of that subspace can therefore be chosen outside every root hyperplane. Its chamber stabilizer is trivial, so \(w=1\). Now prescribe any first jet at a regular geometric point \(x\), multiplied by \(|W|\), and prescribe zero first jets at all other orbit points. The squared maximal ideals of these distinct points are comaximal, so the Chinese remainder theorem gives a polynomial with these jets. Averaging it gives an invariant with the chosen differential at \(x\). Thus the quotient differential is surjective, and it is an isomorphism since both smooth spaces have dimension \(r\), by B.1. Maps between smooth schemes, Lemma 3.A proves étaleness. This establishes the quotient differential assertion used here. The space \(\mathfrak r^f\) is transverse to that orbit at \(x_*\). Use the invariant form. The orthogonal complement of the orbit tangent is \(\mathfrak r^{x_*}\). In an \(sl_2\) summand \(\operatorname{Sym}^{2m}\), its weight-zero line after the rotation sending \(h\) to \(e+f\) has nonzero highest coefficient: rotating \(X^mY^m\) gives a nonzero multiple of \((Y^2-X^2)^m\). It therefore pairs nondegenerately with the lowest line. On each multiplicity space the invariant pairing is nondegenerate, and the same nonzero scalar multiplies it. This proves the transversality, including the central summands. Thus \(d(\chi|_\Sigma)\) is invertible at \(x_*\). Choose the homogeneous generators from B.1 and Chevalley restriction. The restricted polynomial \(p_i|_\Sigma\) has slice weight \(2d_i\), because the quotient is invariant under conjugation and \(p_i\) has ordinary polynomial degree \(d_i\). Its Jacobian determinant has weight \(2\sum_i d_i-2\sum_i(m_i+1)=0\), by (B.1.1) and (B.2.4). All slice variable weights are positive. Hence this determinant is a constant, nonzero by the preceding differential calculation. At the slice origin \(e\) its linear part is an invertible graded map. Consequently the two multisets \(d_i\) and \(m_i+1\) agree. This gives a polynomial inverse without an unproved general assertion about constant Jacobians. Order the common positive weights. In a block of weight \(d\), the quotient coordinates are an invertible linear combination of the variables of weight \(d\), plus polynomials in variables of strictly smaller weights. A nonlinear monomial of total weight \(d\) can only contain such smaller weights. Solve these finite blocks in order. This constructs the inverse polynomial, also on nonreduced parameter algebras. Finally every element has centralizer dimension at least \(r\): regular semisimple elements form a dense open, and the corresponding rank bound on \(\operatorname{ad}x\) is the vanishing of polynomial minors. Choose an \(h\)-homogeneous complement to \(\ker\operatorname{ad}e\). For \(x=e+v\in\Sigma\), the highest nonzero \(h\)-weight of a vector in that complement has nonzero image under \(\operatorname{ad}e\), with weight two higher. All \([v,-]\) terms have at most its original weight. They cannot cancel that term. Thus \(\operatorname{ad}x\) has rank at least \(\dim\mathfrak r-r\), and the opposite inequality just proved makes it equal. The regular minor is invertible locally on the whole slice, so the centralizer Lie kernel is a vector bundle there. ∎ ### B.3. Regular fibres and the full centralizer scheme **Lemma B.3.1.1.** Every nilpotent element of a reductive Lie algebra in characteristic zero belongs to the nilpotent radical of a Borel. The nilpotent locus is irreducible of dimension \(\dim\mathfrak r-r\), and its regular elements form one orbit. **Proof.** Jordan decomposition and its preservation in faithful modules for the semisimple summand are proved in The Killing form and Cartan's criteria, Theorem 6.2 and Complete reducibility, Theorem 5.1. The central torus acts diagonally in an algebraic module; its component is semisimple and commutes with those Jordan parts. This gives the stated reductive version. For a nilpotent \(n\), its finite polynomial exponential is an actual additive subgroup of \(R\). One can check containment schematically: the right-invariant derivation associated with \(n\) preserves the Hopf ideal of the closed group \(R\); every Taylor coefficient of a defining equation on \(\exp(tn)\) is therefore zero. The exponential is polynomial because the represented \(n\) is nilpotent. This additive group has a fixed point on the projective flag variety \(R/B\), constructed in The flag scheme and the Weyl group, Theorem 1.1. Here is the needed fixed-point argument. A nonconstant additive orbit map from \(\mathbf A^1\) into a projective variety is represented by polynomial homogeneous coordinates after trivializing its pulled-back line on \(E[t]\). Homogenizing to their largest degree extends it to \(\mathbf P^1\), with a defined value at infinity. Translation equivariance holds on the dense affine line and therefore on the entire projective line by separatedness. Infinity is fixed by every translation, so its image is fixed. A constant orbit already provides a fixed point. Thus \(n\) lies in a conjugate Borel Lie algebra. In a faithful weight-triangular module its torus part would give diagonal eigenvalues. Nilpotence makes all of them zero, and the faithful torus weights span its Lie dual. Hence that part is zero. It follows that the proper incidence map \(R\times^B\mathfrak n\to\mathfrak r\) covers the nilpotent locus. Its source is an irreducible vector bundle of dimension \(2N=\dim\mathfrak r-r\), so its image is irreducible and has dimension at most that number. Supporting field geometry, Theorem G.3.4 and Theorem G.4.2 proves smoothness of each characteristic-zero finite-type stabilizer and local closedness of its orbits. The principal \(e\) therefore has group centralizer dimension \(r\), by the \(sl_2\) Lie calculation of B.2, so its orbit has exactly that dimension. The bound is an equality. Any regular nilpotent orbit has the same dimension and is open in the irreducible nilpotent locus, so must be the same orbit. For the openness assertion, a constructible orbit contains a nonempty open of its closure; translating that open covers the orbit and makes the whole orbit open in its closure. Two distinct full-dimensional orbits would give disjoint nonempty opens of the same irreducible variety. ∎ **Theorem B.3.1.2.** Each regular quotient fibre has exactly one geometric orbit. The map \[ p:R\times\Sigma\longrightarrow\mathfrak r_{\rm reg}, \qquad(g,\sigma)\longmapsto\operatorname{Ad}_g\sigma \tag{B.3.1} \] is smooth surjective. The closed group scheme \[ J=\{(\sigma,g)\in\Sigma\times R: \operatorname{Ad}_g\sigma=\sigma\} \tag{B.3.2} \] is smooth over \(\Sigma\), has relative dimension \(r\), and is commutative. Its fibres need not be connected. It includes the full original \(Z(R)\). **Proof.** Write a geometric element \(x=s+n\) in Jordan form, with commuting semisimple and nilpotent parts. The semisimple part lies in a torus. This can also be seen directly in a faithful module: in a diagonal basis the formal exponential of \(s\) has character exponents. Distinct exponential series are linearly independent by their derivative Vandermonde matrix. Its algebraic closure is therefore the torus defined by the integral character relations whose derivative on \(s\) is zero; the same Hopf-ideal Taylor argument places that torus in \(R\). Maximal-torus conjugacy is proved in Tori, maximal tori and their conjugacy. Let \(D\) be the torus just obtained. Its character relation lattice is exactly the integral characters whose derivative at \(s\) is zero; this lattice is saturated, since the characteristic is zero. In a faithful diagonal module two weights restrict to the same character of \(D\) exactly when their eigenvalues on \(s\) agree. Commuting with \(s\) means preserving precisely these equal-eigenvalue blocks, while commuting with the universal \(D\)-action means preserving exactly its character blocks. The two matrix conditions are therefore identical over every parameter algebra. Intersecting with the original closed group gives the scheme equality \(C_R(s)=C_R(D)\). Centralizers of subtori, Theorem 2.2 proves that this is a smooth connected reductive group. Choose an integral cocharacter of \(D\) outside the finitely many hyperplanes for roots nontrivial on \(D\); an integral point exists because its cocharacter lattice spans the rational vector space. Its zero roots are exactly those with \(\alpha(s)=0\). Weyl conjugation makes this cocharacter dominant. The positive/negative cocharacter limits and the Levi construction of The flag scheme and the Weyl group, §1 identify the centralizer as the corresponding root Levi \(L\). In particular its rank is the original \(r\), and its roots are precisely \(\alpha(s)=0\). This proves the full centralizer assertion without dropping any component or infinitesimal equation. The centralizer of \(x\) is \(C_L(n)\), since a commuting operator preserves the polynomial Jordan parts. Thus \(x\) is regular exactly when \(n\) is regular in that Levi. B.3.1.1 applies to the Levi. Invariant polynomials have the same values on \(s+n\) and \(s\): put \(n\) in a Levi Borel radical and contract it to zero with its positive coroot cocharacter, fixing \(s\). Chevalley restriction and finite-orbit separation for \(W\) determine the semisimple conjugacy class from these values. The single regular nilpotent orbit of its Levi determines the remaining regular orbit. This proves uniqueness of a regular orbit in a quotient fibre. B.2 gives a slice point for every quotient value, and every slice point is regular. The map (B.3.1) is consequently geometrically surjective. At a slice point, \(d\chi|_{\mathfrak r^f}\) is an isomorphism, so \(d\chi\) has rank \(r\). Invariance places the orbit tangent in its kernel; both have dimension \(\dim\mathfrak r-r\), so they are equal. Conjugation gives the same equality at every regular point. The orbit and slice tangents therefore make the differential of (B.3.1) surjective everywhere. The differential criterion of Lemma 3.A cited in B.2 gives actual smooth surjectivity, of relative dimension \(r\). Base changing this map along the slice gives \(J\). This is the full scheme centralizer, because equality of two quotient values forces their slice coordinates to be equal by the polynomial inverse of B.2, even over nonreduced rings. More explicitly the actual descent relation is \[ (R\times\Sigma)\times_{\mathfrak r_{\rm reg}}(R\times\Sigma) \simeq R\times J, \tag{B.3.3} \] with right multiplication by the centralizer on the first factor. Thus \(J\) is smooth over \(\Sigma\). Its generic fibre is a torus. The two multiplication maps on \(J\times_\Sigma J\) agree there and hence everywhere: its affine coordinate algebra is flat over the integral slice, so is torsion free and injects into its generic localization. Every coordinate difference therefore vanishes already on the whole source. This proves commutativity as a scheme identity. Central elements satisfy (B.3.2). No special fibre has been replaced by its identity component. ∎ ### B.4. The extension across the missing locus **Proposition B.4.1.1.** The nonregular complement in \(\mathfrak r\) has codimension at least two. Restriction of polynomial functions to the regular locus is an isomorphism, and the same is true for maps between free bundles. **Proof.** Use the finitely many root Levi types for semisimple centralizers. If a type \(L\) has semisimple rank \(s\), its semisimple centralizer parameters have dimension \(r-s\). Its nilpotent locus has dimension \(\dim L-r\), by B.3.1.1. The corresponding Jordan stratum is covered by \(R\times^L(Z(\mathfrak l)_{\rm reg}\times\mathcal N_L)\), of dimension at most \(\dim R-s\). If \(s\ge2\), this already gives codimension at least two. If \(s=0\), all elements are regular. If \(s=1\), its derived Lie algebra is \(sl_2\), whose nonzero nilpotents have centralizer dimension one and whose only nonregular nilpotent is zero. The bad stratum in this case has dimension at most \(\dim R-(r+2)+(r-1)=\dim R-3\). The finite union of these bounds proves the assertion. This dimension argument concerns geometric Jordan strata; the regular scheme and its centralizers are the full smooth schemes of B.3. Let a regular function on this open locus be \(u/v\) in reduced form in the polynomial fraction field. If an irreducible factor divided \(v\), its height-one generic point would be in the regular locus, since the missing set has codimension at least two. Regularity there would force cancellation of that factor, a contradiction. Thus \(v\) is a unit and the function is polynomial. Uniqueness follows from density. Apply this coefficientwise to maps between finite free bundles. ∎ ### B.5. Exact Kostant restriction and full free-module morphisms Let \(p_1,\ldots,p_r\) be the chosen invariants. Their differentials on the coadjoint space give sections \(\zeta_i\) of the stabilizer Lie bundle. Equivalently they are gradients under the invariant form. The differentials have rank \(r\) on the slice, and invariance makes their values commute with the slice element. Hence they form the entire \(\operatorname{Lie}J\) bundle. B.3 makes this an abelian bundle. Put \[ \mathcal B=\operatorname{Sym}_{E[\Sigma]}(\operatorname{Lie}J) =E[p_1,\ldots,p_r,z_1,\ldots,z_r], \qquad |p_i|=2d_i,\quad |z_i|=2d_i-2. \tag{B.5.1} \] The tangent-bundle underlying algebra is thus supplied with the displayed deformation grading. In the Hom notation below the superscript records the degree of a homogeneous module map between these coherent graded objects. **Theorem B.5.1.1.** For an \(R\)-equivariant coherent sheaf \(F\) on \(\mathfrak r^*\), restriction to \(\Sigma\), with the stabilizer Lie action, gives an exact functor \(\kappa(F)\) to coherent \(\mathcal B\)-modules. On the free generators \(A\otimes V\) it gives \[ M(V)=E[\Sigma]\otimes V,\qquad z_i\cdot v=\rho_V(\zeta_i)v. \tag{B.5.2} \] It is fully faithful on these generators, including all homogeneous morphism degrees: \[ \operatorname{Hom}^*_{A,R}(A\otimes V,A\otimes W) \xrightarrow{\sim} \operatorname{Hom}^*_{\mathcal B}(M(V),M(W)). \tag{B.5.3} \] The comparison preserves actual composition and the prescribed full group. **Proof.** Pullback to \(R\times\Sigma\) along the smooth surjection \(p\) is exact. Equivariance identifies it with the pullback of \(F|_\Sigma\) along the faithfully flat projection to the slice. Consequently restriction to the slice is exact on equivariant coherent sheaves, even though an arbitrary closed restriction is not flat. The residual action of \(J\) gives its Lie action, hence the module over its symmetric algebra. It is coherent: it is already finite over the Noetherian base \(E[\Sigma]\). This proves exactness and (B.5.2). A \(\mathcal B\)-linear map between the displayed modules is a polynomial family \(\phi_\Sigma\) commuting with \(\operatorname{Lie}J\). On the geometric generic fibre \(J\) is a torus. Commuting with its Lie action implies commuting with its full torus action: rational torus modules split into actual characters, and distinct characters have distinct derivatives in characteristic zero. This retains characters differing on finite centres as well. It follows that \(\phi_\Sigma\) commutes with the full \(J\) everywhere. The required coordinate equalities hold on the generic fibre of the smooth flat group scheme \(J\); its torsion-free coordinate algebra makes them hold as scheme equations on the whole \(J\). Define the family on \(R\times\Sigma\) by \(\rho_W(g)\phi_\Sigma(\sigma)\rho_V(g)^{-1}\). Full \(J\)-equivariance makes it compatible with the actual relation (B.3.3). Faithfully flat descent gives an \(R\)-equivariant free-bundle map on \(\mathfrak r_{\rm reg}\). B.4 extends its coefficients uniquely to polynomials on all of \(\mathfrak r\). Equivariance extends too, because its two matrix formulas agree on the dense \(R\times\mathfrak r_{\rm reg}\). Restriction reverses this procedure, proving (B.5.3). All steps are functorial matrix equalities, so preserve composition. For the grading, give a vector in \(V\) its principal \(h\)-weight, in addition to any specified external grading. The gradient identity under (B.2.3) is \[ \zeta_i(t\cdot\sigma) =t^{\,2(d_i-1)}\operatorname{Ad}_{h(t^{-1})}\zeta_i(\sigma). \tag{B.5.4} \] Its root component of \(h\)-weight \(b\) has coefficient degree \(2(d_i-1)-b\), while its action raises a vector's principal degree by \(b\). Thus its total operator degree is exactly \(2d_i-2\), as in (B.5.1). An equivariant homogeneous polynomial map of degree \(k\) similarly satisfies \(\phi(t\cdot\sigma)=t^k\rho_W(h(t^{-1}))\phi(\sigma)\rho_V(h(t))\). The coefficient and principal-degree changes cancel to give degree \(k\). Conversely decompose the unique polynomial extension into its finitely many homogeneous parts. Their restriction preserves these degrees; faithfulness forces every part of a different degree to be zero. This proves the graded comparison, not just the ungraded Hom equality. There is also the coefficient tensor comparison \[ M(V)\otimes_{E[\Sigma]}M(W)\simeq M(V\otimes W), \tag{B.5.5} \] where \(z_i\) acts primitively as \(z_i\otimes1+1\otimes z_i\). This follows directly from the differential of the representation tensor product. Its associativity and unit are those of that tensor product. A derived convolution or symmetry comparison requires its own enhancement and the conventions of Lesson 8; (B.5.5) is the stated coefficient-level comparison. ∎ ![The actual polynomial Kostant slice gives smooth full-centralizer descent and unique polynomial extension of all free-generator morphisms](assets/kostant-slice-descent.png) Lemmas and Theorems B.1.1.1–B.5.1.1 prove the slice inverse, smooth maps, full scheme relation, codimension bound, grading and composition shown. The invariant-ring argument and positive-weight inverse provide actual polynomial maps over every parameter algebra. Editable SVG source. ### B.6. Rank one: full operators, morphisms and centre **Proposition B.6.1.1.** For \(R=SL_2\) use the trace form \(B(X,Y)=\operatorname{tr}(XY)\), the invariant \(p(X)=\frac12\operatorname{tr}(X^2)\), and \[ X(c)=\begin{pmatrix}0&1\\c&0\end{pmatrix},\qquad E[\Sigma]=E[c],\qquad \mathcal B=E[c,z],\quad |c|=4,\ |z|=2. \tag{B.6.1} \] The unit, standard and adjoint generators have respectively \(z\)-actions \(0,X(c),T=\operatorname{ad}X(c)\). For the adjoint basis \((e,h,f)\) of degrees \(2,0,-2\), \[ T=\begin{pmatrix}0&-2&0\\-c&0&1\\0&2c&0\end{pmatrix}, \qquad T^3=4cT. \tag{B.6.2} \] The two unit-adjoint Hom modules are freely generated over \(E[c]\) in degree two by \[ i=(1,0,c)^{\mathsf t},\qquad \ell=(c,0,1). \tag{B.6.3} \] Their actual compositions are \[ \ell i=2c,\qquad i\ell=2cI-\tfrac12T^2,\qquad Ti=\ell T=0. \tag{B.6.4} \] The adjoint endomorphism algebra is \(E[c,T]/(T^3-4cT)\), and the standard one is \(E[c,X]/(X^2-cI)\). **Proof.** The chosen invariant's gradient under the trace form is \(X\), so these are exactly the operators in (B.5.2). Applying the brackets to \(e,h,f\) gives the three columns of (B.6.2); multiplying gives its relation. The vector \(f\) is cyclic over \(E[c]\): the columns \(f,Tf,T^2f\) have determinant \(2\). Thus a commuting endomorphism is uniquely a polynomial of degree at most two in \(T\), determined by its value on that cyclic vector. There are no denominators or extra endomorphisms at \(c=0\). For the standard module the vector \(v_-\) is cyclic with columns \(v_-,Xv_-\) of determinant \(-1\), proving its endomorphism algebra. Solving \(Ti=0\) and \(\ell T=0\) gives (B.6.3) as the entire free Hom modules. Their grades follow from the displayed basis degrees and \(|c|=4\). Direct multiplication gives (B.6.4), with the stated trace normalization. These are the full compositions supplied by B.5, not their dimensions. The full universal centralizer is \[ J=\operatorname{Spec}E[c,u,v]/(u^2-cv^2-1), \qquad g=\begin{pmatrix}u&v\\cv&u\end{pmatrix}. \tag{B.6.5} \] Every commuting matrix has this form, by the same cyclic-vector calculation, and the equation is its actual determinant-one condition. At \(c=0\) it is \(\mu_2\times\mathbf G_a\), with additive coordinate \(v/u\). This fibre has two components and retains the original centre. Globally there is no nonzero map from the unit to the standard Kostant module: \(X(c)\) has zero kernel over the domain \(E[c]\). At \(c=0\) its Lie-action kernel is one dimensional, but the central element \(-I\) acts on that line by \(-1\), so there is still no map from the unit commuting with the full centralizer group. Fibrewise Lie commutation alone would lose precisely this condition; the generic-to-flat-group argument in B.5 does not. The unit–adjoint block descends to \(PGL_2\); the standard generator in this example belongs to the stated \(SL_2\). ∎ ![The SL2 Kostant generators have explicit composition constants and retain the full finite centre on the nilpotent fibre](assets/sl2-kostant-compositions.png) Proposition B.6.1.1 and Exercises B.7.2–B.7.3 compute the exact operators, cyclic bases, products, scheme equations and nilpotent coefficients. The full centralizer condition distinguishes fibrewise Lie commutation from the global morphism theorem. Editable SVG source. ### B.7. Exercises **Exercise B.7.1 (medium).** Verify the positive-weight inverse argument on \(F(u,v)=(u,v+u^2)\), with weights \(2,4\). Evaluate it on \(E[\epsilon]/(\epsilon^3)\), with \(u=\epsilon,v=\epsilon^2\), retaining every coefficient. **Solution.** The two target coordinates have weights \(2,4\); its linear blocks are both the identity and its only nonlinear term uses the smaller weight. The inverse is \((u,w)\mapsto(u,w-u^2)\). At the specified parameter \(F(\epsilon,\epsilon^2)=(\epsilon,2\epsilon^2)\), and the inverse recovers \(\epsilon^2\). Passing to its point would erase that quadratic coefficient. The argument uses the positive-weight triangular blocks proved in B.2, not a general claim that any constant-Jacobian polynomial map is invertible. **Exercise B.7.2 (advanced).** For \(SL_3\), take \(e=E_{12}+E_{23}\), \(f=2(E_{21}+E_{32})\), and slice \(X=e+uf+vf^2\). Compute its quotient coordinates, the centralizer Lie basis, and the full centre of the nilpotent group fibre. **Solution.** The matrix is \[ X=\begin{pmatrix}0&1&0\\2u&0&1\\4v&2u&0\end{pmatrix}. \tag{B.7.1} \] Multiplication gives \(\frac12\operatorname{tr}X^2=4u\) and \(\frac13\operatorname{tr}X^3=\det X=4v\). Its quotient inverse is \(u=p_2/4,v=p_3/4\), with degrees four and six. The columns \(e_3,Xe_3,X^2e_3\) have determinant \(-1\), on every parameter algebra. Hence every commuting matrix is \(aI+bX+dX^2\). Its trace-zero part has basis \(X\) and \(X^2-\frac{8u}{3}I\), of operator degrees two and four, respectively. These are the gradients of the two chosen invariants under the trace form. At \(u=v=0\), \(X^3=0\) and \(\det(aI+bX+dX^2)=a^3\). The group equation is therefore \(a^3=1\). Its centre factor is the whole \(\mu_3\); after dividing by \(a\), logarithm coordinates \(b/a\) and \(d/a-b^2/(2a^2)\) give the two additive factors. Thus the full regular nilpotent centralizer is \(\mu_3\times\mathbf G_a^2\), with three geometric components in characteristic zero. B.3 never assumed this fibre connected. **Exercise B.7.3 (advanced).** At \(c=\epsilon\in E[\epsilon]/(\epsilon^2)\), verify that \[ \begin{pmatrix}1+\epsilon/2&1\\ \epsilon&1+\epsilon/2\end{pmatrix} \tag{B.7.2} \] is an actual \(SL_2\) centralizer point. Explain the difference between the nilpotent fibre's Lie-action Hom and the global free-module Hom. **Solution.** It has the form (B.6.5) with \(u=1+\epsilon/2,v=1\), and \(u^2-\epsilon v^2=1\) exactly. Multiplying with \(X(\epsilon)\) on either side gives the same matrix; its infinitesimal coefficient is retained by the scheme equation. At \(c=0\), the standard module has a nonzero vector killed by \(z=X(0)\), so its Lie-action Hom from the unit has dimension one. Its \(-I\) action makes the full group Hom zero. Over \(E[c]\) the determinant of \(X(c)\) is the nonzero polynomial \(-c\), so its kernel and the global free-module Hom are zero. Formation of that kernel does not commute with this specialization. There is no contradiction with the global fully faithful restriction theorem. These proofs construct the full algebraic polynomial generators and their faithful restriction, with their precise grading and composition. Identifying these restriction modules with geometric equivariant cohomology, and proving geometric full faithfulness and enhanced convolution compatibility, remain the next steps toward the derived equivalence. ## Appendix C. Contracting loop charts and an integral cohomology criterion We construct opposite-loop open charts over every parameter algebra, prove their actual contraction comparison, and give a cell criterion for equivariant Ext to equal graded module Hom. The criterion requires surjectivity to the fixed-point stalks. Section C.9 proves a sufficient weight implication with its compact-support estimate stated explicitly as a hypothesis. The earlier geometric proofs are Lesson 1, Theorem 2.C and Lemma 2.0, Lesson 2, Lemma 2.1, Theorem 3.1 and Lemma 7.2, Lesson 6, Lemmas A.1.1.1–A.1.2.1, and Lesson 5, §§O–P and Q.1. These provide the root coordinates, lattice families, affine homogeneous-quotient hull, global supported Kummer class, trace and affine-line propagation used below. Free reading is [Ginzburg, *Pointwise purity, derived Satake, and Symplectic duality*, arXiv:2508.15958v2](https://arxiv.org/abs/2508.15958v2), §§4.1–4.2. The proofs here specify the actual integral maps. ### C.1. Orthogonal supports give integral Hom vanishing Let \(S=E[u_1,\ldots,u_r]\), \(Q=\operatorname{Frac}(S)\), and let \(A\) be a commutative \(S\)-algebra. Suppose \[ Q\otimes_S A=\prod_{x\in\Xi}Q \] for a finite set \(\Xi\). Write \(e_x\) for its orthogonal idempotents. Let \(M,N\) be finite \(A\)-modules. Assume their localizations have disjoint idempotent supports and that \(N\) is \(S\)-torsion free. **Lemma C.1.1.1.** Then \(\operatorname{Hom}_A(M,N)=0\). **Proof.** Let \(e\) be the sum of the idempotents supporting \(QM\), so \(e\) acts as the identity there and as zero on \(QN\). For an \(A\)-linear map \(f\), its localization is \(QA\)-linear. For every \(m\in QM\), \[ Qf(m)=Qf(em)=eQf(m)=0. \] For each \(m\in M\), this says that \(f(m)\) is killed by a nonzero element of \(S\). Torsion freeness of \(N\) makes \(f(m)=0\). Thus \(f=0\). No denominator has been used to construct a global map, and no surjectivity is inferred from localization. The same proof applies in every homogeneous map degree. ∎ This is the exact role of fixed-point localization in the geometric criterion: disjoint supports kill cross terms integrally because their targets are free over \(H_T^*(\mathrm{pt})\). The proof does not say that an arbitrary rational morphism extends. ### C.2. Two module filtrations give the required left-exact Hom row Keep \(S,A,Q\). Let \(P,F\) be \(A\)-modules, together with exact sequences \[ 0\longrightarrow P_c\xrightarrow{a}P\xrightarrow{b}P_Z\longrightarrow0, \qquad 0\longrightarrow P_!\xrightarrow{c}P\xrightarrow{d}P_U\longrightarrow0, \] \[ 0\longrightarrow F_Z\xrightarrow{i}F\xrightarrow{j}F_U\longrightarrow0. \] Assume \(P_c,P_U,F_U\) have localized support on a subset \(\Xi_U\), whereas \(P_!,P_Z,F_Z\) have support in its complement. Assume \(F_Z\) and \(F_U\) are \(S\)-torsion free. In geometry these are the compact open, ordinary open and closed restriction sequences; their exactness is a separate geometric assertion. **Lemma C.2.1.1.** Every \(g:P\to F\) induces a unique \(A\)-linear map \[ \beta(g):P_U\longrightarrow F_U, \qquad \beta(g)d=jg. \] Moreover the sequence \[ 0\longrightarrow\operatorname{Hom}_A(P_Z,F_Z) \xrightarrow{\alpha}\operatorname{Hom}_A(P,F) \xrightarrow{\beta}\operatorname{Hom}_A(P_U,F_U) \] is exact, where \(\alpha(f)=ifb\). Surjectivity of \(\beta\) is not an assumption or a conclusion of this lemma. **Proof.** Lemma C.1.1.1 gives \[ \operatorname{Hom}_A(P_!,F_U)= \operatorname{Hom}_A(P_c,F_Z)= \operatorname{Hom}_A(P_Z,F_U)=0. \] The first equality makes \(jgc=0\), so the quotient universal property for \(d\) gives the unique map \(\beta(g)\). Since \(b\) is surjective and \(i\) is injective, \(\alpha\) is injective. For \(f:P_Z\to F_Z\), the composite \(ji=0\) makes \(\beta\alpha(f)=0\). Conversely suppose \(\beta(g)=0\). Then \(jg=0\), so \(g=i\widetilde g\) for a unique map \(\widetilde g:P\to F_Z\). The second displayed Hom vanishing gives \(\widetilde ga=0\). Hence \(\widetilde g=fb\) for a unique \(f:P_Z\to F_Z\), and \(g=\alpha(f)\). This proves exactness without applying an unjustified exactness assertion to a Hom functor. ∎ ### C.3. A surjective upper comparison supplies all lower lifts Consider a commutative diagram whose top row is short exact and whose bottom row is the row of C.2: \[ \begin{array}{ccccccccc} 0&\to&E_Z&\xrightarrow{s}&E&\xrightarrow{t}&E_U&\to&0\\ &&\downarrow h_Z&&\downarrow h&&\downarrow h_U\\ 0&\to&H_Z&\xrightarrow{\alpha}&H&\xrightarrow{\beta}&H_U. \end{array} \] **Lemma C.3.1.1.** If \(h_Z\) and \(h_U\) are isomorphisms, then \(h\) is an isomorphism and \(\beta\) is surjective. **Proof.** To prove injectivity, let \(h(e)=0\). Commutativity and injectivity of \(h_U\) give \(t(e)=0\), so \(e=s(z)\). Then \(0=\alpha h_Z(z)\). Injectivity of \(\alpha\) and \(h_Z\) gives \(z=0\), hence \(e=0\). For surjectivity, choose \(g\in H\). Lift \(\beta(g)\) through the isomorphism \(h_U\) to \(u\in E_U\), and through the surjection \(t\) to \(e\in E\). Then \[ \beta(g-h(e))=0. \] Exactness of the lower row gives \(g-h(e)=\alpha(z')\). Lift \(z'\) through \(h_Z\), giving \(z\). Commutativity now gives \[ g=h(e+s(z)). \] Thus \(h\) is surjective. For any \(v\in H_U\), lift it through \(h_U\) and \(t\) as above; commutativity gives \(v=\beta(h(e))\), proving surjectivity of \(\beta\). ∎ This is the complete induction chase needed for the actual cohomology map. Its top row must come from the actual geometric restriction triangle. Weights may force its connecting maps to vanish, but weight purity alone does not prove the separate module-filtration surjectivity used in C.2. ### C.4. A complete two-fixed-point example Let \[ S=E[u],\quad |u|=2,\qquad A=S[h]/(h(h-u)),\quad |h|=2. \] It is free over \(S\) with basis \(1,h\). Over \(Q\) its two characters are \(h=0\) and \(h=u\), with idempotents \[ e_0=1-h/u,\qquad e_\infty=h/u. \] They satisfy \(e_0^2=e_0\), \(e_\infty^2=e_\infty\), \(e_0e_\infty=0\) and sum to one, directly from \(h^2=uh\). For the rank-one modules \(S_0,S_\infty\) carrying these respective characters, \[ \operatorname{Hom}_A(S_0,S_\infty)=0= \operatorname{Hom}_A(S_\infty,S_0). \] Indeed a map is multiplication by \(a\in S\), and commutation with \(h\) requires \(ua=0\). The polynomial domain makes \(a=0\). At \(u=0\), both specialized modules are \(E\) with trivial \(h\)-action over \(E[h]/h^2\), so their specialized-module Hom is \(E\). The specialization of the global Hom remains zero. This computes the failure of Hom to commute with this base change. It explains why one cannot forget the equivariant parameter and infer the desired geometric full faithfulness from specialized module dimensions. ### C.5. Euler classes and cellular fixed-point localization Let \(k\) be algebraically closed, let \(\ell\) be invertible in \(k\), and let \(E/\mathbf Q_\ell\) be finite. Let \(T\) be a split torus with character basis \(\chi_1,\ldots,\chi_r\). Put \[ S=H_T^*(\operatorname{Spec}k,E)=E[u_1,\ldots,u_r], \qquad |u_i|=2,\qquad Q=\operatorname{Frac}(S). \tag{C.5.1} \] The notation for the polynomial variables suppresses their coefficient lines, rather than changing the maps: \(c_1(\chi_i)\) lies in \(H_T^2(\mathrm{pt},E(1))\). A degree-\(2d\) Euler action therefore has its actual domain \(M(-d)[-2d]\). Every restriction and cup product below retains that line. Here is the finite model of (C.5.1). Choose the right action on \(P_m=(\mathbf A^{m+1}\setminus0)^r\) so that the associated \(\chi_i\)-line over \(B_m=(\mathbf P^m)^r\) is \(\mathcal O(1)\) on its \(i\)-th factor. The closed affine paving and affine trace give projective-space cohomology in even degrees only. The \(a\)-fold hyperplane intersection has positive trace \(1\) on \(\mathbf P^a\); hence the powers of its Kummer class give its basis. Products of those intersections give \[ H^*(B_m,E)=E[u_1,\ldots,u_r]/(u_1^{m+1},\ldots,u_r^{m+1}). \tag{C.5.2} \] Equivalently this follows by applying the same affine-cell induction to the product, with its coordinate hyperplane classes. Its monomials are paired with the complementary coordinate linear subvarieties; their intersection matrix is the identity. The trace and section orientations are the actual comparisons proved in Lesson 6, A.1.1.1–A.1.2.1, and Lesson 5, O–P. The restriction of (C.5.2) stabilizes in each fixed degree. This proves (C.5.1). For \(\alpha=\sum a_i\chi_i\), additivity of the Kummer connecting homomorphism on line-bundle transition units gives \(c_1(\alpha)=\sum a_i u_i\). **Lemma C.5.1.1.** If \(V=\bigoplus_{j=1}^d k_{\alpha_j}\) is a \(T\)-representation with every \(\alpha_j\ne0\) and \(d>0\), then the nonzero polynomial \[ e_T(V)=\prod_{j=1}^d\alpha_j\in S \tag{C.5.3} \] annihilates \(H_T^*(V\setminus0,K)\) and \(H_{T,c}^*(V\setminus0,K)\) for every actual equivariant bounded constructible \(K\) there. **Proof.** Over \(B_m\), the associated vector bundle is the sum of the character lines \(L_{\alpha_j}\). On its total space the \(j\)-th coordinate zero divisor has its global supported Kummer class. Lesson 6, Lemma A.1.2.1, proves that forgetting support gives \(c_1(L_{\alpha_j})\), with the section trace and all transition units fixed. Take the cup product of these supported classes. Cup product with supports lands in the intersection of their supports: this is the tensor product of the localization counits, followed by multiplication of the constant coefficient objects. Their common support is the zero section. Forgetting support gives the product of the pulled-back line classes. Restriction to the complement of the zero section is consequently zero. Thus \(e_T(V)=0\) in the equivariant constant-coefficient cohomology of \(V\setminus0\). The action of a constant-coefficient cohomology class on \(K\) is its representing map tensored with \(1_K\), followed by the unit evaluation. A zero class therefore acts by zero on both ordinary and compact images; projection formula makes this the \(S\)-action just specified. This argument applies at every finite \(B_m\) and hence in each stable degree. Finally each \(\alpha_j=\sum a_{ij}u_i\) is a nonzero linear polynomial: its integral character is nonzero and \(E\) has characteristic zero. The polynomial domain makes their product nonzero. No weight bound or stalk-surjectivity assertion has been used. ∎ Let \(X\) be a proper finite-type \(T\)-scheme with a finite closed filtration whose successive strata are \(T\)-equivariantly isomorphic to such representations \(V_\lambda\); zero-dimensional strata are allowed. Let \(P\subset X\) be the reduced closed union of the origins of those cells, and suppose these are all its geometric fixed points. We use the actual closed point immersion \(i:P\to X\), without replacing a possibly larger nonreduced fixed-point scheme by a statement about its moduli functor. **Theorem C.5.2.1.** For every actual equivariant bounded constructible \(F\), restriction induces \[ QH_T^*(X,F)\xrightarrow{\sim} \bigoplus_{x\in P}QH_T^*(x,i_x^*F). \tag{C.5.4} \] For the constant object this is an algebra isomorphism \(QH_T^*(X,E)=\prod_{x\in P}Q\). It is the localization map that gives the idempotents in Lemma C.1.1.1. **Proof.** Remove \(P\) from each term of the closed filtration. Its successive strata are the punctured \(V_\lambda\), with an empty stratum when \(V_\lambda=0\). Lemma C.5.1.1 kills the localized equivariant compact cohomology of each stratum. The closed–open compact-support triangles and exactness of \(Q\otimes_S-\) therefore kill that of \(X\setminus P\), by finite induction. Apply equivariant cohomology to \[ j_!j^*F\longrightarrow F\longrightarrow i_*i^*F \longrightarrow . \tag{C.5.5} \] At each classifying approximation the base \(B_m\) is proper, and properness of \(X\) identifies the first image with the relative compact image of its open complement. Its localization is zero by the preceding induction. The second arrow thus gives (C.5.4). The finite support \(P\) is a disjoint union of points, so its image is the displayed direct sum. For the constant object, restriction preserves units and cup products, and each point has the ring \(S\); this proves the algebra assertion. In particular a complex supported on one closed union of cells has localized module supported on the corresponding fixed points. Applying Lemma C.1.1.1 to modules with disjoint such supports and torsion-free targets gives integral Hom vanishing. The theorem supplies no assertion that rational maps lift to global maps. ∎ ### C.6. An opposite-loop chart over every parameter algebra Let \(O_R=R[[t]]\), \(F_R=R((t))\), and \[ L^{--}H(R)= \ker\bigl(H(R[t^{-1}])\xrightarrow{t^{-1}=0}H(R)\bigr). \tag{C.6.1} \] This includes negative coefficients in the nilradical of \(R\). We first work with \(H=GL_n\). The bounded lattice scheme of Lesson 2, §2, represents lattices \[ t^NO_R^n\subset L\subset t^{-N}O_R^n \] whose finite quotient \(L/t^NO_R^n\) is a locally direct summand of \(t^{-N}O_R^n/t^NO_R^n\). On its index-zero rank stratum that quotient has rank \(nN\). The nonnegative-power projection gives the map of finite vector bundles \[ \bar\pi_+:L/t^NO_R^n\longrightarrow O_R^n/t^NO_R^n. \tag{C.6.2} \] Let \(\Omega_N\) be its determinant-invertible open. On other rank strata use the empty open. **Theorem C.6.1.1.** The opens \(\Omega_N\) are compatible under increasing \(N\). Their union \(\Omega\) is the image of the open immersion \[ L^{--}GL_n\longrightarrow\operatorname{Gr}_{GL_n}, \qquad h\longmapsto hO^n. \tag{C.6.3} \] For every \(R\), a lattice in \(\Omega_N(R)\) has a unique representative \[ h=1+\sum_{j=1}^N h_jt^{-j}, \qquad h^{-1}=1+\sum_{j=1}^N k_jt^{-j}. \tag{C.6.4} \] **Proof.** For \(L\in\Omega_N(R)\), the full projection \(\pi_+:L\to O_R^n\) is an isomorphism of \(R\)-modules. Indeed an element in its kernel maps to zero under (C.6.2), hence belongs to \(t^NO_R^n\), where projection is the identity. For a prescribed power series \(f\), lift its class modulo \(t^N\) through (C.6.2), and correct the remaining positive tail by an element of \(t^NO_R^n\subset L\). This proves surjectivity. Let \(l_i\) be the unique inverse images of the standard constant vectors. Their columns give a matrix \(h=1+\sum_{j=1}^N h_jt^{-j}\). All its columns belong to \(L\), so \(hO_R^n\subset L\). We must prove invertibility over \(R[t^{-1}]\); a field-point factorization alone would not suffice. At every prime of \(R\), pass to its residue field \(\kappa\). The bounded lattice construction commutes with this passage, as does the inverse of (C.6.2). The matrix \(\bar h\) has determinant a polynomial in \(t^{-1}\) with constant coefficient \(1\), so it is invertible over \(\kappa((t))\). The inclusion \(\bar hO_\kappa^n\subset L_\kappa\), whose latter index is zero, gives \[ \operatorname{val}_t(\det\bar h)\ge0. \] This is the field lattice index calculation proved by elementary divisors in Lesson 1, Lemma 2.0: the difference is the nonnegative length of their quotient. On the other hand a polynomial in \(t^{-1}\) with constant coefficient \(1\) has valuation at most zero, and valuation zero precisely when all its other coefficients vanish. Therefore \(\det\bar h=1\). Write \(\det h=1+b\), with \(b\) a negative polynomial. Every coefficient of \(b\) vanishes at every prime, hence is nilpotent. To see the last assertion directly, a nonnilpotent element would remain nonzero in its localization and a prime of that nonzero localization would avoid it. There are finitely many coefficients. If their nilpotence exponents are \(a_1,\ldots,a_s\), their ideal has zero power \(1+\sum(a_i-1)\), by the pigeonhole principle for its monomials. Thus \(b\) is nilpotent and \[ (1+b)^{-1}=1-b+b^2-\cdots \] is a finite negative polynomial. The adjugate formula now gives \(h^{-1}\in GL_n(R[t^{-1}])\), with value \(1\) at infinity. For a finite negative matrix \(a\) define \(T_a(f)=\pi_+(af)\) on \(O_R^n\). Multiplication by another negative matrix keeps every discarded strictly negative term strictly negative. Consequently \[ T_aT_b=T_{ab}. \tag{C.6.5} \] In particular \(T_h^{-1}=T_{h^{-1}}\). Thus \(\pi_+:hO_R^n\to O_R^n\) is an isomorphism. Since it is also an isomorphism on \(L\), their inclusion is equality. Now \(t^NO_R^n\subset hO_R^n\) implies \(h^{-1}t^NO_R^n\subset O_R^n\); every entry of the negative polynomial \(h^{-1}\) has pole order at most \(N\). This proves (C.6.4). Conversely a negative loop with its inverse bounded by \(N\) defines a bounded lattice. Formula (C.6.5) proves that its projection is an isomorphism, hence that (C.6.2) is an isomorphism. Its columns are recovered by the prescribed unique constant inverse images, proving uniqueness. The constructions commute with every ring map. At a larger bound, (C.6.2) adds the same positive-power quotient to both sides, with identity as its induced map. The determinant-invertible conditions are therefore compatible. The finite negative matrices and their inverses in (C.6.4) are represented by affine schemes with their polynomial inverse equations. The preceding natural inverses identify them with \(\Omega_N\). Their union is exactly the ind-open \(\Omega\), which proves the asserted open immersion as a statement about the entire functor. ∎ For a connected reductive \(G/k\), choose the faithful closed embedding \(G\hookrightarrow H=GL(V)\) of Lesson 2, Lemma 7.2. That lemma proves that the actual fppf quotient \(H/G\) is an open subscheme of an affine finite-type scheme \(A\); it keeps the original \(G\), including its centre. **Theorem C.6.2.1.** The natural map \[ L^{--}G\longrightarrow\operatorname{Gr}_G \tag{C.6.6} \] is an open immersion. Its image is the inverse image of \(\Omega\) under \(\operatorname{Gr}_G\to\operatorname{Gr}_H\). **Proof.** On an fppf cover, a point of that inverse image is represented by \(g\in G(F_R)\). Theorem C.6.1.1 supplies the unique matrix factorization \[ g=hk,\qquad h\in L^{--}H(R),\quad k\in H(O_R). \tag{C.6.7} \] The equality \(g\in G(F_R)\) gives equality of right cosets \(kG=h^{-1}G\) in \((H/G)(F_R)\). Compose their maps with the open immersion \(H/G\subset A\). A function of the affine hull evaluated on \(h^{-1}G\) lies in \(R[t^{-1}]\); evaluated on \(kG\) it lies in \(R[[t]]\). Their common value lies in \[ R[t^{-1}]\cap R[[t]]=R\quad\text{inside }R((t)), \tag{C.6.8} \] an equality of coefficient series for every ring, including nonreduced rings. Evaluation at \(t^{-1}=0\) is the identity coset because \(h^{-1}(\infty)=1\). Hence every affine-hull coordinate is the corresponding constant identity-coset coordinate. The immersion into \(A\) makes the entire quotient map that constant map. The fiber of the quotient torsor over the identity coset is precisely \(G\); therefore \(h^{-1}\in G(R[t^{-1}])\), and then \(k=h^{-1}g\in G(O_R)\). The latter can also be tested on the closed equations of \(G\), since \(O_R\to F_R\) is injective. No passage from a formal section to a polynomial section has been asserted without proof: (C.6.8) proves it coordinate by coordinate in the fixed affine hull. Two negative representatives of the same Grassmannian point differ by a positive loop. In the matrix embedding their ratio belongs to both \(H(R[t^{-1}])\) and \(H(R[[t]])\), so is constant, and its value at infinity is \(1\). They are equal. This uniqueness makes the local negative representatives descend: their finitely many matrix coefficients agree on overlaps, their inverse equations and the equations of \(G\) descend, and the pole bound can be taken on each bounded lattice stage. Thus (C.6.6) identifies the full negative-loop functor with the specified ind-open inverse image. ∎ ### C.7. The exact contracting action on a Schubert support For \(\lambda\in X_*(T)\), translate the preceding chart: \[ U_\lambda=t^\lambda L^{--}G\,K/K,\qquad K=L^+G. \tag{C.7.1} \] It is open and contains \(t^\lambda K\). Its negative coordinates carry the monoid action \[ a_s(t^\lambda h(t^{-1})K) =t^\lambda h(st^{-1})K,\qquad s\in\mathbf A^1. \tag{C.7.2} \] **Proposition C.7.1.1.** Formula (C.7.2) is an actual \(\mathbf A^1\)-action on \(U_\lambda\), commuting with \(T\), whose zero endomorphism is the constant map to \(t^\lambda K\). On \(s\ne0\) it agrees with \[ a_s(gK)=\lambda(s)\,g(s^{-1}t)K. \tag{C.7.3} \] It preserves \(U_\lambda\cap X\) for every closed finite union \(X\) of Iwahori cells stable under \(T\) and rotation, with its full scheme structure. In particular this holds for finite spherical Schubert supports. **Proof.** In a bounded chart, both \(h\) and \(h^{-1}\) are polynomial matrices in \(t^{-1}\) with value \(1\) at infinity. Substitution \(t^{-1}\mapsto st^{-1}\) preserves the inverse equations, the equations of \(G\), and the pole bounds. Thus it defines a regular map over every parameter algebra. Double substitution is multiplication of the parameters and substitution at zero is the identity matrix. This proves the monoid laws and the stated zero map. The left \(T\)-action in these coordinates is conjugation on \(h\), because \(T\) commutes with \(t^\lambda\) and its constant loops lie in \(K\). It commutes with substitution. Finally \[ \lambda(s)(s^{-1}t)^\lambda=t^\lambda \] for invertible \(s\), giving (C.7.3) exactly. There is no missing cocharacter or rotation sign. An Iwahori is preserved by rotation and normalized by \(T\). Their actions therefore preserve each labelled cell and any closed union. To extend this assertion from invertible \(s\) to the whole monoid, test its closed defining equations on an affine chart of \(\mathbf A^1\times(U_\lambda\cap X)\). They vanish after inverting \(s\). Multiplication by \(s\) is injective on \(R[s]\) for every coordinate ring \(R\), so they vanish already before that localization. This proves preservation with nilpotents retained. For the raw spherical IC, its dense spherical-orbit system is the constant rank-one system with its prescribed shift. The action (C.7.3) preserves that orbit and this system. Smooth normalized pullback and the no-boundary characterization of intermediate extension extend this isomorphism uniquely to the IC. Restriction to the identity parameter is the identity, and the two composites on \(\mathbb G_m^2\) agree because they agree on the dense orbit; uniqueness proves the unit and cocycle. These are the genuine heart action comparisons of Lesson 6, B.2.3.1, and preserve the coefficient and Tate lines. The same actual action is inherited by restriction to any stable closed cell union and then to its contracting chart. ∎ ![The determinant-invertible lattice projection constructs an actual negative-loop chart and its contraction while retaining nilpotent parameters](assets/negative-loop-chart.png) Theorems C.6.1.1–C.6.2.1 and Proposition C.7.1.1 give the open condition, normalized inverse, affine-hull coefficient intersection and exact cocharacter/rotation formula. Exercise C.11.1 displays the nonreduced torus chart. Editable SVG source. ### C.8. Contraction gives the actual restriction and counit Let \(f:Y\to B\) be separated of finite type, with a closed section \(z:B\to Y\). Suppose a monoid action \(a:\mathbf A^1_B\times_B Y\to Y\) is over \(B\), is the identity at \(1\), restricts to a group action on \(\mathbb G_m\), and has \(a_0=zf\). Write \(U=Y\setminus z(B)\). **Theorem C.8.1.1.** For an actual naively \(\mathbb G_m\)-equivariant bounded constructible \(F\), the restriction unit and exceptional counit induce \[ Rf_*F\xrightarrow{\sim}z^*F,\qquad z^!F\xrightarrow{\sim}Rf_!F. \tag{C.8.1} \] They also hold for finite cones, shifts and direct summands of such objects. In particular the first is the fixed-stalk comparison on a contracting chart, including its finite torus classifying approximations. **Proof.** We give the action-graph argument for this precise hypothesis, rather than treating the more restricted projective-bundle statement of Lesson 5, Q.2.2.1, as a general contraction theorem. Use that lesson's actual ordinary smooth comparison Q.1.1.1 and affine-line propagation Q.1.3.1–Q.1.3.2, whose proofs apply to the trivial-action base \(B\). Let \(j:U\to Y\), put \(F_0=j_!j^*F\), and put \(C=Rf_*F_0\). Set \[ Q_Y=1_{\mathbf A^1}\times f,\quad q:\mathbf A^1_B\to B,\quad r(s,y)=(s,a_s(y)),\quad D=R(Q_Y)_*r^*p_Y^*F_0. \tag{C.8.2} \] The maps \(Q_Yr\) and \(Q_Y\) are equal. Over \(\mathbb G_m\), \(r\) is an isomorphism, so ordinary smooth comparison and equivariance give \(D|_{\mathbb G_m}=q^*C|_{\mathbb G_m}\). The inverse image of \(U\) under the action is exactly \(\mathbb G_m\times U\): invertible parameters act by automorphisms fixing the section, whereas zero maps into the section. Pullback of open extension by zero and the chosen equivariance isomorphism consequently give \[ r^*p_Y^*F_0 \simeq j_{Y,!}j_Y^*p_Y^*F_0, \tag{C.8.3} \] where \(j_Y:\mathbb G_m\times Y\to\mathbf A^1\times Y\). Ordinary image composition and the affine-line calculation Q.1.3.1 imply \[ Rq_*D =Rf_*R(p_Y)_*j_{Y,!}j_Y^*p_Y^*F_0=0. \tag{C.8.4} \] Q.1.3.2, applied with the already established constant form on \(\mathbb G_m\), now gives \(D|_0=0\). Localization identifies \(D\) with extension by zero of its restriction to \(\mathbb G_m\). This inference uses the affine-line calculation, not ordinary nonproper base change at zero. The ordinary adjunction unit for \(r\), followed by image composition \(Q_Yr=Q_Y\), gives \[ \beta:q^*C\longrightarrow D. \tag{C.8.5} \] It is invertible over \(\mathbb G_m\). Extend its inverse by zero to obtain \(\gamma:D\to q^*C\). The endomorphism \(\gamma\beta\) of \(q^*C\) is the identity at \(1\) and zero at \(0\). Affine-line propagation Q.1.3.1 therefore gives \(C=0\). Apply \(Rf_*\) to the support triangle \[ F_0\longrightarrow F\longrightarrow z_*z^*F \longrightarrow . \] Its specified second arrow is the restriction map, so this proves the first isomorphism of (C.8.1). Verdier duality preserves naive equivariance: on the smooth group parameter its two equal orientation lines cancel, as proved in Q.1.4.1. Apply the first isomorphism to \(D_YF\) and dualize. The actual dual exchanges and supported adjunction of Lesson 5, P.3, identify the resulting map as the section counit \(z^!F\to Rf_!F\), with its original trace normalization. Finally the two arrows are natural transformations of exact functors. Their isomorphism loci are closed under finite cones, shifts and direct summands. ∎ ### C.9. The precise weight implication for stalk surjectivity This section is a conditional weight lemma. Its hypothesis is the compact-support upper weight estimate: \[ \text{if every }H^b(K_{\bar y})\text{ has weights }\le v+b, \quad H_c^n(Y_{\bar k},K)\text{ has weights }\le v+n. \tag{W\_c} \] Here \(Y\) is finite type over a finite field, \(K\) is a mixed bounded constructible complex, and the assertion is required also on the finite classifying approximations being used. This hypothesis is separate from parity and from the statements proved in §§1–8. **Lemma C.9.1.1.** Assume \((W_c)\). Let \(X\) be proper over \(\mathbf F_q\), with a split torus action. Let \(P\) be an actual equivariant mixed complex, let \(x\) be a rational fixed point, and suppose: - the stalks of \(D_XP\) have upper complex weight \(-w\); - every \(H^b(P_x)\) is pure of weight \(w+b\); - \(x\) has an open neighborhood with an actual contracting action commuting with \(T\), and \(P\) has the equivariance of Theorem C.8.1.1. Then the actual restriction \(H_T^n(X,P)\to H_T^n(x,P_x)\) is surjective in every degree. The assertion keeps Frobenius and the coefficient lines, and does not require Frobenius semisimplicity. **Proof.** Work at \(B_m=(\mathbf P^m)^r\). Write \(X_m=X\times^T P_m\), \(U_m\) for the associated contracting neighborhood, \(Z_m=X_m\setminus U_m\), and \(P^{(m)}\) for the descended complex. Theorem C.8.1.1 over the base \(B_m\) identifies the ordinary image from \(U_m\) with that of the fixed section. The cohomology sheaves on that section are the constant \(H^b(P_x)\)-systems: the connected torus acts trivially on each ordinary cohomology system of a point, since its action is an isomorphism of constant systems on connected \(T\) and is the identity at \(1\). The associated systems consequently descend with identity transitions. By (C.5.2), their base degree \(a\) has weight \(a\). The section spectral sequence thus has total-degree-\(n\) terms of weight \(w+n\). Its differentials raise total degree by one and commute with Frobenius, so are zero between these disjoint weights. Its finite filtration makes degree \(n\) pure of weight \(w+n\), without asserting that the filtration splits as a Frobenius representation. The descended dual \(D_{X_m}P^{(m)}\) has upper complex weight \(-w\). Locally it is \(D_XP\) times the smooth base dualizing complex \(E(\dim B_m)[2\dim B_m]\); the shift and Tate line cancel in the complex weight convention. Restriction to \(Z_m\) preserves the stated stalk upper bound. Actual supported duality gives \[ H_{Z_m}^{n+1}(X_m,P^{(m)}) = H_c^{-n-1}(Z_m,i_{Z_m}^*D_{X_m}P^{(m)})^\vee. \tag{C.9.1} \] Hypothesis \((W_c)\) makes the right side have weights at least \(w+n+1\). The localization boundary from \(H^n(U_m,P^{(m)})\), pure of weight \(w+n\), is therefore zero. A Frobenius-linear map between finite-dimensional representations with disjoint eigenvalue weights is zero even for nontrivial Jordan blocks: over a splitting extension its source and target primary eigenvalue factors are disjoint, and the coprime minimal polynomials force the map to vanish. The localization long exact sequence now gives surjectivity from \(X_m\) to \(U_m\), hence to the section. Passing to a stable approximation in each degree proves the assertion. Every used arrow was restriction, duality, or a localization boundary, so the result concerns the actual restriction map. ∎ ### C.10. A cell criterion for the actual Ext-to-Hom isomorphism Let \(X\) have the proper finite linear \(T\)-cell filtration of §5. Let \(A=H_T^*(X,E)\). For actual equivariant bounded constructible objects \(P,Q\), assume: 1. On every cell \(C\), the cohomology sheaves of \(i_C^*P\) and \(i_C^!Q\) are geometrically constant and are concentrated, respectively, in two fixed parities \(\eta_P,\eta_Q\). 2. For every cell origin \(x\), the actual map \(H_T^*(X,P)\to H_T^*(x,P_x)\) is surjective. 3. Restriction from each cell to its origin identifies \(H_T^*(C,i_C^*P)\) with \(H_T^*(x,P_x)\). The third condition also follows from the linear cell contraction whenever that contraction has the indicated actual equivariance. The theorem does not assert that stalk parity supplies the second condition. **Theorem C.10.1.1.** Under these hypotheses, equivariant cohomology induces an isomorphism \[ h:\operatorname{Ext}_{D_T(X)}^*(P,Q) \xrightarrow{\sim} \operatorname{Hom}_A^*(H_T^*(X,P),H_T^*(X,Q)). \tag{C.10.1} \] For a family satisfying these conditions for its pairs, the isomorphisms preserve identities and actual Yoneda composition. **Proof.** We prove a strengthened statement simultaneously for every term \(X'\) of the closed filtration: the comparison for \((i_{X'}^*P,i_{X'}^!Q)\) is an isomorphism with the original \(A\)-linear Hom, and its Ext module is finite free over \(S\), concentrated in parity \(\eta_Q-\eta_P\). The empty term starts the induction. For a last open cell \(u:C\to X'\) and closed complement \(v:Z\to X'\), set \(E'=i_{X'}^*P\) and \(F'=i_{X'}^!Q\). The smaller pair is exactly \((i_Z^*P,i_Z^!Q)\), by composition of the respective restrictions. Its cell hypotheses are inherited. The original surjection to any origin in \(Z\) factors through \(H_T^*(Z,i_Z^*P)\), so the recursive source has the required origin surjectivity. Every module action throughout this strengthened induction is through the original \(A\). First record the cell calculation. A geometrically constant equivariant cohomology system on \(C\) is the constant coefficient system with its vector space: connected \(T\) gives identity on its cohomology, as in C.9. Equivariant cohomology of the constant object on the representation \(C\) is \(S\), and its compact cohomology is \(S(-d)[-2d]\), by the vector-bundle contraction and smooth trace. A complex whose cohomology lies in one parity splits in the geometric equivariant derived category into these shifted constant systems. Indeed each Postnikov attaching obstruction is an odd Ext between constant systems, whereas \(H_T^*(C,E)=S\) has no odd degrees. This proves the assertion inductively through its finite cohomology range. No splitting of Frobenius representations is needed. Thus ordinary and compact cell cohomology are finite free \(S\)-modules of their specified parity. Between the two cell complexes, the actual Ext-to-cohomology map is the usual matrix map with entries in \(S\); it is an isomorphism. To check this without depending on a splitting choice, check constant shifted summands: cup multiplication sends an Ext entry \(a\in S\) to multiplication by \(a\). Finite direct sums give the matrix assertion, and naturality makes the conclusion independent of the chosen summand presentations. The action of \(A\) on these modules factors through \(H_T^*(C,E)=S\). That map is onto, since the base inclusion \(S\to A\) restricts to the identity. Hence their \(A\)-linear Hom equals their \(S\)-linear Hom. Next use compact open–closed induction for the *restrictions of \(P\). All terms have parity \(\eta_P\), so the connecting maps of degree one are zero. This proves finite freeness and the exact first filtration of C.2, \[ 0\to H_{T,c}^*(C,u^*E')\to H_T^*(X',E') \to H_T^*(Z,v^*E')\to0. \tag{C.10.2} \] The properness of \(X'\) identifies ordinary and compact images here. Freeness follows by lifting a basis of the free cell or quotient module to split each \(S\)-module extension. Similarly, closed-costalk induction for \(F'\) uses \(v^!F'\), whose costalk parity is inherited by composition of exceptional restrictions. The open cell terms have parity \(\eta_Q\). It proves freeness and \[ 0\to H_T^*(Z,v^!F')\to H_T^*(X',F') \to H_T^*(C,u^*F')\to0. \tag{C.10.3} \] Hypotheses 2–3 supply surjectivity \(H_T^*(X',E')\to H_T^*(C,u^*E')\). In fact the surjection from \(X\) to the cell origin factors through this map and then through the isomorphism in hypothesis 3. The supported triangle and surjectivity in every degree consequently give \[ 0\to H_T^*(Z,v^!E')\to H_T^*(X',E') \to H_T^*(C,u^*E')\to0. \tag{C.10.4} \] This step uses no unproved exceptional-parity assertion for \(i_{X'}^*P\). The three sequences (C.10.2)–(C.10.4) are exactly those of C.2. Identify their localized supports separately. For the compact open and ordinary open modules \(P_c,P_U,F_U\), the cup action factors through \[ A\longrightarrow H_T^*(C,E)=S \longrightarrow H_T^*(x,E)=S, \tag{C.10.6} \] where \(x\) is the last cell origin. After localization this is the character of that origin in \(\prod_{y\in P}Q\), so these modules are supported there, or are zero. For \(P_!,P_Z,F_Z\), apply Theorem C.5.2.1 to their actual closed extensions from \(Z\) into the proper original \(X\); their localized supports lie among the complementary origins. This does not assert vanishing boundary stalks for an ordinary open direct image. The target modules in C.2 are free by (C.10.3) and the cell calculation. Lemmas C.1–C.2 therefore give the actual left-exact Hom row. Apply \(\operatorname{RHom}(-,F')\) to \(u_!u^*E'\to E'\to v_*v^*E'\to\). Closed and open adjunction identify its two outer terms with \[ \operatorname{Ext}_{D_T(Z)}^*(v^*E',v^!F'), \qquad \operatorname{Ext}_{D_T(C)}^*(u^*E',u^*F'). \tag{C.10.5} \] By the strengthened induction and the cell calculation they are finite free \(S\)-modules, concentrated in parity \(\eta_Q-\eta_P\). The degree-one connecting maps are zero, giving the genuine short exact upper Ext row. Its free cell quotient has a homogeneous \(S\)-basis; lifting that basis splits the sequence as \(S\)-modules. The middle Ext module is therefore finite free with the same parity, proving this part of the strengthened assertion. Freeness has been proved from the Ext sequence, rather than inferred from a Hom submodule of free modules. Its maps and the lower Hom maps form the commutative diagram of C.3: the closed arrow is the counit on the target and unit on the source, and the open arrow is actual restriction. Applying cohomology gives precisely the factorization defining \(\alpha,\beta\) in C.2. The outer comparison maps are isomorphisms by the induction and cell calculation. Lemma C.3.1.1 makes the middle comparison an isomorphism, and also proves the lower row surjective. The empty union starts the induction. At \(X'=X\) this is (C.10.1). Finally \(h\) is induced by the actual cohomology functor. Its map on a composite of derived morphisms is the composite of its maps, with the ordinary grading and shift convention; it sends an identity to an identity. Thus the theorem preserves Yoneda products, rather than only computing their dimensions. ∎ ![Euler localization kills cross-Hom terms, and the actual Ext row lifts every integral module morphism under the cell criterion](assets/localized-support-lifting.png) Lemma C.5.1.1 and Theorem C.5.2.1 prove the displayed Euler annihilation and localized algebra. Lemmas C.1–C.3 give the exact Hom row and integral lifting. Theorem C.10.1.1 verifies those rows and composition under its stated fixed-point restriction hypotheses. Localization is used to prove vanishing; the upper Ext surjection supplies the lifts. Editable SVG source. ### C.11. Three solved exercises **Exercise C.11.1 (basic).** Describe \(\Omega_1\) for \(GL_1\), and compute its contraction over \(R=k[\epsilon]/\epsilon^2\). **Solution.** The lattice quotient is the line generated by \(1+at^{-1}\) in \(t^{-1}O/tO\). Its positive projection is the isomorphism sending that vector to \(1\). Multiplication by \(t\) preserves the line precisely when \[ a=a(1+at^{-1}),\qquad a^2=0. \] Thus the chart is \(\operatorname{Spec}k[a]/a^2\), with normalized negative loop \(1+at^{-1}\) and inverse \(1-at^{-1}\). Over the stated dual numbers, \(a=b\epsilon\) with arbitrary \(b\in k\). The monoid sends \(a\) to \(sa\). Its zero map is the identity lattice, but the chart itself has its nonzero tangent direction. This verifies both the sign and retention of the nilpotent parameter in C.6–C.7. **Exercise C.11.2 (intermediate).** For \(T=\mathbb G_m\), compute the annihilating Euler polynomial for \(V=k_\chi\oplus k_{2\chi}\). Compute the localized support of a closed-extension point module inside a two-fixed-point proper cell scheme. **Solution.** With \(u=c_1(\chi)\), the character-line calculation gives \(e_T(V)=u(2u)=2u^2\), of degree \(4\), with coefficient twist \(E(2)\). It is nonzero over the characteristic-zero field \(E\), also when the geometric characteristic is two; the integral character \(2\chi\) has not become a zero coefficient in \(E\). Lemma C.5.1.1 kills both images from \(V\setminus0\) after inverting \(u\). In the ring \(A=S[h]/h(h-u)\) of §4, the point modules have localized supports \(e_0=1-h/u\) and \(e_\infty=h/u\), respectively. Each idempotent acts as identity on its own point module and zero on the other. Thus C.1 gives their integral cross-Hom vanishing. At \(u=0\) their specialized-module Hom is instead \(E\), exactly as computed in §4; this does not contradict localization or prove that Hom commutes with specialization. **Exercise C.11.3 (advanced).** Let \(X=\mathbf P^1\), with affine open \(j:C=\mathbf A^1_\alpha\to X\) and its nonzero \(T\)-weight \(\alpha\). Show that \(F=j_!E_C\) has even ordinary cell restrictions, while \(H_T^*(X,F)\to H_T^*(C,E)\) is not onto. Explain the relevance to recursive closed restriction of an IC. **Solution.** Its open restriction is \(E_C\) in degree zero and its closed ordinary restriction is zero. Both satisfy even ordinary parity. Properness gives \[ H_T^*(X,F)=H_{T,c}^*(C,E)=S(-1)[-2], \qquad H_T^*(C,E)=S. \] The restriction map is multiplication by the character Euler class \[ S(-1)[-2]\xrightarrow{\alpha}S. \tag{C.11.1} \] To determine this actual arrow, the zero-section orientation maps \(S(-1)[-2]\) to the compact cohomology of the line. Its composite with the line trace is the identity by Lesson 6 A.1.1.1, hence is the compact orientation isomorphism. Forgetting compact support and restricting to the zero section sends that class to \(c_1(k_\alpha)=\alpha\), by A.1.2.1. This proves (C.11.1) with its trace normalization. The map is injective and its cokernel is \(S/(\alpha)\), so it is not onto. The localization boundary places this cokernel in odd supported cohomology at infinity. Thus ordinary parity does not imply exceptional parity or open-restriction surjectivity. For a recursively restricted source \(i_{X'}^*P\), its ordinary cell restrictions are inherited from \(P\), but its exceptional restriction to a smaller closed union is \(v^!i_{X'}^*P\); it need not be the exceptional restriction of \(P\) to that union in the original space. The common-parity theorem for the original IC therefore does not prove the missing condition 2 of C.10. The weight lemma C.9 supplies a distinct sufficient implication only under its explicitly stated hypothesis \((W_c)\). ### C.12. Restriction from the whole Grassmannian The exact earlier proof inputs are Lesson 1, Theorem 2.C, Lesson 2, Theorem 7.1 and Lemmas 7.10-7.12, Lesson 4, Theorem 3.5 and section 5, and Lesson 5, Lemma Q.1.2.1 and the closed-open triangles of Appendices O-P. They supply the root-coordinate recursion, full bounded stages, closure and component classification, and normalized affine-line trace. The restriction and inverse-limit argument are proved below. Continue over an algebraically closed field \(k\), with the original connected reductive group \(G\), maximal torus \(T\), and Borel \(B\). The coefficient field \(E\) has characteristic zero and residue characteristic different from that of \(k\). Put \[ \Lambda=X_*(T),\qquad Q^\vee=\mathbb Z\Phi^\vee,\qquad \Pi=\Lambda/Q^\vee,\qquad S=H_T^*(\mathrm{pt},E). \] The quotient \(\Pi\) includes the free directions contributed by the centre. We use the actual full bounded stages \(Y_N\) of Lesson 2, Theorem 7.1 and §7.6, with their original scheme structures. Ordinary cohomology on this countable ind-presentation means derived compatible sections on those stages. We will compute that complex explicitly below. In particular, we will prove that its restriction to a finite support is onto; this assertion is not built into the definition. #### C.12.1. Cells of bounded dimension in one component Let \(I\subset G(k[[t]])\) be the inverse image of \(B\) under evaluation at zero. For a root \(\alpha\), set \(r(\alpha)=0\) for \(\alpha>0\), and \(r(\alpha)=1\) for \(\alpha<0\). **Lemma C.12.1.1.** The \(I\)-orbit \(C_\lambda=It^\lambda K/K\), for \(\lambda\in\Lambda\), is a \(T\)-linear affine cell of dimension \[ d_I(\lambda)= \sum_{\alpha\in\Phi} \max\bigl(0,\langle\alpha,\lambda\rangle-r(\alpha)\bigr). \tag{C.12.1} \] It is contained in the actual negative-loop open \(U_\lambda\) of C.7. Its contraction there acts on the coordinate indexed by \((\alpha,n)\) as \[ c_{\alpha,n}\longmapsto s^{\langle\alpha,\lambda\rangle-n}c_{\alpha,n}, \qquad r(\alpha)\le n<\langle\alpha,\lambda\rangle. \tag{C.12.2} \] In each fixed class \(\kappa\in\Pi\), only finitely many such cells have \(d_I(\lambda)\le D\), for every integer \(D\). **Proof.** The ordered root-coordinate and height argument of Lesson 1, Theorem 2.C, applies to \(I/(I\cap t^\lambda Kt^{-\lambda})\) as follows. In the big-cell coordinates of \(I\), conjugation by \(t^{-\lambda}\) changes the \(\alpha\)-coordinate exponent \(n\) to \(n-\langle\alpha,\lambda\rangle\). The integral big-cell test in that theorem shows that the conjugated element is integral exactly when each root coordinate has valuation at least \(\langle\alpha,\lambda\rangle\). The torus arc is unchanged. Thus the stabilizer has root thresholds \[ \max\bigl(r(\alpha),\langle\alpha,\lambda\rangle\bigr), \] and contains every torus arc. Here is the coefficient split needed for this quotient. Place first the finitely many coefficients with \(r(\alpha)\le n<\langle\alpha,\lambda\rangle\), and then the common coefficients at or above both thresholds. Each set is closed under addition whenever the sum is a real affine root. Two of the first set cannot sum to a positive imaginary degree: for opposite roots their exponents have sum at least one, while the two strict upper inequalities would make that sum negative. Common coefficients and their opposite-root torus corrections stay in the common set. Use the positive integral affine height from the proof of Theorem 2.C to reorder these products. A commutator changes only coefficients of larger height. Opposite-root interchanges have denominator \(1+uv\) with \(uv\in tR[[t]]\); its inverse is the convergent geometric series, and its torus correction is common. At each height recover the next coefficient uniquely by subtracting the already known corrections. At any bounded height these operations and their inverses are polynomial in finitely many coefficients. This proves, over every test algebra \(R\), the unique factorization into the displayed first coefficients and a stabilizer element. The quotient is their affine space, giving (C.12.1). The refined Cartan decomposition in the same theorem identifies these cells, without repetitions, as the cells of the Grassmannian. A representative of this affine space is an ordered product of \(u_\alpha(c_{\alpha,n}t^n)t^\lambda\). Move \(t^\lambda\) to the left. All remaining factors have exponent \(n-\langle\alpha,\lambda\rangle<0\), are polynomial in \(t^{-1}\), and have value one at infinity. Their product is an actual element of \(L^{--}G(R)\). This proves the claimed inclusion in \(U_\lambda\), including its scheme structure. Substitution \(t^{-1}\mapsto st^{-1}\) gives (C.12.2); every displayed exponent of \(s\) is positive. Constant \(T\) conjugation multiplies this coordinate by the character \(\alpha\). Therefore these are linear \(T\)-coordinates, and the two actions commute. For a positive root put \(m=\langle\alpha,\lambda\rangle\). Its pair with \(-\alpha\) contributes \[ \max(0,m)+\max(0,-m-1) = \begin{cases}m,&m\ge0,\\-m-1,&m<0.\end{cases} \] Consequently \[ |\langle\alpha,\lambda\rangle|\le d_I(\lambda)+1. \tag{C.12.3} \] Fix a representative \(\lambda_0\) of \(\kappa\). Then \(\lambda-\lambda_0\in Q^\vee\). The simple-root pairings give an injective real linear map on the real span of the coroots: its matrix in the simple-coroot basis is the invertible finite Cartan matrix, as proved by the positive definite root-space form in Lesson 1, Theorem 2.C. Inequality (C.12.3) bounds all these pairings, so it bounds \(\lambda-\lambda_0\) in that real vector space. A bounded set meets the free lattice \(Q^\vee\) in finitely many elements, by expressing it in an integral basis. This proves finiteness. If \(G\) is a torus, \(Q^\vee=0\) and each class has exactly one cell; the same argument includes this case. ∎ #### C.12.2. The full stages and reduced Schubert exhaustion Write \(\operatorname{Gr}_\kappa\) for the component with class \(\kappa\). The classification and local constancy of this class were proved in Lesson 4, §5, for the original group and in every geometric characteristic. **Lemma C.12.2.1.** Each component has a nested sequence of projective reduced Schubert varieties \(Z_j\) such that every reduced bounded stage \(Y_{N,\kappa,\mathrm{red}}\) is contained in some \(Z_j\), and every \(Z_j\) is contained in some full bounded stage. The actual restrictions \[ R\Gamma_T(Y_{N,\kappa},E)\longrightarrow R\Gamma_T(Y_{N,\kappa,\mathrm{red}},E) \tag{C.12.4} \] are multiplicative quasi-isomorphisms, compatible with all transitions. **Proof.** Choose a dominant \(\lambda_0\) in the class \(\kappa\); reflecting any representative to a dominant one preserves its class, since \(s_\alpha\lambda-\lambda=-\langle\alpha,\lambda\rangle\alpha^\vee\). Put \[ \eta=\sum_{\alpha>0}\alpha^\vee,\qquad \lambda_j=\lambda_0+j\eta,\qquad Z_j=\overline{\operatorname{Gr}^{\lambda_j}}_{\mathrm{red}}. \] Each simple-coroot coefficient of \(\eta\) is positive: the positive coroots have nonnegative coefficients, and the corresponding simple coroot occurs among them. A simple reflection permutes all positive coroots except its own simple coroot, which it makes negative. Hence \(s_i\eta=\eta-2\alpha_i^\vee\), so \(\langle\alpha_i,\eta\rangle=2\). Thus every \(\lambda_j\) is dominant and \(\lambda_j\le\lambda_{j+1}\). Lesson 4, Theorem 3.5 proves that the \(Z_j\) are nested and projective. If \(\mu\) is any dominant coweight of class \(\kappa\), then \(\lambda_0-\mu\) has integral simple-coroot coefficients. Adding \(j\eta\), whose coefficients are all positive, makes them all nonnegative for sufficiently large \(j\). Thus \(\mu\le\lambda_j\). For a torus the class has only \(\mu=\lambda_0\), and the sequence is constant. We now check the full bounded stages. They are \(K\)-stable: an integral matrix and its inverse preserve both bounding lattices in the faithful representation used in Lesson 2. By its Lemma 7.10 only finitely many dominant Cartan coweights occur in \(Y_N\). A Cartan orbit which occurs is wholly contained there, and so is its reduced closure, because \(Y_N\) is a closed stage. Their finite union, with reduced structure, equals \(Y_{N,\mathrm{red}}\). Indeed their geometric points cover it by Cartan decomposition, over every algebraically closed extension of \(k\). Equality of closed supports in a common bounded stage gives equality of their radical ideals, hence equality of the reduced closed subschemes. In the component \(\kappa\), choose \(j\) dominating all these finitely many coweights. The closure theorem gives the required inclusion into \(Z_j\). Conversely every finite-type Schubert variety has a finite lattice bound, as in the projectivity proof of Lesson 4, and therefore lies in a full \(Y_N\). This is an exhaustion statement about the reductions. For the actual cohomology comparison let \(i:Y_{\mathrm{red}}\to Y\) be any of the closed reductions. Its open complement is empty. The closed–open triangle proved in Lesson 5, Appendices O–P, is therefore the actual unit isomorphism \[ E_Y\xrightarrow{\ \sim\ }i_*i^*E_Y=i_*E_{Y_{\mathrm{red}}}. \] Apply equivariant global sections. This proves (C.12.4). The map is the pullback of constants, so it preserves the unit, cup product, and pullbacks along stage inclusions. The reduction is \(T\)-stable: a product of a reduced \(k\)-scheme with the smooth split torus is reduced, so its action factors through the reduction. The same closed–open argument applies on each finite Borel approximation and is compatible with those approximations. No nilpotent Grassmannian family has been removed from the moduli construction. For example, \(1+\epsilon t^{-1}\) in the torus chart of C.11.1 does not factor through its reduced point. The comparison just proved identifies constant cohomology through an actual closed restriction. It does not identify the two functors of points. ∎ #### C.12.3. Surjective restrictions and the inverse limit **Theorem C.12.3.1.** Let \(X\) be a proper closed finite union of Iwahori cells in \(\operatorname{Gr}_G\), with reduced structure, or a \(T\)-stable closed nilpotent thickening with that reduction. Then the actual restriction of graded rings \[ H_T^*(\operatorname{Gr}_G,E)\longrightarrow H_T^*(X,E) \tag{C.12.5} \] is onto. In a single component, each fixed cohomological degree stabilizes along the sequence of Lemma C.12.2.1 and is finite dimensional over \(E\). No stabilization across all components is asserted. **Proof.** First let \(X\subset Y\) be two projective reduced finite cell unions, and set \(U=Y\setminus X\). Their cells admit filtrations by closed unions with one affine cell at each step. To see this, the closure of an \(I\)-orbit is \(I\)-stable and its boundary has strictly smaller dimension. It is a union of the finitely many cells of \(Y\), by the refined Cartan decomposition. Put all boundary cells before a cell in the order. At each step the new union is the preceding closed union together with that cell's closure, so is closed. Since \(X\) is already a closed cell union, its cells may be put first. The remaining order gives a closed-cell filtration of \(U\). By Lemma C.12.1.1, a cell of dimension \(d\) is a \(T\)-representation of dimension \(d\). Its finite Borel approximation is a rank-\(d\) vector bundle \(v:V\to B_m\). Iterate the actual affine-line trace of Lesson 5, Lemma Q.1.2.1, on a trivializing open to obtain \(Rv_!E_V=E_{B_m}(-d)[-2d]\). The comparison is the smooth trace, with its given normalization, so these local comparisons agree under the vector-bundle transition maps. Equivalently the trace cone vanishes on every trivializing open and hence vanishes globally. Passing to the compatible finite Borel approximations gives the compact cell cohomology \[ H_{T,c}^q(C,E)=S^{q-2d}(-d). \tag{C.12.6} \] Here \(S^a=0\) for \(a<0\) or \(a\) odd. Applying the compact closed–open triangle successively to the filtrations proves that the compact cohomology of \(U,X,Y\) is even and finite free over \(S\). More explicitly, the odd groups vanish by induction; the resulting exact sequence in even degrees has free cell quotient, and lifting its homogeneous basis splits it as a graded \(S\)-module. On \(X,Y\), properness identifies compact and ordinary cohomology. Now use the compact triangle for \(U\subset Y\) with closed complement \(X\). Its boundary from \(H_T^q(X)\) goes to \(H_{T,c}^{q+1}(U)\). For even \(q\) the latter vanishes, and for odd \(q\) all ordinary groups vanish. We obtain the actual graded exact sequence \[ 0\longrightarrow H_{T,c}^*(U,E) \longrightarrow H_T^*(Y,E) \longrightarrow H_T^*(X,E) \longrightarrow0. \tag{C.12.7} \] In particular restriction is onto. It is the pullback of constants and therefore is a ring map; a chosen module splitting is not being used as its definition. Fix a component and a degree bound \(q\ge0\). By Lemma C.12.1.1 there are only finitely many cells in that component of dimension at most \(q/2\). The exhaustion in Lemma C.12.2.1 puts all of them in some \(Z_{j_0}\). For \(j'\ge j\ge j_0\), every cell in \(Z_{j'}\setminus Z_j\) has \(2d>q\). Formula (C.12.6) and its filtration show that this complement's compact cohomology vanishes in degrees at most \(q\). Sequence (C.12.7) consequently makes restriction an isomorphism in all those degrees. Each stabilized group is finite dimensional over \(E\), since a finite free graded \(S\)-module with finitely many homogeneous generators has finite dimensional homogeneous pieces. For completeness, compute the derived compatible-section complex, so an inverse-limit term is not omitted. Write \(C_j=R\Gamma_T(Z_j,E)\) and \(f_j:C_{j+1}\to C_j\) for restriction. Derived compatible sections on this tower are the homotopy fibre \[ R\!\varprojlim_j C_j = \operatorname{fib}\left( \prod_{j\ge0}C_j \xrightarrow{\,1-\mathrm{shift}\,} \prod_{j\ge0}C_j\right), \quad (a_j)\longmapsto(a_j-f_j(a_{j+1})). \tag{C.12.8} \] Formula (C.12.8) specifies the derived compatible-section complex used here, with its canonical maps to the stages. It is invariant under componentwise quasi-isomorphisms: products of \(E\)-vector spaces are exact, so a product of quasi-isomorphisms is a quasi-isomorphism; the two products and their fibre consequently preserve such a replacement. Cohomology of the products is the product of cohomology. The fibre's long exact sequence therefore gives \[ 0\longrightarrow \operatorname{coker}\!\left(1-\mathrm{shift}: \prod_j H_T^{q-1}(Z_j)\to\prod_j H_T^{q-1}(Z_j)\right) \longrightarrow H^q(R\!\varprojlim_j C_j) \longrightarrow\varprojlim_j H_T^q(Z_j) \longrightarrow0. \tag{C.12.9} \] The cokernel is zero. Indeed all the cohomological restriction maps are onto by (C.12.7). For a prescribed sequence \((b_j)\), choose \(a_0=0\) and successively choose \(a_{j+1}\) with \(f_j(a_{j+1})=a_j-b_j\). This solves \((1-\mathrm{shift})a=b\) in each homogeneous degree. Thus the canonical map in (C.12.9) identifies cohomology with the ordinary inverse limit. Apply the same calculation directly to the full bounded presentation. By C.12.2.1 its reduction maps are compatible quasi-isomorphisms; its reduced stages are finite Schubert unions. The preceding cell argument and (C.12.7) show that their cohomological transitions are onto. Thus (C.12.9) also identifies cohomology of derived compatible sections on the full \(Y_{N,\kappa}\) with the ordinary limit of their graded cohomology. These ordinary limits agree canonically with the limit over \(Z_j\): for each stage in either system choose a containing stage in the other and restrict a compatible family. Independence of this choice follows by restricting from a common larger stage; the two resulting operations are inverse by the same argument. This proves the comparison on cohomology without assuming a separate derived cofinality theorem. Cup products are compatible at every stage, and the canonical projection of derived compatible sections to a stage preserves them. One may define their product on a sheaf-cochain model before taking compatible sections; compatibility says exactly that restricting a product restricts both factors. The canonical map in (C.12.9) therefore identifies this product with the product of compatible homogeneous classes, not merely with an abstract isomorphic algebra. If a finite \(X\) lies in \(Z_m\), choose a lift of a homogeneous class in \(H_T^q(X)\) to \(H_T^q(Z_m)\) using (C.12.7). Choose successive lifts to \(Z_{m+1},Z_{m+2},\ldots\) by the same surjectivity, and define the earlier classes by restriction. This is a compatible homogeneous class, and (C.12.9) realizes it in \(H_T^q(\operatorname{Gr}_\kappa)\). This proves (C.12.5) in one component. For the full Grassmannian, the component classification gives a disjoint union indexed by \(\Pi\). Compatible sections give a degreewise product \[ H_T^q(\operatorname{Gr}_G,E) =\prod_{\kappa\in\Pi}H_T^q(\operatorname{Gr}_\kappa,E). \tag{C.12.10} \] One can verify this on the bounded presentation: each stage meets only finitely many classes by Lesson 2, Lemma 7.10, and enlarging stages covers every class. Thus a compatible family is exactly a family of compatible classes on all components. A finite \(X\) meets finitely many components. Lift its class in each of them as above and use zero in the others. This proves the global surjectivity, even when the centre contributes infinitely many classes. Finally, for a nilpotent thickening \(X\) the actual closed restriction to \(X_{\mathrm{red}}\) is the multiplicative isomorphism proved in C.12.2.1. Its composite with (C.12.5) is the already proved surjection onto that reduction, so restriction onto \(X\) is onto as well. ∎ **Corollary C.12.3.2.** For complexes \(P,Q\) supported on such an \(X\), the actual whole-Grassmannian action gives \[ \operatorname{Hom}_{H_T^*(\operatorname{Gr}_G)}^* \bigl(H_T^*(X,P),H_T^*(X,Q)\bigr) = \operatorname{Hom}_{H_T^*(X)}^* \bigl(H_T^*(X,P),H_T^*(X,Q)\bigr). \tag{C.12.11} \] Whenever the hypotheses of C.10 hold on \(X\), its comparison theorem therefore holds with the whole Grassmannian cohomology ring. **Proof.** Both module actions factor through actual restriction to \(H_T^*(X)\): cup product with a supported complex first restricts the constant class to its support. If a homogeneous map commutes with the whole-Grassmannian action, lift any homogeneous class of \(H_T^*(X)\) using C.12.3.1. Commutation with that lift is commutation with the given class. Conversely commutation with every support class implies commutation with their restrictions. These are the same maps, with the same degrees and compositions. The last assertion is now exactly C.10 applied to this equality; no stalk-surjectivity hypothesis has been supplied by the corollary. ∎ #### C.12.4. Two exercises on the component qualification **Exercise C.12.4.1 (basic).** Compute the ring in (C.12.10) for \(G=\mathbb G_m\). Explain why the full Grassmannian does not have finite dimensional degree-zero cohomology, despite degreewise stabilization on each component. **Solution.** Here \(\Lambda=\Pi=\mathbb Z\), \(Q^\vee=0\), and each reduced component is one point \(t^n\). Its full stages may retain the nilpotents displayed in C.11.1, but C.12.2.1 identifies their constant cohomology with that point's cohomology. With \(u=c_1(\chi)\), \[ H_T^*(\operatorname{Gr}_{\mathbb G_m},E) =\prod_{n\in\mathbb Z}^{\mathrm{degreewise}}E[u]. \tag{C.12.12} \] The superscript specifies the graded ring: in each degree one takes the product of that degree's vector spaces, with componentwise multiplication. It is not a direct sum of component rings. In degree zero it is \(\prod_{n\in\mathbb Z}E\), which is infinite dimensional. Restriction to finitely many components is the product projection and is onto. Each individual component is already stable at its reduced point, in every degree. **Exercise C.12.4.2 (intermediate).** For \(G=PGL_2\), choose the coweight coordinate \(n\in\mathbb Z\) with \(\langle\alpha,n\rangle=n\). List the dimensions of the cells \(n=-2,-1,0,1,2\) and the classes in which they lie. Why does dimension zero not identify the two components? **Solution.** The coroot is \(2\), so \(\Pi=\mathbb Z/2\mathbb Z\). Formula (C.12.1) is \[ d_I(n)=\max(0,n)+\max(0,-n-1). \] The listed dimensions are \(1,0,0,1,2\), respectively; their classes are \(0,1,0,1,0\). Thus both components contain a zero-dimensional cell. In the odd component the point \(n=-1\) is the closed cell of the minimal orbit \(\mathbf P^1\), while its other cell, at \(n=1\), has dimension one. In the even component \(n=0\) is the base point. The locally constant coroot class, not dimension, separates them. Within either class, the inequality \(|n|\le d_I(n)+1\) bounds the coweight labels of cells of bounded dimension. ### C.13. Finite jets, constant-group restriction and actual mapping complexes The earlier proof inputs are Lesson 1, Theorem 2.C, Lesson 2, Theorem 7.1, and Lesson 5, Lemmas Q.1.1.1-Q.1.2.1. The finite-jet generator enhancement is Theorem A.4.1.1 above. The full quotient schemes and the actual coefficient and mapping-complex comparisons are proved here. Let \(H_m\) be the algebraic group with \[ H_m(R)=G(R[t]/t^{m+1}),\qquad m\ge0, \] so \(H_0=G\). Constant loops give a section \(G\to H_m\) of evaluation. We work on a finite \(G(k[[t]])\)-stable closed support \(X\) contained in a full bounded stage \(Y_N\). The quotient-stack notation below denotes the actual action-nerve constructible category, or its finite free-frame models, with the maps of those presentations. The enhancements and coefficient comparisons used in A.4 are retained. #### C.13.1. The whole finite-jet quotient **Lemma C.13.1.1.** Evaluation has kernel \[ U_m=\ker(H_m\longrightarrow G)\simeq\mathbf A^{m\dim G} \tag{C.13.1} \] as a scheme, over every parameter algebra. Multiplication identifies \(H_m=U_m\rtimes G\), and right quotient by the constant group identifies \(H_m/G\) with the actual scheme \(U_m\). More generally, \[ U_{n,m}=\ker(H_n\longrightarrow H_m) \simeq\mathbf A^{(n-m)\dim G}\quad(n\ge m). \tag{C.13.2} \] The arc action on \(Y_N\), and hence on \(X\), factors through \(H_m\) whenever \(m+1\ge2N\). **Proof.** The ordered big cell of Lesson 1, Theorem 2.C is a principal open containing the identity, with coordinates \(U^-TU^+\). Any finite jet equal to the identity modulo \(t\) lies in this open: its defining function has an invertible constant term, hence is invertible modulo \(t^{m+1}\), on every algebra \(R\). Its negative and positive root coordinates are uniquely \(\sum_{j=1}^m a_{\alpha,j}t^j\). Choose a character basis for the original split torus. Each of its coordinates is uniquely \[ 1+\sum_{j=1}^m b_{i,j}t^j. \] These expressions are units; their inverses are the finite geometric series in the nilpotent ideal \((t)\). Every displayed coefficient is arbitrary in \(R\). There are \(m(|\Phi|+\operatorname{rank}T)=m\dim G\) coefficients. The big-cell multiplication and inverse coordinate maps give (C.13.1) on arbitrary test algebras. This includes every central torus coordinate. For the kernel in (C.13.2), impose equality to the identity modulo \(t^{m+1}\) instead. Uniqueness of the same big-cell coordinates sets precisely their coefficients of degrees at most \(m\) to zero; their coefficients of degrees \(m+1,\ldots,n\) are arbitrary. This proves (C.13.2), retaining the full kernel schemes. For \(h\in H_m(R)\), let \(g_0=h(0)\). The unique expression \(h=u g_0\), with \(u=h g_0^{-1}\in U_m(R)\), proves the semidirect product. Under right multiplication by \(a\in G(R)\), the expression \(h h(0)^{-1}\) is unchanged, because \((h a)(h(0)a)^{-1}=h h(0)^{-1}\). Thus it gives the quotient map and its section \(u\mapsto u\), identifying the whole right quotient with \(U_m\). The map \(G(R[[t]])\to H_m(R)\) is onto. After dividing a given jet by its constant value, its big-cell coordinates have the displayed form. Extend the root polynomials to formal series and the torus coordinates to the same units in \(R[[t]]\). Their product, times the constant value, is a lift. This argument also proves surjectivity \(H_n(R)\to H_m(R)\), by using the same expressions modulo \(t^{n+1}\). Finally take the faithful matrix embedding and the full lattice bounds from Lesson 2. A matrix \(h=1+t^{m+1}A(t)\), and its inverse, have this congruence form in that embedding. For every bounded lattice \(L\) and every \(\ell\in L\), \[ h\ell-\ell\in t^{m+1-N}R[[t]]^{\dim V} \subset t^N R[[t]]^{\dim V}\subset L \qquad(m+1\ge2N). \] Thus \(hL\subset L\), and the inverse inclusion follows from \(h^{-1}\). The associated-lattice closed immersion of the full Grassmannian in Lesson 2 identifies equality of these lattices with equality of the Grassmannian points, on every algebra. The entire congruence kernel therefore acts as identity on the full \(Y_N\), including its nilpotent parameters. Surjectivity onto \(H_m\) makes the action factor through that finite jet group. ∎ #### C.13.2. Restriction to the constant group **Theorem C.13.2.1.** For an \(H_m\)-action on \(X\), let \[ p:[X/G]\longrightarrow[X/H_m] \] be induced by inclusion of the constant group. For every constructible coefficient complex \(F\) on the target, its actual ordinary unit is an isomorphism \[ F\xrightarrow{\ \sim\ }Rp_*p^*F. \tag{C.13.3} \] Consequently restriction gives canonical isomorphisms \[ \begin{gathered} R\Gamma_{H_m}(X,F)\xrightarrow{\sim}R\Gamma_G(X,p^*F),\\ R\operatorname{Hom}_{H_m}(P,Q) \xrightarrow{\sim}R\operatorname{Hom}_G(p^*P,p^*Q). \end{gathered} \tag{C.13.4} \] They preserve units, cup actions, shifts, composition and, for Weil objects, the actual Frobenius operators. **Proof.** Pull \(p\) back along the smooth action atlas \(X\to[X/H_m]\). The resulting map is \[ X\times(H_m/G)\longrightarrow X. \] Indeed an isomorphism between two trivial \(H_m\)-torsors is an element \(h\); the equality of their maps to \(X\) determines the second point as \(h^{-1}x\), and its change by a constant \(G\)-element divides \(h\) on the right. Lemma C.13.1.1 identifies this quotient with \(U_m\), an affine space. Thus the pulled-back map is an actual product affine-space projection, and the pulled coefficient is the pullback of \(F|_X\). Iterate the ordinary affine-line unit of Lesson 5, Lemma Q.1.2.1. Its \(m\dim G\) iterations show that the ordinary unit on this product projection is an isomorphism, for every such pulled coefficient. Ordinary smooth base change, with its actual counit-transpose comparison in Lesson 5, Lemma Q.1.1.1, identifies this map with the pullback of (C.13.3). Smooth descent is conservative on the action presentation: if a cone restricts to zero on its atlas, every level of its cartesian nerve diagram is zero, so the cone is zero. This proves (C.13.3); no nonproper arbitrary-fibre comparison is used. Apply global sections to its actual unit. Composition of ordinary images identifies the target with \(R\Gamma_G(X,p^*F)\), giving the first map of (C.13.4). For the second, derived adjunction gives \[ R\operatorname{Hom}_G(p^*P,p^*Q) = R\operatorname{Hom}_{H_m}(P,Rp_*p^*Q). \] Under this equality the restriction map is precisely the map induced by the unit of \(Q\). Formula (C.13.3) makes it an isomorphism. This argument needs no asserted internal-Hom pullback formula for singular constructible complexes. Restriction is a functor on the actual coefficient presentations, so it preserves identities, shifts and composition. Pullback preserves multiplication of constant cochains and their action on pulled coefficients, giving the cup and coefficient-ring compatibility. Every map used is an actual unit, pullback, ordinary-image comparison or adjunction, defined over the ground field. If the objects have Weil structures, naturality therefore intertwines those structures and the resulting Frobenius operators. A choice of affine coordinates was used to test the unit, not to define it. ∎ #### C.13.3. Increasing the jet level **Lemma C.13.3.1.** Suppose the action on \(X\) factors through \(H_m\), and let \(n\ge m\). For the quotient-stack map \(q:[X/H_n]\to[X/H_m]\), the actual unit is an isomorphism \[ F\xrightarrow{\ \sim\ }Rq_*q^*F. \tag{C.13.5} \] Thus increasing a sufficient jet level preserves all mapping complexes between objects defined at that level. In the finite-jet spherical generator enhancement of A.4, the restriction maps in C.13.2 consequently identify the actual \(G(k[[t]])\)-equivariant generator mapping complexes with their constant-\(G\) restrictions. **Proof.** The kernel \(V=U_{n,m}\) acts trivially on \(X\). Pulling \(q\) back along the action atlas gives \(X\times BV\to X\): a lift of a trivial \(H_m\)-torsor exists as the trivial \(H_n\)-torsor, and its automorphisms over that torsor are exactly \(V\). This describes the entire quotient stack, retaining all its automorphisms. Every level \(V^s\) of the group nerve is an affine space, by C.13.1.1. For the coefficient pulled from \(X\), the actual ordinary affine-line units give \[ F|_X\xrightarrow{\sim} R(\operatorname{pr}_X:X\times V^s\to X)_* \operatorname{pr}_X^*(F|_X). \] These units commute with faces and degeneracies, since all of those maps preserve the projection to \(X\). The augmented cosimplicial complex of the constant diagrams on the left has identity face and degeneracy maps. Its normalized complex is \(F|_X\) in degree zero and zero in positive simplicial degrees: in degree \(s>0\) a normalized element is in the kernel of a degeneracy which is the identity. The same computation can be made in any bounded coefficient-resolution range and then in the full bounded-below total complex; the affine-space units are quasi-isomorphisms at each level, and the first-quadrant degree filtration makes them a quasi-isomorphism on totalizations. It follows that the ordinary image of the pulled coefficient on \(X\times BV\) is exactly \(F|_X\), with the displayed unit. Descent on the action atlas, or on the compatible free-frame diagrams of A.1, identifies this comparison with (C.13.5). To spell out the ordinary-image calculation, inverse image of the target atlas computes the fibre groupoid just described; its cartesian nerve has the projections \(X\times V^s\to X\). Ordinary image on that presentation is their cosimplicial totalization. Thus the comparison is a calculation on the actual presentation, not an inference from the cohomology of individual geometric fibres. Conservativity of that presentation proves (C.13.5). Global sections and the same derived-adjunction argument as in C.13.2 give cohomology and mapping-complex isomorphisms. The composition \(G\to H_n\to H_m\) is the original constant inclusion. Hence these isomorphisms commute with the constant-group restrictions. The generator enhancement of A.4 is formed from these action and coefficient presentations at sufficient jet levels; the units just proved identify its mapping complexes at every larger level with those at the chosen finite level, and then with (C.13.4). All identifications are the actual restriction functors and retain their compositions and Frobenius structures. This proves the stated generator comparison. It does not assert that every \(G\)-equivariant constructible object on \(X\) comes from an \(H_m\)-equivariant object. ∎ #### C.13.4. Two finite-jet exercises **Exercise C.13.4.1 (basic).** Compute \(U_2\) for \(G=\mathbb G_m\), including its multiplication, over an arbitrary parameter algebra. Explain why its trivial constant cohomology does not mean it is the trivial group scheme. **Solution.** Its units are \(1+at+bt^2\) modulo \(t^3\). Multiplication is \[ (a,b)(a',b')=(a+a',\,b+b'+aa'), \qquad (a,b)^{-1}=(-a,\,a^2-b). \tag{C.13.6} \] Thus the scheme is \(\mathbf A^2\), with a genuine group law in every characteristic. Its ordinary constant cohomology is \(E\) by two affine-line units; the relative nerve calculation in C.13.3 makes \(R\Gamma(BU_2,E)=E\). Neither comparison identifies its points or its group law with those of the trivial group. In particular all \(a,b\), including nilpotent parameters, remain in the quotient scheme \(H_2/G\). **Exercise C.13.4.2 (intermediate).** For \(G=GL_2\), describe \(U_1\) and determine a sufficient jet level for action on a lattice stage with bounds \(t^N O^2\subset L\subset t^{-N}O^2\). **Solution.** Every point of \(U_1\) is uniquely \(1+tA\) modulo \(t^2\), with \(A\in M_2(R)\); its inverse is \(1-tA\), and the group law is addition of \(A\). Hence \(U_1\) is the four-dimensional additive matrix scheme. For the bounded lattice stage, a congruence \(h=1+t^{m+1}A(t)\) changes any lattice vector by an element of \(t^{m+1-N}O^2\). This is inside \(t^NO^2\subset L\) when \(m+1\ge2N\), and the inverse congruence gives equality of lattices. Therefore \(H_m\) acts for \(m\ge\max(0,2N-1)\). This is a sufficient uniform bound; it is not claimed to be the minimum for each individual lattice. ### C.14. The flag basis and actual torus-to-group comparison We use the proved invariant-degree identities of Lemma B.1.1.1 above, the actual Euler and trace comparisons of C.5 and Lesson 6 A.1, and the finite-jet comparisons of C.13. The additional commutative-algebra and flag-coordinate proofs are linked at their point of use. Every change-of-group map below is restriction followed by the specified cup action. Keep the original split connected reductive \(G/k\), with \(k\) algebraically closed, and \(E/\mathbb Q_\ell\) finite, with \(\ell\) invertible in \(k\). Write \[ S=\operatorname{Sym}_E(X^*(T)\otimes_{\mathbb Z}E), \qquad R=S^W,\qquad N=|\Phi^+|. \] A character has polynomial degree one and cohomological degree two. Its Chern class retains its coefficient twist \(E(1)\). Thus a homogeneous class of polynomial degree \(a\) gives the action \(F(-a)[-2a]\to F\). We suppress those lines only when writing graded coefficient rings. #### C.14.1. The finite coinvariant algebra **Lemma C.14.1.1.** Choose homogeneous polynomial generators \(f_1,\ldots,f_r\) of \(R\), of polynomial degrees \(d_i\), as in B.1.1.1. Put \(C=S/(f_1,\ldots,f_r)\). Then \[ \dim_E C=|W|,\qquad C^{>N}=0,\qquad \operatorname{Soc}(C)=C^N\simeq E, \quad N=\sum_i(d_i-1). \tag{C.14.1} \] Every nonzero ideal of \(C\) meets this socle. If \(b_1,\ldots,b_{|W|}\) are homogeneous lifts of an \(E\)-basis of \(C\), then they form an \(R\)-basis of \(S\). They may be chosen with \(b_1=1\). **Proof.** B.1.1.1 proves polynomiality, the sum of the degrees, and \(\prod_i d_i=|W|\), for this Weyl action including its central fixed space. The polynomial ring \(S\) is finite over \(R\): each coordinate \(x_j\) satisfies the monic orbit polynomial \(\prod_{w\in W}(Z-wx_j)\), whose coefficients are invariant. Reducing powers by these finitely many monic equations expresses \(S\) as a finite \(R\)-module. Hence \(C\) is finite dimensional. Its positive ideal \(\mathfrak n_C\) is nilpotent by its finite positive grading, and \(C/\mathfrak n_C=E\); it is therefore local. The list \(f_1,\ldots,f_r\) is a system of parameters in \(S_{(x_1,\ldots,x_r)}\). A polynomial ring over a field is Cohen–Macaulay by Regular sequences, depth and Cohen–Macaulay modules, Theorem 5.3. Its Corollary 4.3 proves that this parameter list is regular. It is regular already in the graded polynomial ring: a nonzero homogeneous element of a successive homogeneous quotient stays nonzero on localizing at \((x_1,\ldots,x_r)\). Indeed multiplication by a polynomial with nonzero constant term has nonzero least-degree component on such an element. A homogeneous annihilator in a kernel would consequently contradict local regularity. The successive multiplication sequences for this regular list give \[ \operatorname{Hilb}_C(z) =\frac{\prod_i(1-z^{d_i})}{(1-z)^r} =\prod_i(1+z+\cdots+z^{d_i-1}). \tag{C.14.2} \] Its degree is \(N\), its top coefficient is one, and its value at one is \(|W|\). We also prove that no smaller-degree socle is hidden. The Koszul complex on a regular sequence is a free resolution of its quotient: start with the two-term injective multiplication complex, and tensor the next such complex; its cone has cohomology only the next quotient because the next multiplication is injective. This induction applies to both the coordinate sequence \(x_1,\ldots,x_r\) resolving \(E\), and \(f_1,\ldots,f_r\) resolving \(C\). Compute their balanced \(\operatorname{Tor}_r^S(E,C)\) using these two resolutions. Equality of the computations follows by taking their finite double tensor complex: resolve one factor first by rows and then the other by columns, using the just proved acyclicity of their free resolutions. In the coordinate Koszul complex tensored with \(C\), its top cycles are precisely the elements of \(\operatorname{Soc}(C)\), with their polynomial degrees increased by \(r\); there is no incoming top boundary. In the \(f\)-Koszul complex tensored with \(E\), every differential is zero because every \(f_i\) has positive degree. Its top term is one line of polynomial degree \(\sum_i d_i=N+r\). Thus the socle is exactly one line in degree \(N\), proving (C.14.1). Every nonzero ideal meets it. Given a nonzero element in that ideal, take the largest power of the nilpotent maximal ideal whose product with the generated ideal is nonzero. Any nonzero element of that product is killed by the maximal ideal and belongs to the socle. Finally the lifts \(b_j\) generate \(S\) over \(R\) by induction on polynomial degree: reduce an element modulo \((f_i)\) in the chosen basis, and express the remainder as \(\sum_i f_i a_i\), with each \(a_i\) of smaller degree. The free graded \(R\)-module on these lifts has Hilbert series \(\operatorname{Hilb}_R(z)\operatorname{Hilb}_C(z)=\operatorname{Hilb}_S(z)\), by (C.14.2). The resulting surjection is a bijection in each finite dimensional homogeneous degree. It is therefore the required free basis. The unique degree-zero basis class is one. All these arguments include the torus case, where the \(d_i\) are one and \(C=E\). ∎ #### C.14.2. Chern classes on the flag variety Let \(F=G/B\). Use the associated line convention \[ \mathcal L_\chi=G\times^B k_\chi,\qquad (g,z)\cdot b=(gb,\chi(b)^{-1}z). \] At \(n_wB\) its \(T\)-character is \(w\chi\). The root-coordinate Bruhat proof in Root data and Weyl groups, Theorems 6.2 and 8.1, and the smooth projective quotient of The flag scheme and the Weyl group, Theorem 1.1, give a closed affine paving of \(F\) indexed by \(W\), and the \(T\)-fixed points \(n_wB\). Its tangent characters there are \(-w\alpha\), for \(\alpha>0\). Put \(\delta=\prod_{\alpha>0}(-\alpha)\in S^N\). **Theorem C.14.2.1.** The actual Chern map gives an isomorphism \[ C\xrightarrow{\sim}H^*(F,E),\qquad [\chi]\longmapsto c_1(\mathcal L_\chi). \tag{C.14.3} \] The class \(c(\delta)\) has trace \(|W|\), with coefficient line \(E(N)\). In particular the images \(c(b_j)\) of C.14.1 form a homogeneous basis of flag cohomology. **Proof.** Bruhat root coordinates are linear for the constant torus and have nonzero weights. The closed-cell trace induction of C.12.3 proves that \(H_T^*(F,E)\) is finite free over \(S\), in even degrees. The same induction without the torus proves that ordinary flag cohomology has one generator for each cell, in its compact degree \(2d\), with twist \(-d\). The actual forgetful restriction gives \[ H_T^*(F,E)\otimes_S E\xrightarrow{\sim}H^*(F,E). \tag{C.14.4} \] Here is the comparison, rather than a general assertion that base change commutes with Hom. At each finite torus model choose a point of \(B_m\) with a frame; pulling back gives \(F\). On each cell the compact trace comparison is \(S(-d)[-2d]\to E(-d)[-2d]\), so is exactly augmentation of its coefficient ring. The compact cell rows are short exact with free cell quotient, both equivariantly and ordinarily. Tensoring the former with \(E\) remains exact because that quotient is free. Induction in the comparison of these rows proves (C.14.4), with its actual pullback. It also proves multiplicativity, since pullback of constants preserves cup product. For any homogeneous \(f\in R\), the equivariant Chern class \(c_T(f)\) restricts at \(n_wB\) to \(w(f)=f\). The base class \(f\in S\) has those same restrictions. C.5.2.1 says that their difference is zero after localization. Freeness over the polynomial domain \(S\) says that a class killed by a nonzero denominator is zero already. Thus \[ c_T(f)=\pi^*f. \tag{C.14.5} \] When \(f\) has positive degree, (C.14.4) sends the right side to zero. This proves that the ordinary Chern map factors through \(C\). Compute the trace of \(c(\delta)\) with its actual orientation. At a fixed point \(p=n_wB\), a translated root chart is an affine neighbourhood with coordinate tangent characters \(-w\alpha\). Compose its coordinate zero-section comparisons from Lesson 6, A.1.1.1–A.1.2.1. The supported point class has restriction \[ e_p=\prod_{\alpha>0}(-w\alpha)=w\delta \] at \(p\), and zero restriction at the other fixed points. Its structural trace is one by composition of the section and structural trace counits. After inverting \(S\setminus0\), C.5.2.1 therefore expresses any equivariant class \(a\) as the sum of these supported point classes with coefficients \(a|_p/e_p\). Applying trace gives \[ \int_F a=\sum_{w\in W}\frac{a|_{n_wB}}{w\delta}. \tag{C.14.6} \] This identity is in the localized coefficient ring and retains the twists; the supported classes, self-intersections and trace defined it. For \(a=c_T(\delta)\), every fraction is one. Its degree-normalized trace is consequently \(|W|\). The trace is constant in degree zero, so its localized equality is its integral equality. Forgetful pullback and trace compatibility give the same number for \(c(\delta)\). It is nonzero in \(E\), regardless of the geometric characteristic. Thus \([\delta]\) is nonzero in \(C^N\), and generates its socle by C.14.1. If the Chern map had a nonzero kernel ideal, that ideal would meet this socle and would kill \(c(\delta)\), which has nonzero trace. Its kernel is therefore zero. Both sides of (C.14.3) have dimension \(|W|\), the right side by its closed affine paving. The map is onto as well. ∎ #### C.14.3. Relative flag decomposition and restriction For a finite-type \(G\)-scheme \(X\) and actual equivariant bounded constructible complexes, write \(\operatorname{res}\) for restriction of the action presentation to \(T\). Cohomology of quotient stacks is computed in fixed ranges by the actual free frames of A.1. Write \(a_j=\deg b_j\). **Theorem C.14.3.1.** Actual restriction identifies \(H_G^*(\mathrm{pt},E)\) with \(R=S^W\). For every \(G\)-equivariant \(F,P,Q\) on \(X\), the maps \[ \begin{gathered} H_G^*(X,F)\otimes_R S \xrightarrow{\sim}H_T^*(X,\operatorname{res}F),\\ \operatorname{Ext}_G^*(P,Q)\otimes_R S \xrightarrow{\sim} \operatorname{Ext}_T^*(\operatorname{res}P,\operatorname{res}Q) \end{gathered} \tag{C.14.7} \] send \(x\otimes s\) to \(s\cup\operatorname{res}(x)\). They retain the shifts and coefficient twists specified above. The Ext comparison preserves identities and Yoneda composition under this scalar extension. **Proof.** First compare \(T\subset B\). Ordered root coordinates identify \(B/T\) with the full affine scheme of the unipotent radical. On a smooth action atlas, the map \([X/T]\to[X/B]\) is the product projection with this fibre. The iterated ordinary affine-line unit and Lemma Q.1.1.1, followed by conservative descent on the atlas, give the actual unit isomorphism, exactly as in C.13.2. This proves the corresponding cohomology and mapping-complex comparisons by adjunction. It keeps the action presentations, rather than replacing them by an arbitrary fibre calculation. Now consider the proper smooth flag map \[ p:[X/B]\longrightarrow[X/G]. \] Its pullback to the action atlas is \(X\times F\to X\). On \([X/B]\) the character lines from \(BB\) give the actual classes \(c(b_j)\). Compose the adjunction unit of a target coefficient \(K\) with cup action by these classes to define \[ \Phi_K: \bigoplus_j K(-a_j)[-2a_j]\longrightarrow Rp_*p^*K. \tag{C.14.8} \] On the atlas, proper base change and projection identify the target with \(K|_X\otimes R\Gamma(F,E)\). The pulled classes are the flag classes of C.14.2.1, so their basis makes this displayed map an isomorphism. One can verify the product coefficient identity by the same compact-cell trace filtration used in C.14.4; its arrows are the proper pullback and projection comparisons. Conservativity on the action atlas proves (C.14.8) globally. In particular this decomposition concerns the actual unit and global character classes, not only the fibre dimensions. Apply global sections for cohomology. For mapping complexes use derived adjunction: \[ R\operatorname{Hom}_B(p^*P,p^*Q) =R\operatorname{Hom}_G(P,Rp_*p^*Q). \] Inserting (C.14.8) gives the direct sum of the shifted and twisted \(G\)-mapping complexes. Its \(j\)-th map is cup action by \(b_j\) after actual restriction. No internal-Hom pullback formula is needed. It remains to identify the coefficient ring used in (C.14.7). Apply this decomposition with \(X=\mathrm{pt}\). Since \(b_1=1\), the actual map \(H_G^*(\mathrm{pt})\to S\) is injective: it is the first summand of the resulting basis decomposition. Its image lies in \(S^W\). Indeed, on a common free \(G\)-frame the right action of \(N_G(T)\) on the quotient by \(T\) preserves its projection to the quotient by \(G\); it factors through \(W\) and transforms character lines by the Weyl action. Pullback from the quotient by \(G\) is invariant under these actual maps. The flag decomposition and (C.14.2) give, degree by degree, \[ \operatorname{Hilb}_{H_G^*(\mathrm{pt})}(z) =\frac{\operatorname{Hilb}_{S}(z)} {\sum_j z^{a_j}} =\prod_i(1-z^{d_i})^{-1} =\operatorname{Hilb}_{R}(z). \tag{C.14.9} \] There are no odd groups: a degree-zero basis summand would otherwise produce an odd group in \(S\), which has none. The same summand proves finiteness of every degree. Thus the invariant inclusion has equal dimensions in each degree and is onto \(R\). It is a ring map, so identifies the coefficient rings multiplicatively. Lemma C.14.1.1 makes \(S=\bigoplus_j Rb_j\). Therefore the cohomology and Ext direct-sum comparisons just constructed are precisely (C.14.7). Restriction is an actual functor and cup coefficients have even degrees, so \[ (s\cup\operatorname{res}u)\circ (t\cup\operatorname{res}v) =st\cup\operatorname{res}(u\circ v). \tag{C.14.10} \] There is no odd coefficient sign. Identities map to identities. This proves the asserted composition compatibility; it does not claim a monoidal enhancement from graded Ext alone. All maps are defined over the ground field and intertwine Weil structures when supplied. ∎ #### C.14.4. Weyl descent of actual morphisms Let \(A_G=H_G^*(X,E)\), \(A_T=H_T^*(X,E)\), and \(M_G=H_G^*(X,P)\), \(N_G=H_G^*(X,Q)\). The actual \(G\)-equivariances give a Weyl action on their torus restrictions: representatives in the normalizer conjugate the torus action, and the action cocycle identifies the pulled coefficients. Representatives differing by \(T\) give the same map on equivariant cohomology and Ext, by that cocycle. These are semilinear actions over \(S\). **Corollary C.14.4.1.** These actions yield \[ \begin{gathered} \operatorname{Ext}_G^*(P,Q) =\operatorname{Ext}_T^*(\operatorname{res}P,\operatorname{res}Q)^W,\\ \operatorname{Hom}_{A_G}^*(M_G,N_G) =\operatorname{Hom}_{A_T}^*(M_G\otimes_R S,N_G\otimes_R S)^W. \end{gathered} \tag{C.14.11} \] The maps are actual restriction and scalar extension, and retain composition. If the comparison of C.10 holds for the torus restrictions of \(P,Q\) on \(X\), it therefore holds for their \(G\)-equivariant Ext and \(A_G\)-module Hom. For a sufficiently large finite jet action, C.13 gives the same assertion for the original spherical generator Ext. **Proof.** C.14.3 identifies each torus group with scalar extension from \(R\), and identifies \(A_T=A_G\otimes_R S\) as rings: the multiplication comparison is the restriction and cup map. Under these identifications \(W\) acts trivially on the \(G\)-factor and ordinarily on \(S\), since all restricted classes came from a \(G\)-presentation. For every \(R\)-module \(L\), the invariant part of \(L\otimes_R S\) is exactly \(L\). To prove this without a flat Hom-base-change assertion, average the \(W\)-action using \(|W|^{-1}\sum_w w\). This projector is \(R\)-linear on \(S\) with image \(R\), and is the identity on that submodule. On \(L\otimes_R S\) its image is therefore \(L\otimes_R R=L\); this inclusion is injective because the same projector splits it. Apply this first to the Ext scalar extension to get the first equation. For an \(A_T\)-linear invariant map \(h\) in the second line, \(h(m\otimes1)\) is invariant and hence belongs to the canonically embedded \(N_G\). It defines a homogeneous map \(h_0:M_G\to N_G\). Applying the actual restriction of each \(a\in A_G\) proves \(h_0\) is \(A_G\)-linear. Every other value of \(h\) is determined by \(S\)-linearity: \(h(m\otimes s)=s\,h_0(m)\). Conversely scalar extension of an \(A_G\)-linear map gives such an invariant \(A_T\)-linear map. These inverse constructions preserve composition. They require neither projectivity of \(M_G\) nor a claim that arbitrary Hom commutes with specialization. Finally the cohomology comparison map of C.10 is natural under these normalizer maps, because it applies actual cohomology to actual morphisms. Its Weyl invariants therefore give the comparison of the first and second lines of (C.14.11). The finite-jet and constant-group restriction of C.13 identifies the original generator Ext with this one, preserving the coefficient action and composition. The hypotheses of C.10, including its stalk-surjectivity requirement, remain required. ∎ **Corollary C.14.4.2.** On the full Grassmannian, constant cohomology satisfies \[ H_G^*(\operatorname{Gr}_G,E)\otimes_R S \xrightarrow{\sim}H_T^*(\operatorname{Gr}_G,E). \tag{C.14.12} \] For every finite closed \(G(k[[t]])\)-stable support \(X\) whose reduction is a union of Iwahori cells, actual restriction \(H_G^*(\operatorname{Gr}_G)\to H_G^*(X)\) is onto. The comparison of C.10, when its stalk hypotheses hold, therefore identifies the original spherical generator Ext with module Hom over the whole \(G\)-equivariant Grassmannian ring. **Proof.** Use the original full bounded stages \(Y_N\); they are \(G\)-stable by the matrix-bound argument of C.12.2. C.14.3 applies at every such stage and commutes with its restriction maps. Its decomposition has the fixed finite basis \(b_j\), independent of \(N\). Finite direct sums and fixed shifts commute with the two products and their fibre in the derived compatible-section model (C.12.8). Thus the same flag decomposition holds for derived compatible sections on the full bounded presentation. The canonical cohomology map gives (C.14.12), with the same actual cup products and Weyl action. Under this comparison \(W\) acts on the scalar factor \(S\) alone. Averaging as in C.14.4 identifies its invariants with \(H_G^*(\operatorname{Gr}_G)\), also for degreewise products over all components. The restriction in C.12.3 is a Weyl-equivariant surjection for this \(G\)-stable \(X\). Given an invariant target class, lift it by that surjection and average the lift; the averaged class is an invariant lift. Hence the \(G\)-restriction is onto. Reduction and the actual closed-restriction unit retain the assertion for the stated nilpotent support structures. Consequently an \(H_G^*(X)\)-module map between supported cohomology modules is exactly a map linear over the whole ring: lift each homogeneous support class through the surjection, as in C.12.3.2. Combine this equality with C.14.4.1 and the actual jet comparison of C.13. The result retains composition and the original group and components. The unconditional stalk-surjectivity input, geometric identification of these modules with Appendix B, and enhanced convolution remain separate requirements. ∎ #### C.14.5. Two change-of-group exercises **Exercise C.14.5.1 (basic).** For the rank-one Weyl action \(u\mapsto-u\) compute \(R,C\), a free \(R\)-basis of \(S\), and the invariant part of \(L\otimes_R S\) for an arbitrary \(R\)-module \(L\). **Solution.** With polynomial degree \(|u|=1\), \(S=E[u]\), \(R=E[u^2]\), \(C=E[u]/(u^2)\), and \(S=R\oplus Ru\). Thus \(L\otimes_R S=L\oplus Lu\); its Weyl action is \((a,b)\mapsto(a,-b)\). Since two is invertible in \(E\), its invariants are exactly the first copy \(L\), including when \(L\) has \(u^2\)-torsion. The quotient \(C\) has its one-dimensional socle generated by \(u\). This verifies both the free basis and the averaging argument without a torsion-free hypothesis on \(L\). **Exercise C.14.5.2 (intermediate).** For \(G=SL_2\), let \(\chi(\operatorname{diag}(z,z^{-1}))=z\) and \(u=c_1(\chi)\). Determine the sign of \(\mathcal L_\chi\) on \(G/B=\mathbf P^1\) in our associated-line convention, and check the tangent-root trace. **Solution.** The upper Borel fixes the line spanned by \(e_1\). The map \([g,z]\mapsto z\,ge_1\) identifies \(\mathcal L_\chi\) with the tautological line: replacing \((g,z)\) by \((gb,\chi(b)^{-1}z)\) leaves that vector unchanged. Therefore \(c(u)=-h\), where \(h=c_1(\mathcal O(1))\) has trace one. The positive root is \(\alpha=2\chi\), and \(\delta=-\alpha=-2u\); hence \(c(\delta)=2h\), with trace \(2=|W|\). At the two fixed points the tangent Euler classes are \(-2u\) and \(2u\), respectively. Formula (C.14.6) for \(c_T(\delta)\) sums two fractions equal to one. This checks the sign and also retains the nonprimitive integral root character. ### C.15. Relative tame boundary comparison with complete coefficients Let \(Y_0\) be smooth and integral over a finite field of characteristic \(p\), let \(p_0:\overline X_0\to Y_0\) be a smooth proper curve, and let \(D_0\subset\overline X_0\) be a Cartier divisor finite étale over \(Y_0\). Write \(j:U_0=\overline X_0\setminus D_0\hookrightarrow\overline X_0\) and \(i:D_0\hookrightarrow\overline X_0\). Fix a finite extension \(E/\mathbb Q_\ell\), with \(\ell\ne p\), valuation ring \(O\), and uniformizer \(\pi\). Let \(L\) be a lisse \(E\)-sheaf of rank \(r\ge1\). Tameness means that the finite reductions of a stable lattice are tame at the generic boundary DVRs. All comparisons below concern the actual ordinary extension \(j_*L=\mathcal H^0Rj_*L\). The local finite-cover input has a complete earlier proof: Specialization and tame ramification, Theorem 2.3, identifies tame covers along a regular Cartier divisor with finite Kummer covers after passage to a strict local scheme. The coefficient inputs are The pro-étale site and l-adic complexes, Theorem 7.10 and Corollary 7.11, and Semi-infinite geometry and MV weight functors, M.6.2.1 and O.2.1.1. Their completed pullback and ordinary-image proofs retain the actual coefficient comparisons. We use them rather than commuting an infinite inverse limit with an ordinary étale stalk. #### C.15.1. The boundary invariant and its geometric fibre **Lemma C.15.1.1.** The actual restriction comparison \[ (j_*L)|_{\overline X_{\bar y}} \longrightarrow j_{\bar y,*}(L|_{U_{\bar y}}) \tag{C.15.1} \] is an isomorphism for every geometric \(\bar y\to Y_0\). At a geometric point \(d\) of the boundary its completed fibre is \(V^{I_d}\), where \(V\) is the local fibre of \(L\) and \(I_d\) is tame inertia. This statement uses the full \(E\)-system. **Proof.** Locally along \(D_0\) there is a relative parameter \(z\) with \(D_0=(z)\). Indeed, trivialize its Cartier ideal near a point. Since \(D_0\to Y_0\) is étale, the image of \(dz\) generates the relative cotangent line along \(D_0\). After restricting to a smaller open, \((p_0,z)\) is étale over \(\mathbb A^1_{Y_0}\): the differential is invertible and both sides are smooth of the same relative dimension. Thus \(z\) has valuation one also on each geometric curve fibre. The total space and divisor are regular. For the strict henselization \(A\) at \(d\), Theorem 2.3 just cited says that every finite tame cover of \(\operatorname{Spec}A[1/z]\) is a disjoint union of \[ \operatorname{Spec}A[1/z,t]/(t^m-z),\qquad (m,p)=1. \tag{C.15.2} \] Each displayed cover is connected: the valuation at the prime \((z)\) gives the degree \(m\) root-uniformizer calculation in that theorem. Its automorphisms are \(t\mapsto\zeta t\), for \(\zeta\in\mu_m(A)\); strict henselianity supplies these roots of unity. A common multiple of two exponents dominates both covers. Maps and automorphisms are determined by these root equations, so this describes the finite tame cover category, including its morphisms. Its pointed automorphism group is \(I_d=\varprojlim_{(m,p)=1}\mu_m=\widehat{\mathbb Z}^{(p')}(1)\). Pull the equations to the strict local curve at the same geometric point. They remain connected Kummer covers of exactly degree \(m\), because the pulled parameter still has valuation one. The same description supplies every finite tame cover there and the same root maps between covers. Consequently restriction induces the identical tame inertia group and the identical action on every pulled finite coefficient fibre. This assertion applies to the entire compatible tower; no single finite cover is asserted to trivialize that tower. There is a global stable lattice on each connected component of \(U_0\). Indeed \(U_0\) is smooth and hence geometrically unibranch. The cited Corollary 7.11 identifies its coefficient representation with a continuous representation of the profinite étale fundamental group. Its image is compact. The translates of a basis lattice lie in one larger lattice by compactness, and their \(O\)-span is a finite torsion-free \(O\)-module containing the starting lattice. It is a stable lattice. Let \(M\) be its local fibre and \(M_n=M/\pi^nM\). Finite ordinary extension at \(d\) has degree-zero fibre \(M_n^{I_d}\). To see this directly, a finite section on the punctured strict local scheme is exactly a fibre vector fixed by the finite covering action. The Kummer comparison therefore identifies this fibre and its restriction to the curve fibre, with the actual restriction map. Now apply O.2.1.1 to the lattice sheaf and M.6.2.1 to its geometric-point pullback. The complete fibre of its ordinary derived image is the coherent derived limit of the finite derived-image fibres. They are in degrees \(\ge0\); hence the degree-zero Milnor sequence has zero \(\varprojlim^1\mathcal H^{-1}\) term and gives \[ \mathcal H^0(d_O^*Rj_*L_O) =\varprojlim_n M_n^{I_d} =M^{I_d}. \tag{C.15.3} \] The last equality follows because compatible fixed vectors are precisely fixed vectors in \(M=\varprojlim_nM_n\). It is an inverse-limit assertion, not an assertion about reduction of \(M^{I_d}\). Here \(d_O^*\) denotes actual completed ringed pullback; degree-zero cohomology is taken after that pullback. Actual rational ordinary image and actual rational pullback are the localizations of these integral constructions, by O.2.1.1 and M.6.2.2. Localization is exact. Choose a dense tame generator \(\tau\), and let \(T\) act on \(M\). Continuity makes invariance under \(\tau\) equivalent to invariance under \(I_d\), since the stabilizer equation is closed. Thus \[ M^{I_d}\otimes_O E =\ker(T-1:M\to M)\otimes_O E =\ker(T-1:V\to V)=V^{I_d}. \tag{C.15.4} \] Finite-level restriction was the same Kummer action, so its coherent limit and localization identify the actual map (C.15.1) with the identity on this invariant fibre. On the open curve it is already the identity. Ordinary rational inverse image is exact on constructible sheaves. On a lisse stratum the locally free \(E\)-coefficient model pulls back to the same degree-zero model. The finite closed-open filtration and the actual identities \(g^*i_*=i'_*g^*\), \(g^*j_!=j'_!g^*\) of M.6.1.2 extend this conclusion to the entire constructible sheaf, because those extensions and finite extensions of sheaves are exact. Closed restriction also has the coefficient identity M.3.4. Geometric fibres detect an isomorphism, using this finite stratal coefficient description. Therefore (C.15.1) is an isomorphism of ordinary sheaves. ∎ #### C.15.2. A finite logarithmic coefficient On \(\mathbb G_m\) define a rank-\(n\) logarithmic system \(\operatorname{Log}_n\) with basis \(e_0,\ldots,e_{n-1}\) and \(Je_i=e_{i-1}\), \(Je_0=0\). Its tame inertia action is \[ a\longmapsto\exp(aJ),\qquad a\in\mathbb Z_\ell(1), \tag{C.15.5} \] after choosing a generator of the Tate line. The remaining prime-to-\(p\) inertia factors act trivially. This is an actual complete coefficient system. The basis \(l_i=i!e_i\) is an \(O\)-lattice, and \[ \exp(aJ)l_i=\sum_{k=0}^{i}\binom{i}{k}a^k l_{i-k}. \tag{C.15.6} \] Its matrices are integral and continuous, so every finite reduction is a finite continuous action. The compatible \(\ell\)-power root covers of the coordinate construct this action geometrically. The integral classification in Theorem 7.10 and completion of its finite reductions construct the lattice sheaf; rationalization gives \(\operatorname{Log}_n\). There is a Weil structure with geometric Frobenius \(Fe_i=q^ie_i\). For completeness, the root covers based at the section \(1\in\mathbb G_m\) give the Kummer quotient of its arithmetic fundamental group. Its geometric subgroup is \(\widehat{\mathbb Z}^{(p')}(1)\), and the section splits the projection to \(\widehat{\mathbb Z}\). Arithmetic Frobenius acts on roots of unity by \(q\), so geometric Frobenius conjugates inertia by \(q^{-1}\). The diagonal action just defined has \(FJF^{-1}=q^{-1}J\) and therefore respects that semidirect-product relation. It extends continuously to the arithmetic group: \(q\) is an \(\ell\)-adic unit, so its integer powers have finite images in every congruence quotient and extend to \(\widehat{\mathbb Z}\). This also proves continuity on the integral lattice (C.15.6). Thus \(J:\operatorname{Log}_n\to\operatorname{Log}_n(-1)\) is an actual Weil morphism. The inclusions \(e_i\mapsto e_i\) give compatible inclusions \(\operatorname{Log}_n\hookrightarrow\operatorname{Log}_{n+1}\). On a parameter chart, pull this system back by \(z\) and put \[ B_n=\mathcal H^0i^*Rj_*(L\otimes z^*\operatorname{Log}_n), \qquad \mathcal N=-1\otimes J:B_n\to B_n(-1). \tag{C.15.7} \] These are constructible rational sheaves and actual morphisms by O.2.1.1 and M.6. In particular \(B_1=i^*j_*L=:B\). **Lemma C.15.2.1.** If \(n\ge r\), the completed geometric fibre of \(B_n\) is the generalized eigenvalue-one part \(V_1\) of tame monodromy on \(V\). Under this identification \(\mathcal N\) is \(N=\log(T|_{V_1})\), with its Tate twist. The actual inclusion \(B\hookrightarrow B_n\) identifies \(B\) with \(\ker\mathcal N\). These identifications commute with geometric curve fibres, inclusions in \(n\), and Frobenius. **Proof.** Normalize the chosen \(\ell\)-component of \(\tau\) to one. Lemma C.15.1.1 applied to the tensor coefficient gives the fibre \(\ker(T\otimes\exp J-1)\). The primary decomposition for the polynomial \(T\) separates \(V_1=\ker(T-1)^r\) from its complementary primary parts. On each other part, after a finite splitting field, the eigenvalues of \(T\otimes\exp J\) are the eigenvalues of \(T\), none equal to one. Its difference from one is invertible there, and faithful scalar extension proves that the invariant kernel has no such part over \(E\). On \(V_1\) the finite logarithm \(N=\log T\) is nilpotent, with \(N^r=0\), and \[ T\otimes\exp J=\exp(N\otimes1+1\otimes J). \] For a nilpotent operator \(A\), \(\exp A-1=A\,h(A)\), where \(h(0)=1\); the polynomial \(h(A)\) is invertible by a finite geometric series. Hence the required kernel is \(\ker(N\otimes1+1\otimes J)\). There is an explicit inverse pair \[ \Psi_n:V_1\xrightarrow{\sim}(B_n)_d^{\mathrm{comp}}, \qquad \Psi_n(v)=\sum_{i=0}^{n-1}(-N)^iv\otimes e_i, \tag{C.15.8} \] and extraction of the coefficient of \(e_0\). Indeed a tensor \(\sum v_i\otimes e_i\) is in the kernel exactly when \(Nv_i+v_{i+1}=0\) for \(i0}. \tag{C.17.1} \] C.14 identifies these actual coefficient rings and makes \(S\) finite free and faithfully flat over the positive graded polynomial ring \(R_0\). For a genuine spherical perverse \(P\), set \(\omega_G(P)=H_G^*(X,P)\), independent of the sufficiently large proper support \(X\), and \(\omega(P)=H^*(X,P)\). The module grading is cohomological. The full Satake equivalence and general rational IC parity were proved in Reconstruction and the dual reductive group, B.4.1.1 and C.4.1.1. We use its corrected symmetric structure. The tensor symmetry on the graded modules below is the ordinary flip; the base ring has only even degrees. The unadjusted fusion symmetry instead corresponds to the usual Koszul flip. #### C.17.1. Free modules and actual augmentation **Lemma C.17.1.1.** Actual forgetting of equivariance induces a natural isomorphism \[ \omega_G(P)\otimes_{R_0}E\xrightarrow{\sim}\omega(P). \tag{C.17.2} \] The module \(\omega_G(P)\) is finite graded free over \(R_0\). A homogeneous basis of \(\omega(P)\) can be lifted to a homogeneous \(R_0\)-basis; this choice is not asserted to be canonical. Its basis degrees are the actual ordinary cohomology degrees. **Proof.** First take \(P\) supported on one component. Semisimplicity makes it a finite sum of raw IC objects, all with that component's parity: \(h_G(\alpha_i^\vee)=2\), so dimension parity is constant modulo \(Q^\vee\). On each Iwahori cell its ordinary cohomology is constant and has this parity by Lesson 11 C.4.1.1 and the root-section comparisons. The equivariant constant-system splitting in C.10 therefore computes compact cell cohomology as its shifted coefficient space tensored with \(S\), with the affine trace's degree and Tate line. Proper compact-cell localization has zero degree-one connecting maps. Consequently all its sequences are short exact, its equivariant terms are finite free over \(S\), and the ordinary sequences have exactly the same basis degrees. The actual cell forget map is \[ H_{T,c}^*(C,i_C^*P)\otimes_S E \xrightarrow{\sim}H_c^*(C,i_C^*P). \tag{C.17.3} \] This is the vector-space coefficient map and the same orientation trace on both sides. Induct through the finite closed cell filtration. Tensoring each equivariant sequence with \(E=S/S_{>0}\) stays exact because its quotient is free; the ordinary sequence and (C.17.3) then identify the middle map. Since \(X\) is proper, this proves \[ H_T^*(X,P)\otimes_S E=H^*(X,P) \tag{C.17.4} \] through actual forgetting. It also proves, degree by degree, \(\operatorname{Hilb}_{H_T(P)}=\operatorname{Hilb}_S\operatorname{Hilb}_{\omega(P)}\), allowing finite Laurent degree shifts. C.14.3 gives \(H_T(P)=\omega_G(P)\otimes_{R_0}S\) by actual restriction and cup product. Forgetting the torus and the group are compatible with the same coefficient augmentation. Tensor this comparison with \(E\) over \(S\) and use (C.17.4) to get (C.17.2). Put \(M=\omega_G(P)\). It is a finite \(R_0\)-module: C.14.4 makes it the invariant submodule of the finite \(R_0\)-module \(H_T(P)\), and \(R_0\) is Noetherian. It is bounded below in degree because it embeds there. Choose homogeneous lifts of a basis of \(M/\mathfrak mM=\omega(P)\), giving a map \(L\to M\) from the finite graded free module \(L\) with those degrees. Its cokernel is finite graded, bounded below, and has zero reduction modulo \(\mathfrak m\). Such a cokernel is zero: a lowest nonzero homogeneous degree could not be generated by elements of strictly positive degree acting on lower degrees. Thus \(L\to M\) is onto. Let \(b_j\) be the homogeneous \(R_0\)-basis of \(S\), of degrees \(2a_j\), from C.14. Then \[ \operatorname{Hilb}_{H_T(P)} =\operatorname{Hilb}_M\sum_jz^{2a_j},\qquad \operatorname{Hilb}_S =\operatorname{Hilb}_{R_0}\sum_jz^{2a_j}. \tag{C.17.5} \] Together with the preceding cell calculation, these equalities give \(\operatorname{Hilb}_M=\operatorname{Hilb}_{R_0}\operatorname{Hilb}_{\omega(P)}=\operatorname{Hilb}_L\). Each degree is finite dimensional; the surjection with equal dimensions is therefore an isomorphism in every degree. This proves freeness without a polynomial-module classification theorem. For several components, use their finitely many supporting component projectors and apply the argument to each summand. Actual augmentation and naturality are preserved by these direct sums; bases still need not be chosen naturally. ∎ #### C.17.2. The relative external cup map **Lemma C.17.2.1.** Let \(P,Q\) be such spherical objects on proper supports \(X,Y\), with diagonal constant-group action on \(X\times Y\). Actual external cup gives an isomorphism \[ H_G^*(X,P)\otimes_{R_0}H_G^*(Y,Q) \xrightarrow{\sim} H_G^*(X\times Y,P\boxtimes Q). \tag{C.17.6} \] It retains the actual coefficient maps, associativity, and the graded external-product signs. **Proof.** On the quotient product over \(BG\), the two structural coefficient maps from \(R_0\) agree. Thus the ordinary external cup map is balanced over \(R_0\), defining (C.17.6). Construct it on the finite classifying models; its action-nerve and model comparisons are those of C.13–C.14. First restrict to \(T\). On two cells, split the bounded constant-parity coefficient complexes as in C.10. The external cup map is an isomorphism on constant shifted coefficients: the product vector-bundle trace is the iterated trace, so it identifies the two compact orientations, retaining the Tate lines. Finite sums give the same statement for the cell coefficients. Induct through the closed cell filtrations of \(X\) and \(Y\). All compact localization sequences are short exact by their component parities, and every equivariant term is free over \(S\). Tensoring one such sequence with the other term is exact. The corresponding compact product-cell sequence is the same actual localization sequence under external tensor and proper-support image. The cell isomorphism and induction therefore identify its middle map. Repeating in the two variables proves \[ H_T(P)\otimes_S H_T(Q) \xrightarrow{\sim}H_T(P\boxtimes Q). \tag{C.17.7} \] Finite component decompositions handle objects of different component parities. By C.14.3, scalar extension of (C.17.6) from \(R_0\) to \(S\) is exactly (C.17.7). Its kernel and cokernel vanish after this extension. Faithful flatness of \(S/R_0\) makes them vanish before extension, proving (C.17.6). The maps were actual cup and tensor maps throughout. Associativity and the external Koszul symmetry can be checked on those maps before descent, so the result retains them. ∎ #### C.17.3. Canonical transport of a complex along affine parameters **Lemma C.17.3.1.** Let \(Z\) be separated of finite type over \(k\), and let \(p:\mathbb A^r\times Z\to Z\). A bounded rational constructible \(K\) is in the essential image of actual \(p^*\) if every ordinary \(\mathcal H^nK\) is isomorphic to \(p^*L_n\) for a constructible sheaf \(L_n\) on \(Z\). For such \(K\), the actual counit and every section restriction give isomorphisms \[ p^*Rp_*K\xrightarrow{\sim}K,\qquad Rp_*K\xrightarrow{\sim}s^*K . \tag{C.17.8} \] They are natural and commute with restriction along affine parameter maps over \(Z\). For an arbitrary base change \(g:Z'\to Z\), the ordinary-image comparison for a complex in this essential image is also an isomorphism. **Proof.** Iterating Semi-infinite geometry and MV weight functors, Lemma Q.1.2.1, gives the actual unit \(C\xrightarrow{\sim}Rp_*p^*C\) for every bounded rational constructible \(C\). Derived adjunction consequently makes \(p^*\) fully faithful on these complexes. Rational pullback is t-exact on constructible objects: on a lisse coefficient stratum its finite free model stays in the same degrees, and the actual closed-open pullback identities of Lesson 5 M.6 extend this assertion through the finite stratification. Induct on the finite ordinary cohomology range of \(K\). The first truncation is a shifted \(p^*L_n\). Suppose the lower truncation is \(p^*C\). The next attaching map has the form \[ p^*L_{n+1}[-n-1]\longrightarrow p^*C[1]. \tag{C.17.9} \] Full faithfulness lifts it uniquely to a map on \(Z\). Its cone, with the truncation shift, constructs a complex on \(Z\) pulling back to the next truncation. Thus \(K\) is pulled back from some bounded constructible \(C'\). Actual ordinary image is constructible by Lesson 5 O.2.1.1, and the unit identifies \(Rp_*K\) with \(C'\). The adjunction identity makes its displayed counit an isomorphism. Pull that counit along a section to obtain the second map in (C.17.8). For a parameter map \(f:\mathbb A^b\times Z\to\mathbb A^r\times Z\) over \(Z\), \(f^*p^*C'\) is the same pullback of \(C'\). The unit, counit and their pasting identities therefore give the same comparisons for the two projections. For a base change \(g\), the pulled complex is \(p'^*g^*C'\). Its ordinary image is \(g^*C'\) through the affine-space unit. Under the two unit identifications the actual map \(g^*Rp_*p^*C'\to Rp'_*p'^*g^*C'\) is the identity of \(g^*C'\): transpose it under the right adjunction and use the counit identity. Naturality then proves the assertion for \(K\) and its morphisms. This proves both compatibilities. It did not replace a cohomology-sheaf comparison by an arbitrarily chosen derived splitting; the actual counit supplies the canonical complex map. ∎ #### C.17.4. Convolution and the corrected symmetry over the coefficient ring **Theorem C.17.4.1.** The functor \(\omega_G\) takes the genuine rational Satake heart to finite graded free \(R_0\)-modules. It is exact and faithful and has canonical tensor comparisons \[ \omega_G(P)\otimes_{R_0}\omega_G(Q) \xrightarrow{\sim}\omega_G(P*Q). \tag{C.17.10} \] They preserve associativity, units, all finite ordered fusion block comparisons, and the corrected ordinary symmetric constraint. Actual augmentation (C.17.2) is the ordinary Satake cohomology tensor comparison. The functor preserves rigid duals: \[ \omega_G(P^\vee)= \operatorname{Hom}_{R_0}(\omega_G(P),R_0) \tag{C.17.11} \] with the dual grading. This is a statement about the actual graded cohomology tensor functor; a dual-Lie-algebra identification or a monoidal dg equivalence is not inferred from it. **Proof.** Use the ordered moving-chain construction on the affine line from Fusion and the symmetric tensor structure, C.1–C.6, with unshifted factors \(P_1,\ldots,P_r\). Its map \(a_r:Y_r\to B_r=\mathbb A^r\) is proper, its coefficient \(A_r\) has the finite product charts of that lesson, and its full-collision fibre is the actual convolution chain. Constant \(G\) acts on all away frames diagonally; this commutes with the translated coordinates \(z-x_i\) and the genuine formal-coordinate coefficient comparisons. Thus the source, coefficient and endpoint maps have a genuine constant-\(G\) action. Use the actual action nerve retained in C.13: its classifying-group level is \(Z_n=G^n\), its source level is \(Z_n\times Y_r\), and its base level is \(Z_n\times B_r\). The coefficient at that level is the pullback of \(A_r\), with the genuine action identifications on the faces. Put \[ K_{r,n}=R(\mathrm{id}_{Z_n}\times a_r)_* \operatorname{pr}_{Y_r}^*A_r . \] The map being pushed forward is proper. Its product-base-change comparison identifies this complex with the pullback of \(K_{r,0}=Ra_{r,*}A_r\) from \(B_r\). Lesson 8 C.4.1.1 proves that every \(\mathcal H^qK_{r,0}\) is constant, with the specified external-product normalization, full coefficients and all degrees. Thus every \(\mathcal H^qK_{r,n}\) is pulled from \(Z_n\). This uses a proved cohomology-sheaf statement and proper base change; no derived splitting has yet been chosen. For \(p_n:B_r\times Z_n\to Z_n\), Lemma C.17.3.1 supplies the actual complex comparison \[ K_{r,n}\xleftarrow{\sim} p_n^*Rp_{n,*}K_{r,n}. \tag{C.17.12} \] These maps retain the nerve, including its higher comparisons. A face or degeneracy induces a map \(g:Z_n\to Z_b\). Equivariance and the group action give the cartesian coefficient comparison for its proper source diagrams; the action face is cartesian because multiplication by the specified group element is an invertible map of the fibre. Proper base change gives the corresponding map between the \(K\)'s. The base-change part of C.17.3.1 applies to these pulled complexes and makes the ordinary-image and counit square commute. Its adjunction proof identifies this with the actual comparison, including its compositions. Consequently (C.17.12) is an isomorphism of the complete cartesian nerve diagrams, not merely a collection of isomorphisms on their cohomology groups. Restrict that diagram to a chosen all-distinct point and to the full collision. Proper base change identifies the two restrictions with the action-nerve complexes for the equivariant proper cohomology of the external product and the convolution. Both are the diagram \(Rp_{n,*}K_{r,n}\), through the same counit. Taking actual derived equivariant sections therefore gives a canonical isomorphism of their equivariant cohomology complexes: it applies the existing derived totalization to an isomorphism of the whole diagram, with no interchange of cohomology and an infinite limit. The coefficient maps from the constant nerve diagram of \(E\) commute with the counit and external cup. Thus it is \(R_0\)-linear. Its cohomology followed by the external cup isomorphism (C.17.6) gives (C.17.10). For three or more factors, use this same construction on \(\mathbb A^r\times Z_n\) at every nerve level. A block collision repeats coordinates and commutes with the projection to \(Z_n\). Lemma C.17.3.1 makes all corresponding section maps restrictions of the same counit. The actual proper block and unit maps are those proved in Lesson 8 C.3 and C.6, and are constant-\(G\)-equivariant by their bundle and away-frame constructions. Their comparisons, coefficients and cocycles therefore give the same maps on the nerve. They give the associator, unitors, and every ordered block compatibility in (C.17.10), without choosing different paths for different parenthesizations. On the distinct open, the unadjusted permutation is the derived external shuffle. Its cohomology sign is \((-1)^{ab}\) on degrees \(a,b\), by Lesson 8 C.5. Naturality of the same counit transports this sign to a collision. For objects on components of parities \(\epsilon_P,\epsilon_Q\), Lemma C.17.1.1 shows that every degree of their equivariant cohomology has these parities, since \(R_0\) is even. The corrected Satake symmetry multiplies the geometric shuffle by \((-1)^{\epsilon_P\epsilon_Q}\). The two signs cancel, giving the ordinary flip on the graded \(R_0\)-modules. Finite component decompositions retain all permutation and braid equations. Forgetting \(G\) uses the same coefficient, proper pullback, counit and external cup maps. Hence (C.17.2) specializes (C.17.10) to the ordinary cohomology tensor comparison, with its proved component adjustment. Semisimplicity of the genuine Satake heart makes every short exact sequence split, so this additive functor is exact. Its augmentation is the faithful ordinary fibre functor; therefore it is faithful. Freeness was proved in C.17.1. Finally the full genuine Satake equivalence of Lesson 11 B.4.1.1 supplies its rigid evaluation and coevaluation maps. Applying the established monoidal comparisons retains their two triangle identities. Thus \(\omega_G(P^\vee)\) is a dual of the finite graded free \(\omega_G(P)\); evaluation identifies that dual with the graded \(R_0\)-linear Hom into \(R_0\). This proves (C.17.11). ∎ #### C.17.5. Two coefficient-functor exercises **Exercise C.17.5.1 (basic).** In rank one, let \(S=E[u]\), \(|u|=2\), \(R_0=E[p]\), \(p=u^2\), and suppose the actual ordinary cohomology has basis degrees \(-2,0,2\). Determine the free equivariant module's Hilbert series and explain why a basis modulo \((p)\) determines a free basis after lifting. **Solution.** The torus Hilbert series is \[ \frac{z^{-2}+1+z^2}{1-z^2}. \] The basis \(1,u\) of \(S/R_0\) has degree polynomial \(1+z^2\). Division gives the group-equivariant series \[ \frac{z^{-2}+1+z^2}{1-z^4}. \] Lift the three homogeneous vectors modulo \((p)\). The induced map \(R_0[2]\oplus R_0\oplus R_0[-2]\to M\) is onto by the lowest-degree argument in C.17.1. Source and target have the displayed same finite dimension in each degree, so its kernel is zero. This proves freeness, while allowing different choices of lifts. **Exercise C.17.5.2 (intermediate).** Suppose \(P,Q\) have odd component parity. Compute the unadjusted and corrected exchange on homogeneous classes in \(\omega_G(P)\otimes_{R_0}\omega_G(Q)\). Explain why the affine-parameter complex comparison remains compatible with a further collision of three ordered factors. **Solution.** All coefficient degrees of \(R_0\) are even, so all degrees from \(P,Q\) stay odd. The unadjusted external exchange sends \(a\otimes b\) to \(-b\otimes a\). The corrected component scalar is also \(-1\), giving \(b\otimes a\), the ordinary flip. For three factors, write \(K_n=p_n^*C_n\) via the actual counit on \(\mathbb A^3\times G^n\). Both orders of two successive block merges are restrictions of this same pulled complex along maps over \(G^n\); their restrictions and counits compose identically. They commute with every nerve face and degeneracy. Proper block comparisons therefore give the same equivariant cohomology tensor map, with the ordered shuffle sign already accounted for. No independent derived splitting is chosen at a diagonal. ### C.18. A graded tensor trivialization and its Lie operators Keep \(R_0\) from C.17.1, and write \(H=\widehat G_E\) for the split dual reductive group of Reconstruction and the dual reductive group, B.4.1.1. In this section \(H\) is over the characteristic-zero coefficient field \(E\), independently of the characteristic of the geometric field \(k\). A grading has its ordinary tensor symmetry, as in C.17. The ordinary cohomology functor, denoted \(\omega\), retains its grading. #### C.18.1. Algebraizing an equivariant lift **Lemma C.18.1.1.** Let \(R=E[p_1,\ldots,p_s]\), with every \(|p_i|>0\), and let \(\mathfrak m=(p_1,\ldots,p_s)\). Let \(P\to\operatorname{Spec}R\) be a smooth affine morphism of finite presentation, equipped with an algebraic \(\mathbb G_m\)-action extending the degree action on \(R\). Every fixed \(E\)-point \(x\) over the origin extends to a \(\mathbb G_m\)-equivariant section. This section need not be unique. **Proof.** The case \(s=0\) is the given fixed point, so assume \(s>0\). Put \(R_n=R/\mathfrak m^n\). We construct compatible equivariant sections \(\sigma_n:\operatorname{Spec}R_n\to P\), starting with \(x\) at \(n=1\). The ideal \[ I_n=\mathfrak m^n/\mathfrak m^{n+1} \subset R_{n+1} \tag{C.18.1} \] is square zero and is annihilated by \(\mathfrak m\). First any \(\sigma_n\) has a lift without imposing equivariance. Its image is the single topological point \(x\). Use the standard smooth chart at that point from Smooth morphisms, Proposition 1.1: \[ \bigl(R[X_1,\ldots,X_b]/(F_1,\ldots,F_c)\bigr)_{h\Delta}, \qquad \Delta=\det(\partial F_i/\partial X_j)_{i,j\le c}. \tag{C.18.2} \] Lift the values of the \(X_j\) arbitrarily to \(R_{n+1}\). The errors \(F_i\) lie in \(I_n\). The displayed Jacobian matrix is invertible there: its reduction is a unit, and units lift across nilpotent ideals by the finite geometric inverse. Correct the first \(c\) values by the solution of the linear system whose right side is minus those errors. Taylor expansion has no terms of order two because \(I_n^2=0\), so all equations now vanish. The localization elements remain units. This gives a lift on the chart and hence on \(P\). Write \(A=\mathcal O(P)\). The difference between two lifts is an \(R\)-derivation \(A\to I_n\); conversely adding such a derivation to one lift gives another. Since \(\mathfrak mI_n=0\), the derivation factors through the relative cotangent space at \(x\). Thus the lift set is a nonempty affine space under \[ D_n=\operatorname{Hom}_E (\Omega_{A/R}\otimes_A E_x,I_n). \tag{C.18.3} \] This is finite dimensional. The action of \(\mathbb G_m\) on this affine space is algebraic and its linear part is the natural action on \(D_n\): use finitely many homogeneous generators of \(A\) and the finite-dimensional graded algebra \(R_{n+1}\) to write its values and equations. Every nonempty affine space with such an action has a fixed point. Here is the argument over \(E\). After choosing an origin, write the action as \(v\mapsto\rho(t)v+c(t)\). Its group law says \(c(tu)=c(t)+\rho(t)c(u)\). Therefore \[ (v,a)\longmapsto(\rho(t)v+a\,c(t),a) \] is a rational linear representation on \(D_n\oplus E\), with trivial quotient \(E\). A rational \(\mathbb G_m\)-representation is a direct sum of its integer weights: expand its coaction in the Laurent monomials \(t^j\), and coassociativity makes their coefficients mutually orthogonal projectors with sum one. The weight-zero projector applied to any vector mapping to \(1\) still maps to \(1\), because the quotient has weight zero. It therefore gives a fixed vector \((v,1)\). The corresponding lift is equivariant. This constructs \(\sigma_{n+1}\) extending the specified \(\sigma_n\). The inverse limit gives a graded \(R\)-algebra map \(A\to\widehat R=\varprojlim R_n\). It is polynomial rather than merely formal. Indeed, if \(d=\min_i|p_i|\), then \(\mathfrak m^n\) has no part of degree less than \(nd\). A compatible homogeneous value of weight \(j\ge0\) stabilizes in the finite-dimensional space \(R_j\) once \(nd>j\); a negative weight has value zero. Choose finitely many homogeneous \(R\)-algebra generators of \(A\). Each of their values consequently lies in \(R\). Every relation vanishes in all \(R_n\), hence in \(R\), since \(\bigcap_n\mathfrak m^n=0\) by the same degree argument. These values define a graded map \(A\to R\) splitting the structural map and specializing to \(x\). Its corresponding section proves the lemma. ∎ #### C.18.2. The tensor-isomorphism torsor, constructed from regular functions **Lemma C.18.2.1.** Let \(R\) be a commutative \(E\)-algebra. Let \(\mathcal C=\operatorname{Rep}_E(H)\), with its ordinary forgetful functor \(\omega_0\), and let \[ F:\mathcal C\longrightarrow\operatorname{Mod}_R \] be an exact \(E\)-linear symmetric tensor functor taking finite-dimensional representations to finite free \(R\)-modules. The tensor-isomorphism functor \[ \mathcal P(B)= \operatorname{Isom}^{\otimes} (B\otimes_R F,\ B\otimes_E\omega_0), \qquad R\to B, \tag{C.18.4} \] is represented by a faithfully flat, smooth affine \(H_R\)-torsor of finite presentation. **Proof.** Extend \(F\) to filtered unions by their module colimits. This extension is exact: filtered colimits of module sequences are exact, and each element of a kernel or image is already represented at a finite stage. Tensor products commute with these colimits. Give \(O=E[H]\) the left regular action \(h\cdot f(g)=f(h^{-1}g)\). It is a union of finite-dimensional subrepresentations. To see this directly, expand a function's coaction in a finite linearly independent set in its second tensor factor; coassociativity shows that the span of its first coefficients is stable. Set \[ A=F(O). \tag{C.18.5} \] Its multiplication and unit come from those of the equivariant algebra \(O\). The algebra \(A\) is finitely presented over \(R\). Choose finitely many algebra generators of \(O\) and a finite-dimensional stable space \(U\subset O\) containing them. The kernel of \(\operatorname{Sym}_E U\to O\) is a finitely generated ideal, by Noetherian and Artinian rings, Theorem 2.1. Choose a finite-dimensional stable space \(D\) in that kernel containing ideal generators. The equivariant sequence \[ (\operatorname{Sym}U)\otimes D \longrightarrow\operatorname{Sym}U \longrightarrow O\longrightarrow0 \tag{C.18.6} \] is exact at the last two terms. Since \(E\) has characteristic zero, symmetric powers are the images of the usual averaging projectors on tensor powers. Exactness and the symmetric tensor comparisons therefore give \[ A=\operatorname{Sym}_R F(U)/(F(D)). \tag{C.18.7} \] The modules \(F(U),F(D)\) are finite free, so this is a finite presentation. It is faithfully flat. Finite representations of \(H\) are semisimple by the full genuine Satake equivalence and semisimplicity proved in Lesson 11 B.4.1.1. Thus \(O\) is the union of finite semisimple subrepresentations, and \(A\) is the filtered colimit of their finite free images, so is flat. The invariants of the left regular action on \(O\) are exactly \(E\): an invariant regular function has the same value as at the identity on every test point, by left translation. On each finite subrepresentation containing \(E\), semisimplicity gives the canonical projection onto its invariant summand; there are no maps from a nontrivial simple to the trivial representation. These projections agree on overlaps and give an \(H\)-equivariant retraction \(O\to E\). Applying \(F\) makes \(R\) a module direct summand of \(A\). For every \(R\)-module \(M\), \(M\to A\otimes_R M\) consequently has a retraction. Flatness is therefore faithful. The equivariant scheme isomorphism \[ H\times H\longrightarrow H\times H,\qquad (g,h)\longmapsto(g,g^{-1}h) \tag{C.18.8} \] changes diagonal left translation into left translation on the first factor and trivial action on the second. Applying \(F\) to its coordinate map gives an algebra isomorphism \[ A\otimes_R A\simeq A\otimes_E E[H]. \tag{C.18.9} \] Its right \(H\)-coaction comes from right translation on \(O\), which commutes with left translation. The identity, multiplication and inverse equations in (C.18.8) show that (C.18.9) is precisely the torsor comparison \(P\times H_R\simeq P\times_R P\), for \(P=\operatorname{Spec}A\). Together with faithful flatness this gives an \(H_R\)-torsor. We check that it represents (C.18.4), rather than relying on a torsor identification alone. For a representation \(V\), the vector-valued function \[ v\longmapsto\bigl(g\longmapsto g^{-1}v\bigr) \tag{C.18.10} \] is an equivariant map \(V\to O\otimes_E\omega_0(V)\), with trivial action on the second factor. Multiplication by \(O\) extends it to the equivariant isomorphism \[ O\otimes V\simeq O\otimes\omega_0(V),\qquad f(g)v\longmapsto f(g)g^{-1}v. \tag{C.18.11} \] Its inverse uses \(gv\). Applying \(F\) gives \(A\otimes_R F(V)\simeq A\otimes_E\omega_0(V)\). Substituting an algebra map \(A\to B\) gives the isomorphisms of (C.18.4). The equation \(g^{-1}(v\otimes w)=g^{-1}v\otimes g^{-1}w\) proves their tensor compatibility, and the identity representation proves unit compatibility. They are natural in \(V\). Conversely extend a tensor isomorphism to \(O\) by filtered unions and compose it with evaluation at the identity to obtain an algebra map \(A\to B\). Naturality applied to (C.18.10) recovers its isomorphism on every \(V\). Applied to \(O\), (C.18.11) and the coordinate counit recover the original algebra map. The two constructions are inverse on every \(B\), proving representability. Finally \(P\to\operatorname{Spec}R\) is smooth. Its geometric fibre over any point has a point, since it is a nonzero finitely generated algebra over an algebraically closed field; The Nullstellensatz and Jacobson rings, Theorem 1.3, proves that the residue field of a maximal ideal is that field. Choosing such a point in the torsor comparison identifies that fibre with the smooth group \(H\). The morphism is flat and finitely presented, and has smooth geometric fibres, so Smooth morphisms, Theorems 2.1 and 3.1, makes it smooth. This completes the construction. ∎ #### C.18.3. A tensor basis for actual equivariant cohomology **Theorem C.18.3.1.** There exists a graded \(R_0\)-linear symmetric tensor isomorphism \[ \tau_P:\omega_G(P)\xrightarrow{\sim} R_0\otimes_E\omega(P) \tag{C.18.12} \] natural in the genuine rational Satake heart. It specializes at the coefficient origin to the actual augmentation of C.17.2. The grading on the right is the sum of the coefficient grading and the ordinary cohomology grading. This simultaneous tensor choice is not asserted to be canonical. **Proof.** Identify the heart with \(\operatorname{Rep}_E(H)\) by Lesson 11 B.4.1.1 and its specified ordinary fibre functor. C.17 supplies the exact symmetric tensor functor \(F=\omega_G\) with finite free values. Forget the gradings temporarily and apply C.18.2. Its tensor-isomorphism torsor is smooth, affine and finitely presented over \(R_0\). The gradings of \(F(V)\) and \(R_0\otimes\omega(V)\) give semilinear algebraic \(\mathbb G_m\)-actions covering the positive coefficient-degree action on \(\operatorname{Spec}R_0\). The ordinary grading of \(\omega\) is a tensor automorphism family, hence a cocharacter of \(H=\operatorname{Aut}^{\otimes}(\omega)\) by the all-test-algebra reconstruction of Lesson 11. These two graded actions act by conjugation on the tensor-isomorphism functor. This is an algebraic action on its representing scheme: (C.18.10)–(C.18.11) express it on the finite generating coefficient spaces in (C.18.7); the degree actions and the ordinary cocharacter are algebraic there. Tensor and naturality equations are preserved. Actual augmentation gives an \(E\)-point of this torsor over the origin. It is fixed, because C.17.2 and its tensor comparison preserve the ordinary grading. By C.14 the coefficient ring is a polynomial ring in strictly positive homogeneous degrees. Lemma C.18.1.1 therefore extends that point to an equivariant section. Its corresponding tensor isomorphism preserves the gradings: equivariant module maps for \(\mathbb G_m\) preserve the weight projectors of the coaction. It is precisely (C.18.12), and its origin value is the specified augmentation. ∎ #### C.18.4. Tensor derivations in dual Lie-algebra coordinates Fix one \(\tau\) from C.18.12, and let \(\vartheta:\mathbb G_m\to H\) be the ordinary cohomology grading cocharacter. A tensor derivation of degree \(j\) means a natural family of \(R_0\)-linear degree-\(j\) maps \(d_P:\omega_G(P)\to\omega_G(P)\) such that \(d_{\mathbf1}=0\) and, under C.17.10, \[ d_{P*Q}=d_P\otimes1+1\otimes d_Q. \tag{C.18.13} \] The formula is the ordinary tensor formula for the corrected symmetry. **Lemma C.18.4.1.** Under \(\tau\), such families are exactly elements of \(R_0\otimes_E\mathfrak h\), where \(\mathfrak h=\operatorname{Lie}H\). A family of degree \(j\) is an element of total degree \(j\), with the coefficient grading on \(R_0\) and the \(\operatorname{Ad}\vartheta\)-weight grading on \(\mathfrak h\). Commutators of families correspond to Lie brackets. **Proof.** Work first without gradings and set \(B=R_0[\epsilon]/(\epsilon^2)\). The maps \(1+\epsilon d_P\) are invertible with inverse \(1-\epsilon d_P\). Naturality, the unit condition and (C.18.13) say exactly that they are a tensor automorphism of the base-changed ordinary fibre functor, reducing to the identity over \(R_0\). By the same all-test-algebra reconstruction, \[ \ker\bigl(H(B)\to H(R_0)\bigr) =\mathfrak h\otimes_E R_0 \tag{C.18.14} \] as an additive first-order group. Explicitly a coordinate map \(E[H]\to B\) reducing to the identity's counit has the form \(f\mapsto e(f)+\epsilon\,\delta(f)\). Multiplicativity says \(\delta(fg)=e(f)\delta(g)+e(g)\delta(f)\); thus \(\delta\) is a linear functional on the cotangent space at the identity with values in \(R_0\). That cotangent space is finite dimensional, so these functionals are exactly \(\mathfrak h\otimes_E R_0\). This also proves the asserted equality without a field-point restriction. For \(\xi\) so obtained, its action in \(V\) is the differential of the representation \(H\to GL(V)\). Coordinate comultiplication on a tensor product gives the sum of the two differentials, proving the converse and (C.18.13). These actions recover every \(d_P\) by (C.18.14). Give \(\epsilon\) degree \(-j\). Requiring \(1+\epsilon d\) to preserve the grading is exactly requiring \(d\) to have degree \(j\). A vector \(x\in\mathfrak h\) of \(\operatorname{Ad}\vartheta\)-weight \(w\) sends \(\omega(V)^n\) into \(\omega(V)^{n+w}\), by differentiating the conjugation equation for \(\vartheta\). Hence \(a x\), for homogeneous \(a\in R_0\), acts in degree \(|a|+w\). The weight decomposition of \(\mathfrak h\) proves the total-degree assertion. Differentials of representations carry the Lie bracket to the matrix commutator, proving its compatibility on all objects; the coordinate first-order description makes these families faithful on \(\mathfrak h\). ∎ In particular, a geometric cup operator can be put into these coordinates once its actual tensor derivation formula has been proved. C.18.4 does not assume that formula for an arbitrary class. Identifying the specific cup operators, their normalizations and their Kostant-slice coordinates remains the next step. #### C.18.5. Two graded-section exercises **Exercise C.18.5.1 (basic).** Let \(R=E[p]\), \(|p|=4\), and \[ A=R[x,y,z]/(y-x^2-p),\qquad |x|=2,\quad |y|=4,\quad |z|=-2. \] Find every graded section through the origin of \(\operatorname{Spec}A\to\operatorname{Spec}R\). Explain why the presence of a negative coordinate does not obstruct C.18.1. **Solution.** The ring is \(R[x,z]\), so the map is smooth and the origin is fixed. A graded value of \(x\) must be in \(R_2=0\), and a graded value of \(z\) in \(R_{-2}=0\). Thus \(x=z=0\), and the relation forces \(y=p\). This is the unique graded section. In the formal construction all negative-degree values are zero; every fixed nonnegative degree stabilizes to its finite polynomial part. Negative weights on the total space therefore cause no formal-series obstruction. **Exercise C.18.5.2 (intermediate).** Let \(H=SL_2\), \(R=E[p]\) with \(|p|=2\), and let the standard representation have basis \(v_+,v_-\) of degrees \(1,-1\). Show that a tensor trivialization with fixed augmentation can still be nonunique. **Solution.** For every \(a\in E\), the \(SL_2(R)\)-matrix \[ g_a(p)= \begin{pmatrix}1&0\\ap&1\end{pmatrix} \tag{C.18.15} \] acts by \(v_+\mapsto v_++ap\,v_-\), \(v_-\mapsto v_-\). The added term has degree \(2-1=1\), so the matrix is graded, and at \(p=0\) it is the identity. Its determinant is one and its inverse is \(g_{-a}\). More intrinsically, for \(\vartheta(t)=\operatorname{diag}(t,t^{-1})\), \[ \vartheta(t)g_a(t^2p)\vartheta(t)^{-1}=g_a(p). \] Applying any representation proves its compatibility with that representation's semilinear degree action. These group elements therefore give graded natural tensor automorphisms on all representations. Composing a trivialization with \(g_a\) preserves its specified augmentation, while distinct \(a\)'s already act differently on the standard representation. ∎ ### C.19. The determinant cup operator is a regular Lie element Fix the closed faithful representation \(\rho:G\hookrightarrow GL(V)\) used in Loop groups and the affine Grassmannian, Theorem 7.1 and Lemmas 7.2–7.12. Its finite-stage closed embeddings retain nilpotents in every geometric characteristic. Write \(b=\dim_kV\), \(O_A=A[[t]]\), and \(F_A=A((t))\), for a parameter algebra \(A\). The coefficient field remains the finite characteristic-zero extension \(E/\mathbb Q_\ell\). The representation's integral torus weights will be used over \(E\); no trace-form nondegeneracy over \(k\) is assumed. #### C.19.1. Relative determinants on the full lattice functor **Lemma C.19.1.1.** The formula \[ \mathscr D_\rho(L)= \det_A(L/t^NO_A^b)^{-1}\otimes \det_A(O_A^b/t^NO_A^b) \tag{C.19.1} \] defines a line bundle on \(\operatorname{Gr}_G\), independent of a sufficiently large bound \(N\), on the full scheme functor. It has genuine positive-loop equivariance and a canonical equivariant trivialization at the unit lattice. For three commensurable lattices its relative version has associative isomorphisms \[ \mathscr D(L_2,L_0)= \mathscr D(L_1,L_0)\otimes \mathscr D(L_2,L_1). \tag{C.19.2} \] Determinants may be retained as graded lines: \(\det M\) has integer degree \(\operatorname{rk}M\), and their interchange has the sign prescribed by these degrees. **Proof.** Lesson 2 Lemma 2.1 makes every bounded lattice quotient finite projective over \(A\). In particular \(L/t^NO_A^b\) is finite projective: it is the kernel of the quotient between the finite bounding window and its projective quotient by \(L\), so that exact sequence splits over \(A\). This argument is valid before reduction and after every base change. For a finite projective module \(M\), define \(\det M=\bigwedge^{\operatorname{rk}M}M\) separately on its open-and-closed rank loci. If \(0\to M'\to M\to M''\to0\) is exact with all terms finite projective, a local splitting and an ordered basis give \(\det M=\det M'\otimes\det M''\). A second splitting changes the concatenated basis by a block triangular matrix of determinant one. Thus the comparison is independent of that splitting and glues. For a longer filtration the two successive comparisons both concatenate the same ordered bases, proving associativity. Interchanging two blocks has wedge sign \((-1)^{\operatorname{rk}M'\operatorname{rk}M''}\), explaining the graded-line convention. If a common lower lattice \(U\) is replaced by \(U'\subset U\), the exact sequences for both lattices add the same determinant \(\det(U/U')\). In the relative ratio \[ \mathscr D(L_i,L_j)= \det(L_i/U)^{-1}\otimes\det(L_j/U) \tag{C.19.3} \] these extra factors cancel by their evaluation. The filtration calculation proves compatibility for successive smaller \(U'\). Any two bounds have a smaller common bound, so this gives a choice-free relative line. Cancellation of the middle determinant proves (C.19.2), with the indicated ordering and graded interchanges. Its three-factor associativity is the same cancellation and ordered-basis equality. This constructs (C.19.1) on every finite window and identifies its restrictions through all transitions. A positive integral matrix preserves \(O_A^b\) and \(t^NO_A^b\). It acts on both finite quotients and hence on their determinant ratio. The action law is the determinant action law, and the bound comparisons are equivariant. On a fixed window the action factors through a sufficiently deep genuine jet quotient, since a matrix congruent to one modulo \(t^{2N}\) acts trivially on \(t^{-N}O_A^b/t^NO_A^b\). At \(L=O_A^b\), evaluation of the two identical determinant factors gives the canonical unit frame, also preserved by that action. All these maps commute with base change. Descent through the full faithful-representation lattice charts therefore gives the stated line on \(\operatorname{Gr}_G\), with its actual equivariance. ∎ #### C.19.2. The moving line and independent-disc factorization **Lemma C.19.2.1.** On every ordered moving chain \(Y_r\to\mathbb A^r\) of Fusion and the symmetric tensor structure, C.1–C.6, there is a constant-\(G\)-equivariant determinant line of its last bundle relative to the standard bundle. On the distinct-point locus it is the ordered tensor product of the individual \(\mathscr D_\rho\)'s. On the full collision it is the pullback of (C.19.1) by the convolution endpoint. All ordered block, bound and unit comparisons agree with (C.19.2), on every parameter algebra. **Proof.** In the affine coordinate write \[ f(t)=\prod_{i=1}^r(t-a_i),\qquad u=f(t). \tag{C.19.4} \] The monic construction of Lesson 8 §1 identifies the completed disc ring with a finite free \(A[[u]]\)-algebra, with basis \(1,t,\ldots,t^{r-1}\). The associated last \(V\)-bundle lattice \(L\) and standard lattice \(L_0\) have common finite bounds \(f^ML_0\subset L\subset f^{-M}L_0\). That lesson's operator-kernel and quotient calculation gives finite projective \(A\)-modules \(L/f^ML_0\) and \(L_0/f^ML_0\). Their determinant ratio is therefore a line on the full finite moving scheme. Smaller common lower lattices give the same canonical comparisons proved in C.19.1. It descends through the actual formal-frame charts; the away frame fixes the standard lattice. The construction is constant-\(G\)-equivariant. On the distinct-point locus the ideals \((t-a_i)\) are pairwise comaximal. CRT and completion identify the disc ring with the product of the individual completed discs. Locally on the \(i\)-th disc, the other factors of \(f\) are units, so \(f^ML_0=(t-a_i)^ML_0\) as a submodule. Both finite quotients are the corresponding ordered direct sums. The exact determinant direct-sum maps of C.19.1 identify their relative ratio with the product of the individual relative lines, including its graded interchange signs. For a separated collection of blocks the same CRT argument uses their monic block polynomials. When labels inside a block collide, its polynomial becomes a power of \(t-a\). The corresponding lower lattice is \((t-a)^{M|\text{block}|}L_0\), which is just a larger bound in C.19.1. Thus its specialization is the single-disc line on the actual last bundle. If all labels collide, this is the convolution endpoint's line. Regrouping a chain compares its successive relative lattices by (C.19.2); associativity of determinant filtrations and of CRT makes those comparisons agree before and after further collisions. The empty chain has \(L=L_0\), with its specified unit frame. No section displacement or infinitesimal lattice direction has been discarded. ∎ #### C.19.3. The actual primitive cup map Set \[ c_\rho=c_1(\mathscr D_\rho) \in H_G^2(\operatorname{Gr}_G,E(1)). \tag{C.19.5} \] This notation means the compatible actual Kummer classes on the bounded stages; its action on a supported object is computed on that support. **Theorem C.19.3.1.** Cup with \(c_\rho\) gives a natural degree-two operator \[ d_{\rho,P}:\omega_G(P)\longrightarrow\omega_G(P)(1) \tag{C.19.6} \] with its displayed Tate line. Its actual tensor rule is \[ d_{\rho,P*Q} =d_{\rho,P}\otimes1+1\otimes d_{\rho,Q} \tag{C.19.7} \] under C.17.10. It is zero on the unit and retains all ordered block comparisons. After choosing \(\tau\) in C.18.12 it corresponds to an element \[ \xi_\rho\in R_0\otimes_E\mathfrak h\otimes_E E(1) \tag{C.19.8} \] of total degree two. **Proof.** The Kummer connecting map is additive on line tensors and negates inverse lines. At each finite coefficient level this follows by multiplying or inverting the frame transitions and their root cocycles in the actual exact coefficient sequence; the connecting map is a homomorphism. The completed coefficient construction of The Satake category and IC sheaves, A.1–A.2, retains those same maps and their inverse-limit homotopies. Thus the equality is for the actual complete class, with no interchange of invariants and reductions. It holds also on the equivariant nerve because the line comparisons are equivariant. On \(Y_r\), cup with the moving line is a morphism \(A_r\to A_r(1)[2]\). Proper image gives the same morphism of each \(K_{r,n}\) on \(\mathbb A^r\times G^n\) in C.17.12. That complex is in the essential image of the affine pullback. The full faithfulness proved in C.17.3 therefore lifts this morphism uniquely to \(Rp_{n,*}K_{r,n}\to Rp_{n,*}K_{r,n}(1)[2]\). Its natural counit square commutes. Those lifts commute with every action face and degeneracy by the actual base-change and adjunction comparisons of that lemma. Consequently the operator commutes with the complete nerve transport and its derived sections. At a distinct-point section, C.19.2 and Kummer additivity make this the sum of the external individual cup maps. At a collision it is the convolution endpoint cup map, by proper base change and the line specialization. The external cup map of C.17.2 identifies those two descriptions under C.17.10, proving (C.19.7). Its degree is even, so no external Koszul sign enters this sum. The same proof for \(r\) factors and their block maps proves the ordered block compatibility. The unit frame makes its Kummer class zero on the unit. Tensoring a class with an object commutes with every coefficient morphism, proving naturality. Apply C.18.4 to this tensor derivation. To keep its coefficient line, choose temporarily a nonzero vector in \(E(1)\) and divide the operator by that vector; the lemma gives its unique Lie element. Changing the vector rescales that element inversely, so their tensor is the well-defined (C.19.8). This proves the Tate-valued assertion and its total degree. ∎ #### C.19.4. Fixed-point characters and localized projectors List the torus weights \(\chi_1,\ldots,\chi_b\) of \(V\), with multiplicity, and define \[ B_\rho(\lambda,\mu) =\sum_{j=1}^b \langle\chi_j,\lambda\rangle \langle\chi_j,\mu\rangle,\qquad \psi_\lambda=\sum_j\langle\chi_j,\lambda\rangle\chi_j. \tag{C.19.9} \] The pairing is regarded on the integral coweight lattice and then over \(E\). **Lemma C.19.4.1.** The constant-\(T\) character of \(\mathscr D_\rho\) at \(t^\lambda\) is \(\psi_\lambda\). The pairing \(B_\rho\) is Weyl invariant and nondegenerate over \(E\), including the entire central torus, and \(\psi_\lambda\ne\psi_\mu\) for distinct coweights. Put \(Q=\operatorname{Frac}S\). On a proper spherical support \(X\), the actual restriction identifies \[ Q\otimes_S H_T^*(X,P) \simeq \bigoplus_{t^\lambda\in X^T} Q\otimes_S H_T^*(t^\lambda,i_\lambda^*P). \tag{C.19.10} \] There \(d_{\rho,P}\) is scalar cup with \(c_1(\psi_\lambda)\) on the \(\lambda\)-summand. Its localized Lagrange polynomials are the projectors onto those summands. **Proof.** In a weight basis, \(t^\lambda V(O)\) has \(j\)-th lattice \(t^{\langle\chi_j,\lambda\rangle}O\). At a large bound \(N\), the first quotient in (C.19.1) has \(N-\langle\chi_j,\lambda\rangle\) vectors of character \(\chi_j\), and the second has \(N\). The determinant inverse cancels the \(N\) terms and leaves precisely \(\langle\chi_j,\lambda\rangle\chi_j\). This proves the character and hence its actual Kummer restriction. Lesson 2 Lemma 7.10 proves that these weights generate \(X^*(T)\), including detection of non-smooth finite kernels. On the real coweight space the Gram sum in (C.19.9) is strictly positive on every nonzero vector. Its matrix is integral, with nonzero rational determinant; it therefore stays nondegenerate over the characteristic-zero field \(E\), also when \(k\) has positive characteristic. Normalizer conjugation permutes the weight multiset, proving Weyl invariance. Nondegeneracy makes \(\lambda\mapsto\psi_\lambda\) injective, so distinct coweights give distinct nonzero differences as linear elements of \(S\). To prove (C.19.10) for these coefficients, remove the fixed points from the finite Iwahori-cell filtration of \(X\). On each cell the constant-parity restrictions split equivariantly as in C.10. Its punctured linear cell has localized compact cohomology zero by the supported Euler proof C.5.1: every root weight is a nonzero polynomial in \(S\), and their product becomes invertible in \(Q\). Finite sums and shifts retain this vanishing. Closed–open induction gives \(QH_{T,c}^*(X\setminus X^T,P)=0\). Since \(X\) is proper, its actual localization triangle now gives (C.19.10). Cup with the line restricts on a point to its character class, so acts by the asserted scalar. Choose a Tate vector only to write those scalars as \(\nu_\lambda\in S^2\). For the finitely many fixed labels, the polynomial \[ \prod_{\mu\ne\lambda} \frac{d_\rho-\nu_\mu}{\nu_\lambda-\nu_\mu} \tag{C.19.11} \] has value one on the \(\lambda\)-summand and zero on every other summand. All denominators are nonzero in \(Q\). These are therefore exactly its actual localized projectors; they sum to one and are mutually orthogonal. Each ratio has total degree zero, and a change of Tate vector cancels in it. This proves the claim without extending a polynomial identity from ordinary fixed-point stalk dimensions. ∎ #### C.19.5. A nonzero top cup power on the actual IC object **Lemma C.19.5.1.** Let \(X=\overline{\operatorname{Gr}}^\lambda\) have dimension \(d=h_G(\lambda)\). Then \[ c_\rho^d:H^{-d}(X,IC_\lambda) \longrightarrow H^d(X,IC_\lambda)(d) \tag{C.19.12} \] is nonzero. This is the actual ordinary cup map; no hard Lefschetz theorem for singular IC objects is used. **Proof.** For \(d=0\), \(X\) is its point support and the assertion is the identity, so assume \(d>0\). On its fixed component and a sufficiently large finite window, C.19.1 is the inverse determinant of the tautological subbundle times a constant line. It is therefore the pullback of the Plücker \(\mathcal O(1)\) under the finite closed embedding of Lesson 2 Theorem 7.1. In particular it is very ample on \(X\). Let \(U\simeq\mathbb A^d\) be the dense Iwahori cell and \(i:Z=X\setminus U\to X\). There exists an actual \(T\)-equivariant map \[ u:E_X[d]\longrightarrow IC_\lambda \tag{C.19.13} \] restricting to the identity on \(U\). Here is its lifting proof. On every cell of \(Z\), the ordinary restriction of \(E_Z[d]\) is constant in parity \(d\), and the exceptional restriction of \(i^!IC_\lambda\) is constant in that same parity by Lesson 11 C.4.1.1. On a cell, the constant-system equivariant Ext ring is the even polynomial ring \(S\), as proved in C.10. Thus their odd Ext vanishes. Induct on the closed cell filtration using the source ordinary and target exceptional restrictions, as in C.16.2; the two outer odd Ext groups are zero, so the middle odd group is zero. In particular \(\operatorname{Ext}_T^1(E_Z[d],i^!IC_\lambda)=0\). The target localization triangle \[ i_*i^!IC_\lambda\to IC_\lambda \to Rj_*E_U[d]\to i_*i^!IC_\lambda[1] \] therefore lifts the ordinary open unit to (C.19.13). No costalk-parity assertion for the constant sheaf on the singular \(X\) is needed. Raw Verdier self-duality gives \(DIC_\lambda=IC_\lambda(d)\). Dualize \(u\) and untwist to obtain \[ v:IC_\lambda\longrightarrow\omega_X(-d)[-d]. \tag{C.19.14} \] On \(U\) these two maps have the actual constant unit and smooth orientation normalizations. We supply the finite linear-section argument for their top trace. Embed \(X\) by this very ample line in \(\mathbb P^M\). A general ordered \(d\)-tuple of hyperplanes has finite nonempty intersection with \(X\), entirely inside \(U\), and transverse there. For the avoidance and transversality assertions, a fixed point imposes one independent condition on each hyperplane. The incidence over \(Z\), of dimension at most \(d-1\), has dimension at most \(dM-1\). Over a smooth point of \(U\), hyperplane derivatives surject onto the cotangent space because the map is an immersion. A dependent \(d\)-tuple of derivatives is cut out by a nonzero determinant; its incidence again has dimension at most \(dM-1\). The closures of the images of these two incidences consequently miss a nonempty open in the \(dM\)-dimensional hyperplane parameter space. The dimension bound follows on each component by injection of function fields into those of its dominating incidence component. Nonemptiness of the projective intersection also has an elementary proof. The homogeneous coordinate domain of the affine cone has dimension \(d+1\); away from its vertex its affine charts are line charts times \(\mathbb G_m\). Krull dimension and Noether normalization, Theorem 4.2 and Corollary 4.4, make its vertex maximal ideal have height \(d+1\). An ideal generated by the \(d\) hyperplane forms cannot have that vertex as its only prime, by Dimension theory of Noetherian local rings, Theorem 3.1. If its projective zero set were empty, localization at every degree-one coordinate would be zero; every such coordinate would be nilpotent in the quotient, making the vertex its only prime. This contradiction proves nonemptiness. A tuple outside the two incidence images thus has a nonempty finite reduced intersection \(Z'\subset U\). Each hyperplane section has the global supported Kummer class of its divisor, whose forget-support image is \(c_1(\mathscr D_\rho)\). This is the relative class and orientation construction of Lesson 6 A.1–A.2, pulled from the ambient hyperplane line. Their supported product has support in \(Z'\). At each of those points the \(d\) independent parameters give the iterated normalized section class, of trace \(1\). On that support \(u\) and \(v\) are precisely the constant identity and smooth orientation. Therefore actual supported cup, forget support and proper trace give \[ \operatorname{Tr}\bigl(v(d)\,c_\rho^d\,u(1)\bigr) =|Z'|\ne0 \quad\text{in }E. \tag{C.19.15} \] Here \(u(1)\) is the image of the constant global section in degree \(-d\); the notation \(v(d)\) retains the displayed Tate cancellation. All point traces have the same positive Kummer normalization. The coefficient field has characteristic zero, so this positive integer is nonzero even in bad geometric characteristic. The composite factors through (C.19.12), proving the lemma. ∎ #### C.19.6. The regularity of the geometric Lie element **Theorem C.19.6.1.** Choose a Tate vector and a graded tensor trivialization in C.18.12 to express (C.19.8) as an \(R_0\)-valued Lie element. Its origin value is a regular principal nilpotent, and every geometric value is regular. This assertion holds for the full dual reductive group, including its centre. Identifying its adjoint quotient with the specified coefficient coordinates, and identifying all higher Kostant-gradient operators, are additional steps. **Proof.** The ordinary cohomology grading cocharacter is \(\vartheta=2\rho_G\) in the dual torus, by the weight degrees and dual-root identification of Lesson 11 B.1–B.4. Degree two in C.18.4 implies \[ \xi_\rho(0)=\sum_i a_i e_i, \tag{C.19.16} \] where \(e_i\) are the simple positive root vectors of \(H\). Indeed the Cartan has grading zero, the simple roots have grading two, and a positive root of height at least two has grading at least four; negative roots have negative grading. At the origin every positive-degree coefficient vanishes, leaving exactly the degree-two root spaces. If the derived root system is empty, every Lie element of the torus is regular, so the conclusions already hold. Otherwise choose a strictly dominant integral highest \(H\)-weight \(\lambda\) in the root lattice; it corresponds to an allowed \(G\)-coweight. For example \(2\rho_H\) has this property. On \(\omega(IC_\lambda)\), the degrees \(-d,d\), with \(d=\langle2\rho_G,\lambda\rangle\), are its one-dimensional lowest and highest weight spaces, of weights \(w_0\lambda,\lambda\). This is the full highest-weight and weight-degree theorem of Lesson 11. Every simple-root coefficient in (C.19.16) is nonzero. Suppose \(a_i=0\). Every monomial in its \(d\)-th power then raises the lowest weight by a sum of simple roots omitting \(\alpha_i\). It cannot reach the highest weight. To check that the latter difference uses every simple root, take a reduced word \(w_0=s_{i_1}\cdots s_{i_N}\). Telescoping gives \[ \lambda-w_0\lambda =\sum_{k=1}^N \langle\lambda,\alpha_{i_k}^{\vee}\rangle s_{i_1}\cdots s_{i_{k-1}}\alpha_{i_k}. \] The inversion roots in this reduced word are all positive roots: at each prefix, a simple reflection permutes the positive roots other than its simple root, so the inverse inversion set gains exactly the new positive prefix root; for the longest element that set is the entire positive system. All displayed coefficients are positive by strict dominance. In particular their simple-root coefficients include each simple root with a positive coefficient. The reduced-word and inversion-root facts have the complete root proof in Finite-dimensional Lie theory, §5. Therefore the hypothesized missing \(a_i\) makes the map from degree \(-d\) to degree \(d\) zero. Actual augmentation identifies that map with the ordinary \(c_\rho^d\) of C.19.12, contradicting C.19.5.1. The simple characters are a basis of the character lattice of the adjoint torus. Their nonzero values can therefore rescale every \(a_i\) to one, over an algebraic closure or by the split adjoint torus itself. Thus \(\xi_\rho(0)\) is adjoint-conjugate to the principal \(e=\sum_i e_i\), proved regular in B.2. The central summand is zero at the origin and adds its full rank to the centralizer, so causes no exception. For global regularity, C.18.4 gives the precise homogeneous identity \[ \xi_\rho(t\cdot p) =t^2\operatorname{Ad}_{\vartheta(t^{-1})}\xi_\rho(p). \tag{C.19.17} \] The nonregular locus is closed by the minors of the centralizer matrix and is invariant under conjugation and nonzero scalar multiplication, as in B.3. Its inverse image in \(\operatorname{Spec}R_0\) is therefore a closed subset stable under the positive grading action. If it had a geometric point \(p\), the map \(t\mapsto t\cdot p\) would extend to \(\mathbb A^1\), with value the coefficient origin at zero. Closedness would put that origin in the nonregular inverse image. This contradicts the principal origin value just proved. The inverse image is empty, so every geometric value is regular. ∎ ![Relative determinant cancellation, the actual collision cup sum, and the principal-origin regularity argument](assets/determinant-cup-regularity.png) The common lower lattice retains the full finite quotients. The moving line transports its degree-two Tate-valued cup map through the collision. Its nonzero top IC cup power forces every simple-root coefficient at the origin, and positive grading propagates regularity. Proofs C.19.1.1–C.19.6.1. The disc symbols and the final grading arrow are schematics of those proved comparisons. Editable SVG source. #### C.19.7. Two determinant-operator exercises **Exercise C.19.7.1 (basic).** Take \(G=\mathbb G_m\) and its identity character representation. Compute the determinant character and primitive cup action on the point-supported object at label \(n\), and compare labels \(n,m,n+m\). **Solution.** For \(N>|n|\), the two ranks in (C.19.1) are \(N-n,N\), so their character difference is \(n\chi\). If \(u=c_1(\chi)\), the action on its module \(S\) is multiplication by \(nu\), with its Tate line. Labels add under convolution, and \((n+m)u=nu+mu\) is exactly (C.19.7). At label zero the line has its canonical unit frame and the operator is zero. These supported cohomology objects use the reduced points, while the line was constructed on the full torus Grassmannian including its nilpotent directions. Integrally the operator \(nu\) alone does not recover the degree-zero label operator \(n\) by division by \(u\); that division requires localization. **Exercise C.19.7.2 (intermediate).** Take \(G=GL_2\), its standard representation, and the minuscule \(P=E_{\mathbb P^1}[1]\). Choose a Tate vector for the scalar formulas. Let \(u_1,u_2\) be the two character classes and \(\sigma_1=u_1+u_2,\sigma_2=u_1u_2\). Compute the determinant cup matrix and its normalized rank-one companion form. **Solution.** At the support \(tO^2\subset L\subset O^2\), the tautological line is \(L/tO^2\). Formula (C.19.1) is its inverse times \(\det(k^2)\). If its class is \(a\), put \(x=\sigma_1-a\). The relative flag Chern-basis calculation of C.14 gives \[ H_T^*(\mathbb P^1,E)= S[a]/((a-u_1)(a-u_2)) =S[x]/(x^2-\sigma_1x+\sigma_2). \] The two fixed values of \(x\) are \(u_1,u_2\), agreeing with (C.19.9). The invariant basis \(1,x\) descends through C.14 to the \(R_0=E[\sigma_1,\sigma_2]\)-basis of \(H_G^*(P)\), in degrees \(-1,1\). Actual cup with \(x\) has matrix \[ \begin{pmatrix}0&-\sigma_2\\1&\sigma_1\end{pmatrix}. \tag{C.19.18} \] Set \(y=x-\sigma_1/2\) and \(c=\sigma_1^2/4-\sigma_2\). In the ordered homogeneous basis \((y,1)\), of degrees \(1,-1\), the matrix becomes \[ \frac{\sigma_1}{2}I+ \begin{pmatrix}0&1\\c&0\end{pmatrix}. \tag{C.19.19} \] The second matrix is the rank-one Kostant companion, with \(c\) of degree four; the scalar part retains the full \(GL_2\) centre. At the coefficient origin its raising map is nonzero and its square is zero. The eigenvalues are \(u_1,u_2\), and their difference is nonzero after localization. The basis change is graded and has determinant a unit. These are actual cup matrices on this geometric module; they do not identify all higher operators for a general group. ### C.20. Fixed-point weights and the actual Kostant quotient Retain the coefficient rings, full dual group \(H\), faithful representation \(\rho\), and actual determinant cup element of Sections 17–19. In this section choose one nonzero vector of \(E(1)\) for the scalar Lie and polynomial formulas. The conclusions with their Tate lines follow by undoing that choice. Write \[ L=X_*(T_G)=X^*(T_H),\qquad \mathfrak t_G=X_*(T_G)\otimes E,\qquad \mathfrak t_H=\operatorname{Hom}_{\mathbb Z}(L,E). \tag{C.20.1} \] The trace pairing from C.19.4 defines the isomorphism \[ \beta_\rho:\mathfrak t_G\longrightarrow\mathfrak t_H,\qquad \lambda(\beta_\rho(u)) =\sum_{\chi\text{ a weight of }V_\rho} \langle\chi,\lambda\rangle\,\chi(u). \tag{C.20.2} \] Multiplicities occur in this sum. It is \(W\)-equivariant and invertible, including on the centre, by C.19.4. The cohomological degree of a linear coordinate is two. Thus \(\beta_\rho\) identifies the positively graded quotient bases. We first check which multiplicities occur in the localized geometric operator. ![Compact attracting-slice localization, invariant trace tests and the adapted graded frame](assets/geometric-kostant-quotient.png) The first panel concerns the reduced cohomological support of the finite attracting scheme: invertible character coordinates kill its punctured compact image only after extension to \(Q\). The second uses the resulting actual weight multiplicities to test every invariant polynomial and then descends along the injective coefficient map. The third shows the full smooth conjugation scheme and the precise grading sign that permits a polynomial section through the identity. Proofs C.20.1.1–C.20.4.1 establish these maps; Exercise C.20.5.2 explains why adapting the rational pinning matters. Editable SVG source. #### C.20.1. Compact attraction computes the fixed-point rank **Lemma C.20.1.1.** For a raw spherical \(P=IC_\lambda\), supported on a finite projective Schubert variety \(X\), and any torus label \(\mu\), there is an actual restriction isomorphism \[ Q\otimes_S H_{T,c}^*(S_\mu\cap X,P) \xrightarrow{\ \sim\ } Q\otimes_S H_T^*(t^\mu,i_\mu^*P), \qquad Q=\operatorname{Frac}S. \tag{C.20.3} \] An empty slice gives zero on both sides. In particular \[ \dim_E F_\mu(IC_\lambda) =\sum_j\dim_E H^j(i_\mu^*IC_\lambda). \tag{C.20.4} \] Under the full Satake equivalence, this is the multiplicity of weight \(\mu\) in \(V_\lambda\). **Proof.** The finite projective lattice embedding is \(T\)-equivariant. Choose a strictly dominant constant cocharacter \(\gamma\) which does not annihilate any nonzero difference of the finitely many ambient weights. Such an integral choice exists: the strictly dominant rational cone is open, and finitely many proper rational hyperplanes cannot cover it. The root-parameter calculation of Lesson 5, §7, equation (7.1), applies to this \(\gamma\). Its positive root powers contract every point of \(S_\mu\) to \(t^\mu\); the proved Iwasawa partition and uniqueness of a limit show that this is precisely the corresponding attracting locus. Here is a finite coordinate realization sufficient for localization. Choose an eigen-coordinate \(z_0\) nonzero at the fixed point in the projective embedding. Its chart is \(T\)-stable. If \(\eta_i\) is the weight of the ratio \(z_i/z_0\), attraction to this point sets the negative \(\gamma\)-weight ratios to zero and the zero-weight ratios to their point values. The remaining ratios \(x_i\) have \(\langle\eta_i,\gamma\rangle>0\). Intersecting these equations with the closed equations of \(X\) gives a closed affine attracting scheme \(A\), with distinguished origin. The description holds on every parameter algebra: an equivariant polynomial path has coordinate \(s^{\langle\eta_i,\gamma\rangle}x_i\); a negative power belongs to \(R[s]\) only when its coefficient is zero, including for nilpotent coefficients. The underlying reduced locus is \(S_\mu\cap X\), by the preceding partition. Actual closed-reduction comparison in C.12 and Lesson 5 O.4.2.1 identifies their cohomology and compact images, so we may use this finite scheme. The finitely many invariant opens \(D(x_i)\) cover \(A\setminus\{t^\mu\}\). Each \(\eta_i\) is a nonzero character. On \(D(x_i)\) the function \(x_i\) is an invertible equivariant section of the line with character \(\eta_i\), up to the harmless inverse convention for the function action. Its first Chern class is therefore zero on that open. This is an actual equivariant line trivialization: after tensoring by the inverse character, the section has weight zero and gives the trivial frame on the whole action nerve. The Kummer cocycle of that frame is zero. Consequently the nonzero linear class \(c_1(\eta_i)\), or its negative, acts by zero on the compact equivariant cohomology of any restricted coefficient complex there. The same argument works on every intersection in this finite open cover. Localization to \(Q\) therefore kills all these compact cohomology modules. The compact Mayer–Vietoris triangle follows from the two extension-by-zero maps and their overlap, checked on stalks; induction on the number of opens gives \[ Q\otimes_S H_{T,c}^*(A\setminus\{t^\mu\},P)=0. \tag{C.20.5} \] The compact image of the actual open–closed triangle \(j_!j^*P\to P\to i_{\mu*}i_\mu^*P\) now gives exactly (C.20.3). Localization is flat, so this conclusion preserves the specified restriction map. To compute its two ranks, form the genuine \(T\)-equivariant compact image of \(P|_A\) on the point. Its underlying complex is concentrated in degree \(h_\mu=\langle2\rho_G,\mu\rangle\), with coefficient \(F_\mu(P)\), by Lesson 5, Theorem Q.4.2.1. A complex with one ordinary cohomology sheaf is canonically that sheaf shifted. The equivariant point sheaf here is constant: its action matrix is a degree-zero section on the connected torus, hence is a constant matrix, and the identity section forces that matrix to be the identity. Its cocycle is consequently the constant cocycle. Taking equivariant cohomology gives \[ H_{T,c}^*(A,P)=S\otimes_E F_\mu(P)[-h_\mu]. \tag{C.20.6} \] This computation uses the actual compact image object, whose construction on the action nerve commutes with the smooth atlas comparisons; it does not assert nonproper ordinary base change. At the fixed point, the ordinary stalk of \(IC_\lambda\) has one parity by Lesson 11 C.4.1.1. Its finite equivariant ordinary tower splits as in C.10: an attaching map between constant even-separated terms belongs to an odd group of \(S\), which is zero. Thus its equivariant cohomology is a free \(S\)-module of rank \(\sum_j\dim H^j(i_\mu^*P)\). Comparing ranks in (C.20.3) proves (C.20.4). Finally the full weight identification in Lesson 11 B.1–B.4 identifies \(F_\mu(IC_\lambda)\) with the \(\mu\)-weight space of \(V_\lambda\). This proves the last assertion. ∎ #### C.20.2. Traces test every invariant polynomial **Lemma C.20.2.1.** Let \(H\) be the original split connected reductive group in characteristic zero, with its full character lattice \(L\). In each positive polynomial degree \(n\), the functions \[ x\longmapsto\operatorname{tr}_{V_\lambda}(x^n), \qquad \lambda\in L\text{ dominant}, \tag{C.20.7} \] span \(E[\mathfrak h]^H_n\). No change to a simply connected cover is required. **Proof.** First work on \(\mathfrak t_H\). Integral linear forms \(\mu\in L\) span its linear dual. Their \(n\)-th powers span all degree-\(n\) polynomials, because \[ n!\,\ell_1\cdots\ell_n =\sum_{J\subset\{1,\ldots,n\}} (-1)^{n-|J|} \left(\sum_{j\in J}\ell_j\right)^n. \tag{C.20.8} \] Expansion proves this identity: any monomial omitting an index cancels in the subset sum, and the sole monomial using every index has coefficient \(n!\). Choose the \(\ell_j\) from an integral basis and use linearity. Averaging over the finite Weyl group therefore shows that the invariant polynomials are spanned by \(\sum_{\mu\in W\lambda}\mu^n\) for dominant \(\lambda\in L\), with distinct orbit elements. The formal character of \(V_\lambda\) is the orbit sum of \(\lambda\), with coefficient one, plus a finite combination of orbit sums of strictly lower dominant weights. This is the highest-weight theorem of Lesson 11: the extremal weights \(W\lambda\) have multiplicity one, and all other weights differ from \(\lambda\) by positive-root sums. This triangular relation can be inverted using only representations of the original group. To see termination, pair the dominant weights with the sum of positive coroots of the derived group. The resulting nonnegative integers strictly decrease under a strict dominance drop. All terms keep the same central character. For a torus there are no lower terms. Thus induction expresses every orbit sum as a finite linear combination of these full-group characters. Apply the linear operation \(e^\mu\mapsto\mu^n\) to that character identity. The character of \(V_\lambda\) becomes the torus restriction of (C.20.7), because a torus Lie element acts by the scalar \(\mu(x)\) on its \(\mu\)-weight space. Hence those trace functions span the torus invariants. Chevalley restriction and its full reductive centre, including the descent to every characteristic-zero coefficient field proved in §3 of this lesson, identify the latter with \(E[\mathfrak h]^H\). The trace polynomials are themselves conjugation invariant, so the spanning assertion follows. ∎ #### C.20.3. The quotient is the trace-pairing coordinate map **Theorem C.20.3.1.** The actual geometric element satisfies, for every invariant polynomial \(f\), \[ f(\xi_\rho)=f(\beta_\rho(u)) \quad\text{in }R_0=E[\mathfrak t_G]^W. \tag{C.20.9} \] Here the right side denotes restriction of \(f\) to \(\mathfrak t_H\), followed by \(\beta_\rho\). Thus its adjoint quotient is the specified isomorphism of the coefficient base with \(\mathfrak h/\!/H\). **Proof.** Extend the actual cup action and its graded tensor trivialization to \(S\), and then to \(Q\). C.19.4 gives the actual fixed-point decomposition and projectors. On the block at \(\mu\) the determinant cup element acts by \[ \nu_\mu(u)=\mu(\beta_\rho(u)). \tag{C.20.10} \] These eigenvalues are distinct for distinct labels, since \(\beta_\rho\) is an isomorphism. Lemma C.20.1.1 identifies the dimension of that block with \(\dim V_\lambda[\mu]\). Consequently, for every \(n>0\), \[ \operatorname{tr}_{V_\lambda}(\xi_\rho^n) =\sum_\mu\dim V_\lambda[\mu]\, \mu(\beta_\rho(u))^n =\operatorname{tr}_{V_\lambda}(\beta_\rho(u)^n) \quad\text{in }Q. \tag{C.20.11} \] The first trace is taken on the finite free module \(\omega_G(IC_\lambda)\); under C.18 it is precisely the represented Lie action on \(R_0\otimes V_\lambda\). The actual torus restriction and fixed-point projectors computed its trace, rather than substituting an abstract module with the same rank. Lemma C.20.2.1 gives (C.20.9) for homogeneous invariant polynomials of positive degree. Constants agree as well, and finite sums handle every polynomial. Both sides lie in \(R_0\). Its inclusion into \(S\), and into \(Q\), is injective by C.14, so the equality already holds in \(R_0\). Since \(\beta_\rho\) is invertible and \(W\)-equivariant, its map on the two invariant polynomial rings is an isomorphism. This proves the precise quotient statement over every coefficient parameter algebra, including nonreduced ones. ∎ #### C.20.4. A rational pinning and a graded conjugation **Theorem C.20.4.1.** Adapt the pinning of \(H\) to the nonzero simple coefficients of \(\xi_\rho(0)\). Let \(\sigma_\rho:\mathfrak h/\!/H\to\mathfrak h_{\mathrm{reg}}\) be the resulting Kostant section. There is a graded tensor trivialization, with the same actual augmentation as C.18, in which the determinant cup element is exactly \[ \sigma_\rho\!\left(\beta_\rho(u)\bmod W\right). \tag{C.20.12} \] The construction preserves the original full dual group and centre. It need not give a conjugation to an arbitrarily prescribed old pinning over \(E\). **Proof.** By C.19.6 write \(e_\rho=\xi_\rho(0)=\sum_i a_i e_i\), with every \(a_i\in E^\times\). Set \[ e_i^\rho=a_i e_i,\qquad f_i^\rho=a_i^{-1}f_i,\qquad f_\rho=\sum_i c_i f_i^\rho . \tag{C.20.13} \] Their principal triple has the same \(h=2\rho_H^\vee\) and cocharacter \(\vartheta\) as B.2. These are a full rational pinning, not a claim about an \(H(E)\)-conjugator. Indeed the simple characters form a basis on the adjoint torus, so the tuple \((a_i)\) gives an \(E\)-point there. After a finite extension it lifts to the original split torus: in character bases, Smith normal form reduces the lift to finitely many equations \(z^m=a\), all with \(a\ne0\). The corresponding extensions are separable in characteristic zero. Two lifts differ by the centre and hence give the same conjugation on \(H\). The resulting conjugation automorphism and its inverse descend to \(E\); the matrix coefficients agree on the two overlap projections, so affine faithful-flat descent supplies the descended maps. Denote this automorphism by \(\phi_\rho\). It fixes the torus, the centre, \(\vartheta\), and every invariant polynomial. The last assertion can be tested after that finite extension, where it is ordinary inner conjugation, and then descends injectively. Apply \(\phi_\rho\) to the complete proofs of B.2–B.3. The adapted slice is \(\Sigma_\rho=e_\rho+\ker\operatorname{ad}f_\rho\). Its quotient is the same polynomial isomorphism on all parameter algebras; its positive action is still \(t^2\operatorname{Ad}_{\vartheta(t^{-1})}\). Its inverse defines \(\sigma_\rho\). The full map \(H\times\Sigma_\rho\to\mathfrak h_{\mathrm{reg}}\) is smooth surjective. In the torus case \(\phi_\rho\) is the identity and the slice is all of \(\mathfrak h\); the argument includes it. Let \(B=\operatorname{Spec}R_0\), and put \(s(p)=\sigma_\rho(\beta_\rho(p)\bmod W)\). Consider the affine scheme of finite presentation \[ P=\{(p,g)\in B\times H: \operatorname{Ad}_g s(p)=\xi_\rho(p)\}. \tag{C.20.14} \] It is the base change of the preceding smooth surjection along \(\xi_\rho:B\to\mathfrak h_{\mathrm{reg}}\). To check this scheme equality, the base change initially has a variable slice point \(\sigma\). Applying the quotient to its defining equality forces \(\sigma=s(p)\), by (C.20.9) and the polynomial inverse on the full slice. This argument holds for every parameter algebra. Thus \(P\to B\) is smooth surjective. Its fibres retain the full centralizer, including its centre and all components. Homogeneity of \(\xi_\rho\) and of the slice gives the algebraic action \[ t\cdot(p,g) =\bigl(t\cdot p,\, \vartheta(t^{-1})g\vartheta(t)\bigr) \quad\text{on }P. \tag{C.20.15} \] Substitution in (C.20.14) proves this identity, including its sign: both \(s(t\cdot p)\) and \(\xi_\rho(t\cdot p)\) equal \(t^2\operatorname{Ad}_{\vartheta(t^{-1})}\) of their values at \(p\). The point \((0,1)\) is fixed, since \(s(0)=e_\rho=\xi_\rho(0)\). The base coordinates have positive weights. Lemma C.18.1.1 therefore gives an actual polynomial equivariant section \(g:B\to H\) through the identity, by its finite-order lifts and homogeneous stabilization. No field-point lifting or merely formal section is substituted. If \(\tau\) is the former tensor trivialization, replace it on each representation by \(g^{-1}\tau\). The represented group action is a natural tensor automorphism on every test algebra. Equation (C.20.15) makes the new frame graded; its identity value at the origin retains the actual augmentation. Finally \(\operatorname{Ad}_{g^{-1}}\xi_\rho=s\) is exactly (C.20.12). The higher primitive operators still have to be constructed and identified with the centralizer gradients. ∎ #### C.20.5. Two quotient and pinning exercises **Exercise C.20.5.1 (basic).** For \(GL_2\) with its standard trace pairing, recover the two adjoint-quotient coordinates from the actual matrix C.19.18. Check that the centre survives the comparison. **Solution.** Here \(\beta_\rho\) is the identity on the two diagonal coordinates. The matrix has trace \(\sigma_1\), determinant \(\sigma_2\), and squared trace invariant \(\operatorname{tr}(\xi_\rho^2)=\sigma_1^2-2\sigma_2\). Hence its characteristic polynomial is \(z^2-\sigma_1z+\sigma_2\), exactly the polynomial of the diagonal pair \((u_1,u_2)\). The central scalar in C.19.19 is \(\sigma_1/2\); it is retained by this quotient map. Trace and squared trace generate the same invariant algebra as trace and determinant since \(2\) is invertible in \(E\). **Exercise C.20.5.2 (intermediate).** In \(SL_2\) over a characteristic-zero coefficient field \(E\), let \(a\in E^\times\). Determine when \(aE_{12}\) is conjugate to \(E_{12}\) by an \(SL_2(E)\)-matrix. Construct its adapted principal triple even when that conjugator does not exist. **Solution.** Any such conjugator preserves the unique kernel line \(Ee_1\), so has the form \(\left(\begin{smallmatrix}z&b\\0&z^{-1}\end{smallmatrix}\right)\). Its conjugation sends \(E_{12}\) to \(z^2E_{12}\). A conjugator exists exactly when \(a\) is a square in \(E^\times\). For any \(a\ne0\), however, the vectors \(e_a=aE_{12}\), \(f_a=a^{-1}E_{21}\), and \(h=\operatorname{diag}(1,-1)\) satisfy \([e_a,f_a]=h\), \([h,e_a]=2e_a\), and \([h,f_a]=-2f_a\). They define the adapted slice \(e_a+Ef_a\) with the same invariant quotient. Its adjoint-torus rescaling descends over \(E\) even if its original-torus lift requires adjoining a square root. This is the precise distinction used in C.20.4. ### C.21. Higher primitive cup operators and the generator modules Keep the adapted graded tensor frame of C.20.4. Choose a Tate vector as in C.20 for scalar formulas. For a homogeneous coefficient class \(q\in R_0\) of polynomial degree \(d\geq1\), its cohomological degree and natural twist are \(2d\) and \(E(d)\). We construct a cup class of degree \(2d-2\), with twist \(E(d-1)\). Its normalization will be a directional derivative, including in polynomial degree one. ![The fixed-point curve sign, actual higher gradients and the whole-ring generator morphism comparison](assets/higher-cup-kostant-modules.png) The first panel is the exact torus-fixed calculation on the curve, with its negative lattice degree and trace normalization. The second uses the full framed moving torsor to obtain the primitive operator before identifying its gradient. The third displays the two coefficient descents that recover linearity over every whole-Grassmannian class. Here the diagram abbreviation \(B_{\mathrm{op}}\) denotes the polynomial operator algebra \(\mathcal B_\rho\), while the inverse trace form is \(B_H\). Proofs C.21.1.1–C.21.5.1 supply the maps; the coherent Hom uses the deformation grading of B.5. Editable SVG source. #### C.21.1. The inverse trace form on the full dual Lie algebra **Lemma C.21.1.1.** There is a nondegenerate \(H\)-invariant symmetric bilinear form \(B_H\) on \(\mathfrak h\) whose restriction to \(\mathfrak t_H\) is \[ B_H(v,w)=B_\rho(\beta_\rho^{-1}v,\beta_\rho^{-1}w). \tag{C.21.1} \] For \(f=q\circ\beta_\rho^{-1}\), regarded as an invariant polynomial on \(\mathfrak h\) by Chevalley restriction, its gradient satisfies \[ \lambda\!\left(\nabla_{B_H}f(\beta_\rho(u))\right) =D_\lambda q(u)\qquad(\lambda\in L). \tag{C.21.2} \] **Proof.** The displayed Cartan form is nondegenerate and \(W\)-invariant by C.19.4. It is orthogonal between distinct derived simple components, and between the derived Cartan and the centre. Indeed a reflection fixes the other component and the centre, while negating its simple coroot; invariance makes the pairing with that coroot zero. The simple coroots span the derived Cartan. On one simple component, compare this form with the Cartan restriction of the Killing form. Its Cartan matrix is the root Gram form \(\sum_{\alpha}\alpha\otimes\alpha\): these roots span the rational Cartan dual, and the real Gram form is positive definite. Its determinant is therefore a nonzero rational number, hence remains nonzero over every characteristic-zero \(E\). The Killing form is invariant by the trace-of-a-commutator identity. Distinct root spaces pair only when their roots are opposite, by Cartan invariance. For a normalized root pair its value is \(K(e_\alpha,f_\alpha)=K(\alpha^\vee,\alpha^\vee)/2\), again nonzero by that rational Gram calculation. This also verifies nondegeneracy on the full simple Lie ideal. The comparison endomorphism commutes with every reflection. A reflection's negative eigenspace is its simple-coroot line, so the endomorphism has some scalar on each such line. For adjacent simple coroots, the formula \(s_i(\alpha_j^\vee)=\alpha_j^\vee-m_{ij}\alpha_i^\vee\), with \(m_{ij}\ne0\), forces their scalars to agree. The connected simple Dynkin diagram makes the comparison a single scalar. That scalar is nonzero by nondegeneracy. Extend by this scalar times the Killing form on each simple Lie ideal, and by the prescribed form on the centre. This constructs the required full nondegenerate invariant form; neither a central direction nor a simple factor is discarded. Invariance for the algebraic group follows from the same torus and root-exponential argument as §3: a Lie-invariant polynomial tensor is torus-invariant and root-group-invariant, and those subgroups generate \(H\). Put \(v=\beta_\rho(u)\), and let \(b=\beta_\rho^{-1}\nabla_{B_H}f(v)\) in the Cartan. The gradient lies there: invariant gradients are fixed by the torus stabilizing \(v\), so this holds on the dense regular Cartan and then as a polynomial identity on the entire Cartan. For \(x\in\mathfrak t_G\), the gradient definition gives \[ B_\rho(b,x) =B_H(\nabla f(v),\beta_\rho(x)) =df_v(\beta_\rho(x)) =dq_u(x). \tag{C.21.3} \] By C.20.2, \(\lambda(\beta_\rho(b))=B_\rho(\lambda,b)\). Substituting \(x=\lambda\) in (C.21.3) proves (C.21.2). The equality extends from the regular Cartan to all its points because both sides are polynomials. ∎ #### C.21.2. A framed characteristic class and its primitive cup map **Theorem C.21.2.1.** There is a compatible actual whole-Grassmannian equivariant class \[ z_q\in H_G^{2d-2}(\operatorname{Gr}_G,E(d-1)). \tag{C.21.4} \] Its cup maps on the spherical cohomology modules are natural and \(R_0\)-linear, vanish on the tensor unit, and satisfy \[ z_q|_{\omega_G(P*Q)} =z_q|_{\omega_G(P)}\otimes1 +1\otimes z_q|_{\omega_G(Q)} \tag{C.21.5} \] under the actual tensor comparison C.17. The class is linear in \(q\). The construction uses the full lattice and moving-chain functors before taking their cohomological supports. **Proof.** Use \(\mathbb P^1\) with modification point \(0\). The full gluing equivalence of Lesson 3, Proposition 2.1 and Theorem 3.1, gives on \(\mathbb P^1\times\operatorname{Gr}_G\) the universal right \(G\)-torsor \(\mathcal P\), with its specified frame off \(0\). This assertion can be read stage by stage: pull back to the local loop-frame covers, glue the disc torsor to the framed trivial torsor on the complement, and descend along the exact change-of-frame maps. The gluing equivalence includes morphisms, so these transitions satisfy the cocycle on the full parameter algebras. A framed automorphism which is the identity off \(0\) is the identity: on a local trivialization its defining functions vanish after inverting the curve parameter, which is a nonzerodivisor on the flat torsor. Thus there is no residual automorphism ambiguity in this descent. The constant \(G\)-action is retained. Let \(\mathcal P_{\mathrm{ref}}\) be the trivial curve torsor with its diagonal constant-\(G\) equivariance. Its characteristic class is the base class \(q\); its ordinary bundle being trivial does not make its equivariant class zero. Characteristic classes here mean actual pullback from the \(G\)-torsor classifying nerve: the element \(q\in H_G^{2d}(\mathrm{pt},E(d))\) pulls back along each torsor's classifying map. The off-point frame gives a specified isomorphism of these torsors there. We spell out how that frame supplies a supported class, rather than choosing a lift of a vanishing restriction. In the completed derived mapping models of Lesson 5, M.6 and Lemma P.1.3.1, use the relative-cone construction of Lesson 6 A.2 and represent \(q\) by a closed cochain. Write \(F_b,F_r\) for its two classifying pullback chain maps. On the complement \(U\) the framed torsor isomorphism gives a chain homotopy \(K\) with \[ (F_b-F_r)|_U=dK+Kd. \tag{C.21.6} \] It is the ordinary nerve prism: in a string of composable transition arrows insert the isomorphism in each possible position and take the alternating sum. All inner faces cancel in pairs; the two remaining end faces are the two classifying maps. This verifies (C.21.6) directly. On a common trivializing cover the formula is finite in every nerve degree. Refining that cover preserves the formula, and the same prism for change of cover identifies its augmentations. Étale descent and the completed inverse limits retain these finite identities. Hence it defines the specified homotopy in the genuine equivariant enhancement, with all action faces and degeneracies. In the relative complex \(\operatorname{Cone}(C(\mathbb P^1\times X)\to C(U))[-1]\) of any finite stage, use the pair \[ c_q=\big((F_b-F_r)q,Kq\big),\qquad d(c,u)=(dc,c|_U-du). \tag{C.21.7} \] It is closed by (C.21.6). Replacing \(q\) by \(q+dv\) changes it by the relative boundary of \(((F_b-F_r)v,Kv)\), since \(Kdv=(F_b-F_r)v|_U-dKv\). Thus its supported class is well defined and linear in the original cohomology class. The construction commutes with parameter pullback, change of frame and the constant-\(G\) nerve maps through their actual prism homotopies. Let \(\pi:\mathbb P^1\times X\to X\). Forget the support in (C.21.7), and apply the actual smooth proper curve trace of Lesson 5, Theorems P.2.3.1–P.2.4.1, with the Kummer orientation normalized in Lesson 6 A.1.1. Define \[ z_q=-\pi_*\bigl(q(\mathcal P)-q(\mathcal P_{\mathrm{ref}})\bigr). \tag{C.21.8} \] The minus sign belongs to the lattice convention and is checked below. This trace lowers degree by two and twist by one, proving the degree in (C.21.4). The supported construction remains available to compare sums of modifications. Proper base change and the trace comparisons make these classes compatible on every finite stage and on all action-nerve levels. C.12 identifies these compatible classes with an actual class of the whole ind-Grassmannian, including all its component products. For primitivity, construct the same framed relative class on \(\mathbb P^1\times Y_r\) for the full moving ordered chain of Lesson 8. Its support is the union of the graphs of the moving points. Off the diagonals those graphs are disjoint. Cochains supported on a finite disjoint union split as the direct sum of the individual support complexes: the corresponding closed support functor is the sum of their extension-by-zero functors, an equality checked on stalks. On the open obtained by deleting the other graphs, the full gluing and CRT comparison of Lesson 8 C.1–C.6 identifies the last framed torsor with the single input modification. Consequently the prism class restricts to that input's prism class on each support summand. Curve trace then gives the sum of the input \(z_q\)'s. On a full collision the last framed torsor is the endpoint pullback of \(\mathcal P\); its off-point frame is the endpoint frame. Proper curve base change therefore identifies the same moving class with the endpoint pullback of (C.21.8). These statements concern the actual maps on the full families and their nilpotent parameters. Cup with the moving class gives the actual morphism of the moving coefficient complex, shifted by \(2d-2\) and twisted by \(d-1\). Take its proper image on the full action nerve as in C.17.4. The canonical affine counits of C.17.3 transport this morphism to the base: both its source and its shifted target belong to the fully faithful essential image. Thus its lift is unique. Restriction to distinct sections is the sum just proved, and restriction to collision is the endpoint cup map. All finite block maps, action faces and degeneracies commute with those counits. Derived sections and the actual external cup comparison give (C.21.5). This repeats the morphism-level transport of C.19.3 with its actual higher degree and twist. At the zero modification the framed torsors are identical; their prism difference is zero, proving the unit assertion. ∎ #### C.21.3. The fixed-point derivative and the lattice sign **Lemma C.21.3.1.** Restriction to a torus point is \[ z_q|_{t^\lambda}=D_\lambda q(u)\in S, \qquad \lambda\in L. \tag{C.21.9} \] The natural twist of this equality is \(E(d-1)\). **Proof.** At \(t^\lambda\) the universal bundle has its torus reduction. For a character \(\chi\), its associated line on the curve is \[ \mathcal O_{\mathbb P^1} (-\langle\chi,\lambda\rangle\,0)\otimes k_\chi, \tag{C.21.10} \] where \(k_\chi\) denotes the geometric character line. The negative sign is exactly the full lattice gluing convention of Lesson 3, §3. Its equivariant first Chern class is \(\chi(u)-\langle\chi,\lambda\rangle h\), where \(h=c_1(\mathcal O(1))\). The curve has trivial \(T\)-action. Its affine-line and point filtration, with the normalized supported point class of Lesson 6 A.1 and the compact affine trace, gives the two actual ordinary basis classes \(1,h\). Above degree two its ordinary cohomology is zero, so \(h^2=0\); the trivial equivariant action tensors this calculation with \(S\). Equivalently this is the relative projective Chern basis of C.14 for the trivial rank-two bundle. The actual trace gives \[ H_T^*(\mathbb P^1)=S[h]/(h^2),\qquad \pi_*h=1,\quad\pi_*1=0. \tag{C.21.11} \] The reference line has class \(\chi(u)\). Restriction of \(q\) along the torus classifying map is its actual polynomial in the character Chern classes, by C.14. Hence the two curve characteristic classes differ by \[ q(u-\lambda h)-q(u) =-D_\lambda q(u)\,h. \tag{C.21.12} \] Expansion of each polynomial monomial proves the equality, since \(h^2=0\). Taking the negative trace in (C.21.8) proves (C.21.9). The trace removes the \(E(1)\) of \(h\), leaving exactly \(E(d-1)\). No character, coefficient variable or lattice index is inverted. ∎ #### C.21.4. The higher operators are the full centralizer gradients **Theorem C.21.4.1.** Let \(q_1,\ldots,q_r\) be homogeneous polynomial generators of \(R_0\), of degrees \(d_i\), and put \(f_i=q_i\circ\beta_\rho^{-1}\) on the dual adjoint quotient. In the adapted frame C.20.4, the actual \(z_q\)-operator is \[ \nabla_{B_H}(q\circ\beta_\rho^{-1}) \bigl(\sigma_\rho(\beta_\rho(u)\bmod W)\bigr). \tag{C.21.13} \] In particular the \(z_{q_i}\) act by the entire basis of the full centralizer Lie bundle in B.5. They give the geometric module \(\omega_G(IC_\lambda)\) exactly the Kostant presentation \[ M(V_\lambda)=R_0\otimes_E V_\lambda,\qquad z_{q_i}\cdot v= \rho_{V_\lambda}(\nabla_{B_H}f_i(\sigma_\rho))v, \quad |z_{q_i}|=2d_i-2. \tag{C.21.14} \] The polynomial algebra \(\mathcal B_\rho=R_0[z_{q_1},\ldots,z_{q_r}]\) embeds in the actual whole equivariant cohomology ring; equality with that whole ring is not asserted. **Proof.** Theorem C.21.2.1 and C.18.4 express cup with \(z_q\) as a homogeneous Lie element \(\eta_q\) of total degree \(2d-2\), retaining the Tate line. Cups of these even-degree classes commute with the determinant cup. Reconstruction identifies their commutator with \([\eta_q,\xi_\rho]\), so that bracket is zero. Thus in the adapted slice frame \(\eta_q\) is a section of the full centralizer Lie bundle. Extend to the fraction field \(Q\) of \(S\) and then to an algebraic closure of \(Q\). The vector \(\beta_\rho(u)\) is regular semisimple: its root values are nonzero linear polynomials, since \(\beta_\rho\) is invertible. C.20.3 and B.3 identify \(\xi_\rho\) with its geometric adjoint orbit. Choose a conjugator taking it to \(\beta_\rho(u)\). The corresponding \(\eta_q\) then belongs to \(\mathfrak t_H\). In every \(V_\lambda\), the eigenvalues of this Cartan vector \(\beta_\rho(u)\) distinguish its distinct weights. Therefore its \(\mu\)-weight blocks are precisely the transported actual fixed-point blocks of C.19.4. Actual cup restriction and C.21.9 show that \(\eta_q\) acts on such a block by \(D_\mu q(u)\). Lemma C.21.1.1 gives exactly the same scalar for the gradient in (C.21.13). The weights of a faithful \(H\)-representation span its full character lattice by the all-kernel argument of Lesson 2, Lemma 7.10. Decomposing that representation into its genuine simple modules proves that these scalar equalities determine the Cartan Lie element, including its centre. The two elements agree over the extended fraction field. Invariant gradients are equivariant under \(H\), by invariance of \(B_H\) and the chain rule. Conjugating back consequently identifies \(\eta_q\) with the gradient at the slice point. Both are sections of the finite free module \(R_0\otimes_E\mathfrak h\). Its injection into the extended fraction-field module makes this equality hold already over \(R_0\), proving (C.21.13) on every parameter algebra. The \(f_i\) are polynomial generators of the dual invariant ring by C.20.2. Their differentials form the cotangent basis of the quotient along the polynomial slice isomorphism. B.5 therefore makes their gradients the entire Lie-centralizer basis, with the specified total degrees. This proves the actual action and grading in (C.21.14). The full centre supplies its degree-zero gradient operators when \(d_i=1\). For completeness these actual cup classes have no polynomial relation over \(R_0\). A relation would act as zero on every \(IC_\lambda\), and hence on every localized fixed-point block. Each label \(\mu\in L\) occurs: its dominant Weyl translate has that extremal weight with multiplicity one, and C.20.1 supplies the corresponding nonzero block. At the generic regular Cartan the map \[ \mu\longmapsto(D_\mu q_1(u),\ldots,D_\mu q_r(u)) \tag{C.21.15} \] extends to an invertible linear map over \(Q\); equivalently the quotient differentials are independent there, as checked in B.2. The integral lattice \(L\) is polynomially dense in its affine space: in a lattice basis, repeated one-variable infinite-root arguments prove this for the integer grid. Its image under that invertible map is equally dense. The proposed relation, evaluated at (C.21.15), is therefore the zero polynomial in its operator variables over \(Q\). Injectivity \(R_0\to Q\) makes every coefficient zero already over \(R_0\). This proves the claimed embedding. It leaves the whole ring's degreewise component products intact. ∎ #### C.21.5. Morphisms over the actual whole cohomology ring **Theorem C.21.5.1.** For two spherical IC generators, actual cup linearity gives \[ \operatorname{Hom}_{H_G^*(\operatorname{Gr}_G)}^* (\omega_G(IC_\lambda),\omega_G(IC_\mu)) =\operatorname{Hom}_{\mathcal B_\rho}^* (M(V_\lambda),M(V_\mu)). \tag{C.21.16} \] Consequently C.16 and B.5 give an actual composition-preserving graded comparison \[ \operatorname{Ext}_{L^+G}^*(IC_\lambda,IC_\mu) \xrightarrow{\ \sim\ } \operatorname{Hom}_{A,H}^* (A\otimes V_\lambda,A\otimes V_\mu), \qquad A=E[\mathfrak h^*], \tag{C.21.17} \] with the deformation grading and full group of B.5. This is the geometric generator module and morphism comparison. An enhanced monoidal equivalence and dg formality remain separate requirements. **Proof.** Write \(A_G=H_G^*(\operatorname{Gr}_G)\). Since the \(z_{q_i}\) are actual cup classes, every \(A_G\)-linear map is \(\mathcal B_\rho\)-linear. Conversely let \(F\) be \(\mathcal B_\rho\)-linear. The section \(\xi_\rho=\sigma_\rho\) commutes with itself and hence belongs to the full Lie-centralizer bundle. Its basis from C.21.4 writes it as an \(R_0\)-linear combination of the gradient operators. Thus \(F\) commutes with the actual determinant operator. Extend \(F\) along the actual scalar comparison \(R_0\to S\) of C.14, then to \(Q\). The determinant eigenvalues \(\mu(\beta_\rho(u))\) distinguish labels, so \(F_Q\) preserves each fixed-point block. Every class of the whole \(A_T=H_T^*(\operatorname{Gr}_G)\) acts on that block by its scalar restriction at \(t^\mu\). This follows from the actual fixed-point restriction isomorphism of C.19.4, which is a module map for those whole classes. The scalar is the same on source and target blocks. Hence \(F_Q\) commutes with every such cup class. The underlying source and target torus cohomology modules are finite free over \(S\) by C.17. A matrix commutator which becomes zero over \(Q\) is already zero over \(S\). In particular \(F\otimes_{R_0}S\) commutes with the restrictions of every class of \(A_G\). The faithful flatness of \(S/R_0\) makes the original commutator zero. Thus \(F\) is \(A_G\)-linear, proving (C.21.16). This uses scalar actions on the actual generic blocks, not a polynomial presentation of the whole ind-Grassmannian cohomology. C.16.4 identifies the left side with the original finite-jet spherical generator Ext, through the actual cup comparison preserving Yoneda composition. C.21.4 identifies the right side with the exact modules of B.5, using \(B_H\) to identify the coadjoint coordinates and using the adapted pinning automorphism from C.20.4. Both preserve the original full group. B.5 then gives the right side of (C.21.17), including its specified grading and composition. Composing these isomorphisms proves (C.21.17). None of these graded Hom isomorphisms asserts a dg lift or eliminates higher composition obstructions. ∎ #### C.21.6. Two derivative and centre exercises **Exercise C.21.6.1 (basic).** For \(G=\mathbb G_m\), take \(R_0=E[u]\), label \(n\), and \(q=u^d\). Compute \(z_q\), including \(d=1\), and check primitivity on labels \(n,m\). **Solution.** The curve line at that label has class \(u-nh\), with \(h^2=0\). Thus \(-\pi_*((u-nh)^d-u^d)=dn\,u^{d-1}\). Its degree is \(2d-2\), with twist \(E(d-1)\). At \(d=1\) it is the degree-zero label operator \(n\), obtained without dividing by \(u\). At label \(n+m\) its value is \(d(n+m)u^{d-1}=dn u^{d-1}+dm u^{d-1}\), the actual tensor rule. **Exercise C.21.6.2 (intermediate).** For \(G=GL_2\), use \(q_1=u_1+u_2\) and \(q_2=u_1u_2\). On the standard minuscule module with determinant cup matrix \(M\) of C.19.18, compute the two higher operators and verify the polynomial relation expressing \(M\) in their span. **Solution.** The two labels are \((1,0)\) and \((0,1)\). Their \(q_1\)-derivatives are both one, while their \(q_2\)-derivatives are respectively \(u_2,u_1\). Therefore the two operators are \(I\) and \(\sigma_1I-M\). Equality holds over \(R_0\): localization computes their two distinct eigenspaces, and finite freeness descends the matrix equality. Explicitly \(M=\left(\begin{smallmatrix}0&-\sigma_2\\1&\sigma_1\end{smallmatrix}\right)\), so \(z_{q_2}=\left(\begin{smallmatrix}\sigma_1&\sigma_2\\-1&0\end{smallmatrix}\right)\). The standard trace form gives \(\nabla q_1=I\) and \(\nabla q_2(X)=(\operatorname{tr}X)I-X\), exactly these matrices. Finally \(q_{\rho,2}=\tfrac12(u_1^2+u_2^2)=q_1^2/2-q_2\) has gradient \(X\), and its actual derivative operator is \(\sigma_1 z_{q_1}-z_{q_2}=M\). The degree-zero central operator is essential to this integral formula. ### C.22. Tensor generators, their enhanced envelope and arithmetic purity For the arithmetic statements in this section work over \(k=\overline{\mathbf F}_q\), with finite characteristic-zero coefficient field \(E\) containing \(a\) such that \(a^2=q\). The underlying split group and all the rootwise constructions descend to the finite field. Taking a square power of its prime field makes \(a\) a rational integer, if that is needed to keep the original coefficient field. Let \(A_\lambda\) denote the raw geometric IC with its canonical Weil map multiplied by \(a^{-d_\lambda}\). This is a normalization of the Weil map; it changes neither the underlying geometric object nor its cohomological shift. In the statements not mentioning Frobenius, retain the general geometric coefficient scope of C.21. Write \[ A=\operatorname{Sym}_E(\mathfrak h[-2]). \tag{C.22.1} \] Thus each linear polynomial has cohomological degree two and the differential is zero. Put a representation \(V\) in degree zero in the free equivariant dg module \(F_V=A\otimes V\). On slice restriction, its grading is the coefficient-plus-principal grading proved in B.5. The notation \(\operatorname{Perf}^H(A)\) means the finite shift, cone and retract envelope of these free modules, exactly as in §8.1. ![Actual tensor generators, their arithmetic enhanced envelope and the locally finite weight model](assets/tensor-generators-weight-model.png) The first panel distinguishes the actual graded tensor comparison from the enhanced arithmetic equivalence of finite envelopes. The second records the normalized IC Weil maps \(A_\lambda\), the identity action on multiplicities and the sole Ext eigenvalues; the diagram's IC labels denote those same normalized maps. The third gives the strict, locally finite hypothesis and the exact weight-versus-degree truncation. The target keeps the full \(H\), with polynomial coordinates in cohomological degree two. Proofs C.22.1.1–C.22.4.1 establish these maps. Exercise C.22.5.2 shows why cohomological purity alone does not authorize a termwise locally finite replacement. Editable SVG source. #### C.22.1. The actual tensor comparison on every generator morphism **Theorem C.22.1.1.** Let \(\mathcal E_G^\oplus\) be the graded category of spherical perverse objects, with their actual spherical Ext as morphisms and the corrected Satake convolution symmetry. The comparisons of C.21 extend to a graded symmetric monoidal equivalence \[ \Phi:\mathcal E_G^\oplus\xrightarrow{\ \sim\ }\mathcal F_H, \qquad P\longmapsto F_{\omega(P)}, \tag{C.22.2} \] where \(\mathcal F_H\) has the free modules \(F_V\), their genuine graded \(A,H\)-linear Hom, and tensor over \(A\). For homogeneous generator morphisms \(b,c\), \[ \Phi(b*c)=\Phi(b)\otimes_A\Phi(c). \tag{C.22.3} \] This is a comparison of the actual graded tensor operations. It does not yet identify the enhanced convolution functor. **Proof.** Semisimplicity of the geometric spherical heart makes every object a finite sum of IC generators. Thus C.21.5 extends by matrices to a fully faithful graded functor with the displayed objects. Every finite-dimensional representation of the original \(H\) is a finite sum of its true simple modules, by Lesson 11, so these objects give its whole target. The required naturality is for derived morphisms, rather than just degree-zero maps in the heart. Choose common finite supports and jet bounds for \(b:P\to P'[n]\) and \(c:Q\to Q'[m]\). Genuine equivariant pullback and descent give their external map on the full moving ordered chain. Its target is the moving coefficient for \(P',Q'\), shifted by \(n+m\). The proper images of both coefficient complexes lie in the affine pullback essential image of C.17.3, since the proved constant-cohomology property is unchanged by shifts. The actual affine counits therefore transport this map uniquely on every action-nerve level, including the action face, and on all finite block diagrams. At distinct points its derived sections are the actual external product map. At collision they are the actual convolution map. The external cup comparison in C.17.2 is functorial for these coefficient maps: its construction uses the tensor of representatives, the projection comparison and composed proper counits; each commutes with a coefficient morphism, with the usual tensor sign. Therefore \[ \omega_G(b*c)=\omega_G(b)\otimes_{R_0}\omega_G(c) \tag{C.22.4} \] under the actual tensor isomorphism C.17.4. In particular the argument did not require an arbitrarily chosen lift of a morphism to fusion. There is no hidden odd sign on the unshifted generator category. Distinct component supports have zero Ext. Within one component the IC dimensions have the same parity, since a simple coroot changes the dimension height by two. Their star and costalk cell coefficients thus have matching parity. The actual comparison C.16, with the even coefficient ring of C.10, makes their odd Ext zero. Homogeneous nonzero \(b,c\) have even degrees. The shifted finite envelope will use the usual dg signs. The frame of C.20.4 is tensor on objects and on the heart. Substituting it in (C.22.4) makes the slice module map the coefficient tensor of the two maps in C.21. This equals restriction of their coherent tensor product: on every slice point it is literally the matrix tensor product over the common coefficient ring. The full faithfulness in B.5 makes the inverse polynomial extension equal to that coherent tensor map, proving (C.22.3). The same tensor frame carries the associator, unit and corrected symmetry on the heart to the usual associator, unit and flip of representations. These are the degree-zero structural maps in the graded category. The genuine finite-set comparisons of C.17 preserve every pentagon, unit and permutation identity, and B.5 preserves their compositions. Hence all diagrams for a graded symmetric monoidal functor commute. This proves (C.22.2) with its actual structural maps. ∎ #### C.22.2. The enhanced arithmetic free-generator envelope **Theorem C.22.2.1.** At the stated arithmetic coefficients and geometric base, there is an equivalence of enhanced categories \[ D^b_{L^+G,\mathrm{sph}}(\operatorname{Gr}_G,E) \xrightarrow{\ \sim\ }\operatorname{Perf}^H(A) \tag{C.22.5} \] sending every IC generator to \(A\otimes V_\lambda\). The left category has bounded, finite-supported objects. This theorem does not assert that the chosen enhanced equivalence is monoidal, or that it has been transferred to complex coefficients. **Proof.** The actual arithmetic formality and generator envelope A.4 identify the left side with the finite-cone and retract envelope of the graded Ext category with zero differential. Its hypotheses are the split finite-field geometry, the actual IC stalk/costalk Frobenius of Lesson 12 C.3, and the enhanced Frobenius lift in A.4. The nonresonance proof A.3 permits its Jordan parts. C.22.1 identifies that formal graded category, including composition, with \(\mathcal F_H\). We check that its envelope is the asserted dg module envelope, rather than invoking an ordinary coherent derived category in its place. For a rational \(H\)-equivariant dg \(A\)-module \(M\), \[ \operatorname{Hom}_{A,H}^\bullet(F_V,M) =(V^\vee\otimes_E M)^H . \tag{C.22.6} \] Invariants of rational \(H\)-modules are exact in characteristic zero. For an arbitrary rational module this follows from the finite-dimensional proof: an element and a chosen lift lie in finite-dimensional \(H\)-stable spans, and complete reducibility splits the resulting finite image. An invariant image vector consequently has an invariant lift. Applying this to the cycles and boundaries in an acyclic dg module proves that (C.22.6) is acyclic whenever \(M\) is acyclic. Thus each \(F_V\) computes its derived Hom directly. In particular the actual dg Hom between two free objects is \[ \operatorname{Hom}_{A,H}^\bullet(F_V,F_W) =(A\otimes V^\vee\otimes W)^H \tag{C.22.7} \] with differential zero, the very graded category \(\mathcal F_H\). There are no extra rational group cohomology terms. Its enriched Yoneda comparison is an isomorphism on these free generators, with their actual composition. In each variable it extends through shifts and cones by the corresponding mapping-cone identity, and through retracts by the image of a commuting idempotent. This proves an enhanced equivalence of its finite envelope with the finite shift, cone and retract subcategory of equivariant dg \(A\)-modules. That subcategory is our definition of \(\operatorname{Perf}^H(A)\). Composing this equivalence with A.4 and C.22.1 gives (C.22.5), with the specified generator images. Although C.22.1 is monoidal on the graded generators, the formality coordinate change in A.4 was constructed for composition. A comparison of the enhanced convolution bifunctor and its coherent structural maps is still needed to make this chosen equivalence monoidal. ∎ #### C.22.3. Scalar Frobenius on ordinary IC cohomology and tensor multiplicities **Theorem C.22.3.1.** The actual normalized ordinary cohomology of \(A_\lambda\) has scalar Frobenius \[ F|_{H^n(\operatorname{Gr}_G,A_\lambda)} =a^n\operatorname{id}. \tag{C.22.8} \] For any finite convolution word \(P=A_{\lambda_1}*\cdots*A_{\lambda_r}\), every multiplicity space in its canonical geometric isotypic decomposition \[ P=\bigoplus_\nu A_\nu\otimes_E M_\nu(P), \qquad M_\nu(P)=\operatorname{Hom}_{\mathrm{Perv}}(A_\nu,P), \tag{C.22.9} \] has the identity Frobenius. Consequently every degree-\(n\) Ext between normalized convolution words has sole eigenvalue \(a^n\), with Jordan parts allowed. Associators, units and the corrected permutations on these words retain their actual arithmetic structures. **Proof.** The rootwise source \(\mathcal B_\lambda\) of Lesson 11 C.2.1 is an iterated wall projective-line bundle, beginning with a point, defined over \(\mathbf F_q\). On each last line choose the global character line of C.16.1, whose fibre degree is a nonzero integer \(-m\). The class \(-c_1(\mathcal L)/m\) has fibre trace one. The coefficient field has characteristic zero, so this division is allowed even when the geometric wall central kernel is nonsmooth. For a wall projection \(p\), its classes \(1,-c_1(\mathcal L)/m\) give the actual proper-image comparison \[ E\oplus E(-1)[-2]\xrightarrow{\ \sim\ }Rp_*E. \tag{C.22.10} \] Indeed on every geometric fibre \(\mathbb P^1\) they give its two independent normalized basis classes; actual proper base change checks the cone at those fibres. The line and divisor transitions are defined over the finite field. Kummer Frobenius sends its untwisted degree-two class to \(q\) times that class. Thus (C.22.10) is compatible with the two displayed Frobenius scalars. Iterate (C.22.10). Products of its degree-two classes give an actual basis of \(H^{2j}(\mathcal B_\lambda,E)\), on which Frobenius is \(q^j\) times the identity, and the odd groups vanish. For the raw resolution image \(B_\lambda=Rf_{\lambda*}E[d_\lambda]\), its ordinary degree-\(n\) cohomology therefore has scalar \(q^{(n+d_\lambda)/2}\). Use the actual underlying IC retractions \(I_\lambda\xrightarrow{u}B_\lambda\xrightarrow{v}I_\lambda\), with \(vu=1\) and both dense restrictions the identity, proved in Lesson 11 C.4.2. The compression proof of Lesson 12 C.2 applies also to ordinary cohomology on this proper support. Here is its required transport. If \(s:X\to\operatorname{Spec}k\) is the support map, the field square \(s\sigma_X=\sigma_ks\), actual proper base change and pullback composition identify \(R\Gamma(X,\sigma_X^*K)\) with the pullback of \(R\Gamma(X,K)\) to the geometric field point. Field pullback on its coefficient-module category is canonically the identity, naturally for every underlying derived map. The composite \(v\alpha_{B_\lambda}\sigma^*u\) is the canonical raw IC Weil map: it has the same identity on the dense cell, and intermediate extension excludes an endomorphism with boundary-supported image. With the just-specified global transport, its degree-\(n\) action is therefore \[ H^n(v)\,F_{B_\lambda}\,H^n(u) =q^{(n+d_\lambda)/2}\operatorname{id}. \tag{C.22.11} \] No separate arithmetic equivariance of \(u,v\) was assumed. Multiplication of the Weil map by \(a^{-d_\lambda}\) gives (C.22.8). The actual ordinary cohomology tensor comparison of Lesson 8 and C.17 is arithmetic-compatible: all its gluing, proper maps, normalization traces and external cup maps descend to the finite field. Hence the degree-\(n\) cohomology of \(P\) also has scalar \(a^n\), as a tensor of the scalars in (C.22.8). The geometric semisimple heart gives the canonical evaluation isomorphism (C.22.9). It is arithmetic-equivariant when \(M_\nu(P)\) has the conjugation action defined by the two Weil maps. Frobenius fixes the labels, since their rootwise data are split. Applying \(H^n\) gives a direct sum of \(H^n(A_\nu)\otimes M_\nu(P)\). For each occurring \(\nu\), choose \(n\) with \(H^n(A_\nu)\ne0\); such an \(n\) exists by the proved faithfulness of total cohomology. On this summand the two scalar formulas force \(a^n(1\otimes F_{M_\nu})=a^n1\). Tensoring with a nonzero finite-dimensional vector space is faithful, so \(F_{M_\nu}=1\), proving the assertion even against a possible Jordan part. Decompose both convolution words by (C.22.9). Their actual Ext is the finite sum of IC Ext tensored with coefficient Hom between identity-Frobenius multiplicities. A.2 gives its sole eigenvalue \(a^n\). The structural maps mentioned in the theorem were actual finite-field fusion maps in Lesson 8; normalization uses the same scalar on every ordered input. They consequently retain their descended arithmetic structures under this isotypic description. ∎ #### C.22.4. A multiplicative truncation when the cochain action is locally finite **Theorem C.22.4.1.** Let \(C\) be a dg category, dg algebra, or diagram of complexes over \(E\), with a strict chain automorphism \(F\), fixing every object in the category case. Suppose \(F\) is locally finite on every cochain vector, all its eigenvalues are among \(a^w\) for integers \(w\), and \(a\) is not a root of unity. Suppose \[ H^n(C)\text{ has sole eigenvalue }a^n. \tag{C.22.12} \] Its composition and specified degree-zero multilinear structure maps commute strictly with \(F\). Then there is a natural multiplicative zigzag \[ C\ \longleftarrow\ D(C)\ \longrightarrow\ H^*(C) \tag{C.22.13} \] of quasi-isomorphisms, retaining the ordinary graded structure. For a strictly \(F\)-equivariant higher operation of degree \(-r<0\), its restriction to \(D(C)\) lands in \(D(C)\) and its projection to \(H^*(C)\) is zero. The locally finite and strict cochain hypotheses in this statement must be verified before applying it to an enhanced geometric diagram. **Proof.** Local finiteness means that each vector lies in a finite-dimensional \(F\)-stable space. Primary decomposition there, followed by its unique compatible inclusions, gives the canonical direct sum \[ C=\bigoplus_{w\in\mathbb Z}C_w,\qquad C_w=\{x:(F-a^w)^N x=0\text{ for some }N\}. \tag{C.22.14} \] Distinct integer weights have distinct eigenvalues. The differential preserves these subspaces. Products add their weights: on finite-dimensional primary spaces the tensor automorphism has sole eigenvalue \(a^{w+v}\), since its remaining part is a polynomial without constant term in commuting nilpotent operators. Every \(F\)-equivariant multilinear map therefore sends input weights to their sum. Define, in cohomological degree \(n\), \[ D(C)^n= \bigoplus_{w>n}C_w^n \ \oplus\ \ker(d:C_n^n\to C_n^{n+1}). \tag{C.22.15} \] Weights below \(n\) are omitted. This is a subcomplex. If \(w>n+1\), the differential remains in the full \(w\)-summand; if \(w=n+1\), it lands in the diagonal cycle term; the differential of a diagonal cycle is zero. For a fixed weight \(w\) it is exactly the good subcomplex truncation \(\tau_{\leq w}C_w\). Equation (C.22.12) says that \(H^n(C_w)=0\) for \(n\ne w\), so its inclusion into \(C_w\) is a quasi-isomorphism. Direct sums of vector-space complexes are exact, proving the left arrow in (C.22.13). The right arrow takes a diagonal cycle to its cohomology class and every term of weight \(w>n\) to zero. It is a chain map: an off-diagonal differential can reach the diagonal only as an ordinary boundary. On the sole weight-\(w\) cohomology group it is its defining quotient map, so it too is a quasi-isomorphism. If two inputs have weights at least their degrees, their product has that property. When both weights equal the degrees, both inputs are cycles and the degree-zero chain product is a cycle. Thus (C.22.15) is closed under composition and the degree-zero chain structure maps. The right arrow preserves them: any strict weight excess on an input leaves an excess on the output, and on diagonal cycles it is ordinary cohomological composition. Units lie in degree and weight zero, and are retained. These observations also prove naturality in strictly \(F\)-equivariant chain maps. Tensor comparisons are compatible as well. There is a natural product map \(D(C)\otimes D(C')\to D(C\otimes C')\); both paths to \(C\otimes C'\) are the inclusion of the same tensor of cochains. Tensoring quasi-isomorphisms over a field preserves them, even for unbounded complexes: an acyclic vector-space complex has a contracting homotopy by complements to its cycles, and tensoring that homotopy contracts its tensor with any complex. The two inclusion maps are quasi-isomorphisms, so this product comparison is one. The same multiplication of representatives proves its associative and permutation identities. Finally an operation of degree \(-r\) sends inputs of degrees \(n_i\) and weights \(w_i\ge n_i\) to output degree \(\sum n_i-r\) and weight \(\sum w_i\). For \(r>0\) that output has strictly larger weight than degree. It belongs to the full off-diagonal term of \(D(C)\), and its cohomology projection is zero. Restriction thus retains any supplied operation identities; projection kills their negative-degree terms. Jordan parts have been retained throughout by primary spaces, not discarded by diagonalization. ∎ #### C.22.5. Two cochain-model exercises **Exercise C.22.5.1 (intermediate).** Let \(C=E[x,z]\otimes\Lambda(y)\), with \(|x|=|z|=2,|y|=1\), \(dy=z\) and \(dx=dz=0\). Give all three generators weight two and let \(F\) act by \(a^2\) on them. Compute \(D(C)\) and its map to cohomology. **Solution.** The polynomial generators give diagonal cycle monomials \(x^i z^j\); the monomials \(yx^i z^j\) have weight one above their cohomological degree. These exhaust the basis, so \(D(C)=C\). Their differentials are \(x^i z^{j+1}\). Thus every monomial containing \(z\) is a boundary, the odd terms have no cycles, and \(H^*(C)=E[x]\). The right arrow sends \(x\) to \(x\), \(z,y\) to zero. It is multiplicative and retains the weight-two class of \(x\). The calculation demonstrates why diagonal boundaries are killed, while an off-diagonal primitive is retained until the projection. **Exercise C.22.5.2 (advanced).** In degrees zero and one take \(C^0=C^1=E[t,t^{-1}]\), differential multiplication by \(t-a\), and chain automorphism multiplication by \(t\). Compute its cohomology and show why taking the locally finite subspace of each term is not a valid replacement. **Solution.** The differential is injective in the Laurent polynomial domain and has cokernel \(E\), by evaluation at \(t=a\). Thus \(H^0=0\), and \(H^1=E\) with Frobenius \(a=a^1\), exactly diagonal purity. Yet every nonzero Laurent polynomial has linearly independent translates among sufficiently many powers of \(t\): a finite polynomial relation among them would contradict the domain property. There is no nonzero locally finite vector in either term. Taking those termwise subspaces gives the zero complex and loses \(H^1\). A geometric application must construct a compatible derived locally finite model, rather than making that termwise substitution from cohomological purity alone. ### C.23. Derived primary replacement and coherent arithmetic formality We retain the arithmetic field and normalized Weil objects of §22. Put \[ R=E[t,t^{-1}],\qquad K=E(t),\qquad \mathcal W=\{t-a^w:w\in\mathbb Z\}. \tag{C.23.1} \] An \(R\)-module is the same thing as an \(E\)-vector space with an invertible operator \(F=t\). In every tensor product in this section the base is \(E\), and \(t\) acts diagonally, \(F_{M\otimes N}=F_M\otimes F_N\). It is not a tensor product over \(R\). Let \(\mathsf L_{\mathcal W}\) be the full abelian category of modules that are locally finite for \(F\) and have only the primary eigenvalues \(a^w\). Thus \[ M=\bigoplus_{w\in\mathbb Z} \bigcup_{r\ge1}\ker(F-a^w)^r. \tag{C.23.2} \] The Jordan blocks in (C.23.2) are allowed. A bounded-below derived object will mean one with cohomology zero below some integer; it need not be presented initially by a bounded-below cochain complex. The free corrected author version of [Cirici and Horel, *Étale cohomology, purity and formality with torsion coefficients*](https://geoffroy.horel.org/formality-etale.pdf), §§2 and 4, explains the distinction between locally finite cochains and locally finite cohomology. Here the coefficient field has characteristic zero and \(a\) has infinite multiplicative order. We prove the required derived replacement and coherence statement below; no theorem from that paper stands in place of a proof. #### C.23.1. Resolve first, then take primary vectors **Theorem C.23.1.1.** There is a natural bounded-below complex \(L(C)\) in \(\mathsf L_{\mathcal W}\), for every bounded-below complex \(C\) of \(R\)-modules whose cohomology lies in \(\mathsf L_{\mathcal W}\), and a natural zigzag \[ C\longrightarrow J(C)\longleftarrow L(C) \tag{C.23.3} \] of quasi-isomorphisms. Both \(J(C)\) and \(L(C)\) have injective \(R\)-modules in each term. In particular, this construction applies to the Laurent-polynomial complex of Exercise C.22.5.2, although its original terms have no locally finite vectors. **Proof.** The ring \(R\) is a principal ideal domain: every ideal is obtained by localizing an ideal of the Euclidean polynomial ring \(E[t]\). An \(R\)-module is locally finite precisely when it is torsion. In one direction a finite-dimensional invariant subspace gives an annihilating polynomial with nonzero constant coefficient, since \(F\) is invertible. In the other direction \(Rv\), annihilated by a nonzero Laurent polynomial, is a quotient of a finite-dimensional vector space \(R/(p)\). We need two elementary facts about injectivity. An injective \(R\)-module \(I\) is divisible: a map \((p)\to I\) sending \(p\) to a specified element extends to \(R\). Conversely a divisible module is injective. To prove the converse, extend a map on a submodule to a maximal larger domain by the union of chains. If \(x\) is outside that domain, the ideal \(\{r:rx\text{ belongs to the domain}\}\) is either zero or \((p)\). In the zero case assign any value to \(x\). In the second case choose \(y\) with \(py\) equal to the already prescribed value of \(px\). Then \(f(n+rx)=f(n)+ry\) is well defined and extends the map, contradicting maximality. This proves injectivity without invoking a criterion. Write \(\Gamma_{\rm tor}I\) for the torsion submodule of an injective module. It is divisible: if \(px=v\) in \(I\) and \(qv=0\), then \(qpx=0\), so \(x\) is torsion. It is therefore itself injective. The quotient \(I/\Gamma_{\rm tor}I\) is canonically \(K\otimes_R I\). The kernel of the localization map is precisely torsion; surjectivity follows from divisibility, which supplies division by every nonzero denominator. We have consequently proved \[ 0\longrightarrow\Gamma_{\rm tor}I \longrightarrow I\longrightarrow K\otimes_R I \longrightarrow0. \tag{C.23.4} \] Torsion modules have their canonical primary direct sum decomposition. Indeed an element has an annihilating polynomial with finitely many irreducible factors; Bézout identities for the mutually coprime factor powers give its unique primary pieces. These identities agree when one enlarges an annihilating polynomial. Projection onto the summands in \(\mathcal W\), denoted \(\Gamma_{\mathcal W}\), is exact on torsion modules. Also \(\Gamma_{\mathcal W}I\) is divisible: choose a torsion solution of \(px=v\) as just above, and project that solution onto its \(\mathcal W\)-primary summands. The projection commutes with \(p\) and fixes \(v\). Thus \(\Gamma_{\mathcal W}I\) too is injective over \(R\). Here is a functorial resolution, including the boundedness needed to totalize it. For an \(E\)-vector space \(V\), put \[ I(V)=\operatorname{Hom}_E(R,V),\qquad (t f)(t^j)=f(t^{j+1}). \tag{C.23.5} \] If \(U\) forgets the \(R\)-action, evaluation at \(1\) identifies \(\operatorname{Hom}_R(M,I(V))=\operatorname{Hom}_E(UM,V)\). The latter functor is exact, so \(I(V)\) is injective. Let \(T=IU\), with unit \(\eta_M(m)(r)=rm\) and multiplication \(I\epsilon U:T^2\to T\), where \(\epsilon\) is evaluation at \(1\). The augmented cosimplicial resolution of \(M\) has term \(T^{p+1}M\) in degree \(p\ge0\); its cofaces insert \(\eta\) in each position, and its codegeneracies apply \(I\epsilon U\). The unit and multiplication identities give the cosimplicial identities. After applying \(U\), evaluation at the outermost \(1\) is an extra codegeneracy \(e\). If \(d^i\) is a coface, then \(ed^0=1\) and \(ed^i=d^{i-1}e\) for \(i>0\). Taking the alternating sum gives \(e\delta+\delta e=1\) on the augmented complex. Thus it is exact. Apply this construction degreewise to \(C\), whose terms may be taken zero in degrees below \(b\). Totalize with \[ J(C)^n=\bigoplus_{\substack{p\ge0\\p+q=n}} T^{p+1}C^q,\qquad d_{\rm tot}=\delta+(-1)^p d_C. \tag{C.23.6} \] For each \(n\) the sum has at most \(n-b+1\) nonzero terms. Every term is injective and the total complex is bounded below. The preceding extra codegeneracy contracts the augmented horizontal direction; its cancellation with the vertical differential follows from the signs in (C.23.6). Equivalently filter by the bounded finite diagonals: horizontal exactness says that \(C\to J(C)\) induces an isomorphism on cohomology. This proves the first quasi-isomorphism in (C.23.3). Apply (C.23.4) termwise to \(J(C)\). Localization is exact, so \[ H^n(K\otimes_R J(C))=K\otimes_R H^n(C)=0. \tag{C.23.7} \] The exact sequence of complexes therefore proves \(\Gamma_{\rm tor}J(C)\to J(C)\) a quasi-isomorphism. The primary decomposition of the torsion complex is a direct sum of subcomplexes. Exactness of taking primary summands shows that its summands outside \(\mathcal W\) have zero cohomology, because \(H^n(C)\) belongs to \(\mathsf L_{\mathcal W}\). Hence \[ L(C)=\Gamma_{\mathcal W}J(C) \tag{C.23.8} \] has the claimed inclusion and cohomology. Its terms are injective over \(R\) by the divisibility argument. All constructions commute with \(R\)-linear cochain maps. The resolution is taken before (C.23.8); no assertion has been made about \(\Gamma_{\mathcal W}C\) in an arbitrary original cochain model. ∎ #### C.23.2. The replacement retains tensor coherence Let \(D^+_{\mathcal W}(R)\) be the full enhanced derived category of bounded-below \(R\)-complexes with cohomology in \(\mathsf L_{\mathcal W}\). “Enhanced” here retains mapping complexes and their homotopies, rather than just the degree-zero morphisms in a triangulated category. The tensor product is the diagonal \(E\)-tensor product specified after (C.23.1). **Theorem C.23.2.1.** Inclusion induces a symmetric monoidal equivalence of enhanced derived categories \[ D^+(\mathsf L_{\mathcal W}) \ \xrightarrow{\ \simeq\ }\ D^+_{\mathcal W}(R). \tag{C.23.9} \] Its inverse retains a complete homotopy-coherent diagram of multilinear maps, including units, associators and permutations. It is not the termwise functor \(\Gamma_{\mathcal W}\) on the original diagram. **Proof.** Kernels, cokernels and extensions of modules in \(\mathsf L_{\mathcal W}\) remain in that category. For an extension, lift an element killed by a product of allowed primary powers in the quotient; a second such product kills its image in the submodule, so their product kills the lift. The primary decomposition then proves (C.23.2) for the extension. The category has enough injectives that are also injective over \(R\): embed a module \(M\) in \(I(UM)\) by \(\eta_M\), and take \(\Gamma_{\mathcal W}I(UM)\). Its image contains \(M\); divisibility above proves the target injective over \(R\), hence also in the full subcategory \(\mathsf L_{\mathcal W}\). A bounded-below complex of injective modules is an injective mapping model. Here is the required argument for unbounded source complexes as well. If \(B\) is acyclic and a cochain map \(f:B\to J\) is given, start below the lowest nonzero term of \(J\). Inductively, after choosing \(h^n:B^n\to J^{n-1}\), the map \(f^n-d_Jh^n\) vanishes on \(\ker d_B^n=\operatorname{im}d_B^{n-1}\), by the cochain-map identity and the preceding induction step. It defines a map on \(\operatorname{im}d_B^n\) with values in \(J^n\). Injectivity extends this to \(h^{n+1}:B^{n+1}\to J^n\). These extensions give \(f=dh+hd\). Apply the same argument to every shift of \(J\). Thus its Hom complex from an acyclic complex is acyclic, which is precisely the mapping-model property. For two complexes in \(\mathsf L_{\mathcal W}\), resolve the target by these injectives. The very same Hom complex computes its enhanced derived maps in \(\mathsf L_{\mathcal W}\) and in \(R\)-modules: the inclusion is full, and the chosen injectives work in both categories. This proves full faithfulness on mapping complexes, in every degree. Theorem C.23.1.1 proves essential surjectivity. An object initially bounded below only in cohomology may first be replaced by its good quotient truncation; its canonical quasi-isomorphism supplies the same derived object. The diagonal tensor product preserves \(\mathsf L_{\mathcal W}\). Each finite collection of vectors lies in finite-dimensional primary spaces. On weights \(w,v\), the tensor automorphism has sole eigenvalue \(a^{w+v}\), with a nilpotent remaining part, by the commuting-nilpotent calculation in C.22.4. The tensor product is exact over the field \(E\). It preserves quasi-isomorphisms: a splitting of an acyclic vector-space complex into its boundary pairs supplies a contracting homotopy after tensoring. Tensor products of bounded-below complexes are bounded below. The inclusion in (C.23.9) is therefore an honest symmetric monoidal functor before passing to enhanced derived categories. We explain explicitly why its inverse retains coherence. The functor \(L\) of (C.23.8) preserves quasi-isomorphisms on this subcategory, and (C.23.3) gives a natural derived equivalence \(\zeta_C:UL(C)\simeq C\). The desired tensor comparison is the lift, by full faithfulness, of \[ \zeta_{C\otimes D}^{-1} (\zeta_C\otimes\zeta_D): UL(C)\otimes UL(D)\longrightarrow UL(C\otimes D). \tag{C.23.10} \] It is an equivalence. The unit comparison is obtained in the same way. This lifting uses the full mapping objects, not only their \(H^0\): the preceding identical mapping complexes make the space of a lift, together with its comparison homotopy, contractible. Consequently the associativity and permutation homotopies of (C.23.10) lift too. Their next coherence homotopies lift for the same reason, and so on over the dimension of the supplied diagram. More concretely, a discrepancy on the boundary of a chosen higher homotopy is a cycle in the homotopy fibre of the mapping comparison; that fibre is acyclic, so the discrepancy has a primitive. Differences between choices are filled one degree lower in the same fibre. Each fixed finite-dimensional diagram requires finitely many such steps; the constructions agree as that dimension is enlarged. This proves all coherent identities and their compatible lifts. It proves a symmetric monoidal equivalence, without claiming a strictly symmetric tensor product on the particular coinduced total complex (C.23.6). ∎ #### C.23.3. Diagonal purity formalizes the whole diagram Call a derived \(R\)-complex diagonally pure if it is bounded below, has degreewise finite-dimensional cohomology, and \(H^n\) has sole generalized eigenvalue \(a^n\). These objects are closed under the diagonal \(E\)-tensor product: in each total degree the Künneth sum is finite, and the input eigenvalues multiply to \(a^{n+m}\). **Theorem C.23.3.1.** On the diagonally pure full subcategory, the functor which forgets the \(R\)-action is equivalent, as a symmetric monoidal enhanced functor, to its graded cohomology with zero differential. Therefore a complete diagram of diagonally pure mapping objects with Frobenius-equivariant compositions, tensor maps and structural homotopies is formal together with that entire structure. **Proof.** Use (C.23.9) to lift the derived diagram to locally finite primary complexes. This is a symmetric monoidal lift; it is not a replacement of its members independently followed by unspecified new tensor maps. On complexes in \(\mathsf L_{\mathcal W}\) with diagonal cohomology, C.22.4 gives the two natural maps \[ UC\ \longleftarrow\ UD(C)\ \longrightarrow\ H^*(UC), \tag{C.23.11} \] where \[ D(C)^n=\bigoplus_{w>n}C_w^n \oplus\ker(d:C_n^n\to C_n^{n+1}). \tag{C.23.12} \] Both arrows are quasi-isomorphisms, and \(D\) preserves quasi-isomorphisms on these objects because its cohomology map is the defining cohomology isomorphism. The cochain functor \(D\) is lax symmetric monoidal. Its comparison sends a tensor of representatives to the same tensor in \(D(C\otimes C')\). Input weights at least the input degrees give this inclusion; diagonal inputs are cycles. C.22.4 proved the comparison a quasi-isomorphism. The associative, unit and signed permutation diagrams commute on the representatives themselves. The projection in (C.23.11) is monoidal under the field Künneth isomorphism: diagonal cycles project to their classes, and any positive weight excess projects to zero before or after taking a product. Thus, after passing to derived objects, all comparisons are equivalences and (C.23.11) is a symmetric monoidal equivalence of functors. Compose this equivalence with the coherent inverse of (C.23.9) and with \(\zeta\). Cohomology of that inverse is canonically the original cohomology by (C.23.3), so the result is exactly the asserted formal functor, with no altered graded multiplication. For clarity, a coherent algebraic diagram is made of tensor products of its mapping objects, structure maps out of those products, and specified homotopies among composites. Apply the functors and their monoidal comparisons to every member and every such map. Naturality carries each specified homotopy and its higher faces to the corresponding one. Objectwise equivalences give an equivalence of the entire diagram. On the formal output a strictly equivariant operation of degree \(-r<0\) would have to take eigenvalue \(a^{\sum n_i}\) into eigenvalue \(a^{\sum n_i-r}\); these differ, even with Jordan parts. The map is zero. This is also the representative calculation in C.22.4. Ordinary degree-zero structure maps remain their actual graded cohomology maps. The result is simultaneous formality, not merely an unrelated quasi-isomorphism for each Hom complex. ∎ **Lemma C.23.3.2.** For two diagonally pure objects \(C,D\), every negative-degree derived \(R\)-linear map is zero, and \[ \operatorname{Hom}_{D(R)}(C,D) =\prod_n\operatorname{Hom}_R(H^nC,H^nD). \tag{C.23.13} \] This statement does not assert the vanishing of positive-degree derived \(R\)-linear maps. **Proof.** In a locally finite primary model, decompose into its weight subcomplexes. The weight-\(w\) subcomplex has cohomology only in degree \(w\). Its good subcomplex truncation through \(w\), followed by its cycle quotient in degree \(w\), gives a natural quasi-isomorphism zigzag to \(H^w[-w]\). Direct sums are exact. By (C.23.9), this proves in \(D(R)\) \[ C\simeq\bigoplus_n(H^nC)[-n], \qquad D\simeq\bigoplus_m(H^mD)[-m]. \tag{C.23.14} \] For a finite-dimensional module \(M\) with invertible \(F_M\), the following is a free resolution: \[ 0\longrightarrow R\otimes_E M \xrightarrow{\,t\otimes1-1\otimes F_M\,} R\otimes_E M\longrightarrow M\longrightarrow0. \tag{C.23.15} \] The last map sends \(t^j\otimes v\) to \(F_M^jv\). The middle map is injective by comparing the highest Laurent exponent in a nonzero finite sum. In its cokernel the relation \(t\otimes v=1\otimes F_Mv\), and its inverse using \(F_M^{-1}\), reduces every Laurent sum uniquely to \(1\otimes v\). This proves exactness. Applied to a target module \(N\), its Hom differential, up to a harmless overall sign, is \[ g\longmapsto F_Ng-gF_M. \tag{C.23.16} \] If the two sole eigenvalues are \(a^m,a^n\) with \(m\ne n\), this is invertible: left and right multiplication commute, their nonconstant parts are nilpotent, and their scalar difference is nonzero. Thus both Hom and \(\operatorname{Ext}^1_R(M,N)\) vanish for different weights, and higher Ext vanish by (C.23.15). The direct sum of the shifted two-term resolutions (C.23.15) is a valid projective mapping model for (C.23.14). Each two-term resolution has acyclic Hom into an acyclic complex, and the Hom of their direct sum is the product of those complexes. Products of exact sequences of modules are exact, by choosing a preimage in each coordinate. Hence that product is acyclic. This argument avoids assuming that an arbitrary bounded-below complex of projectives is such a mapping model. For the \(n,m\) terms, a degree-\(k\) map is computed by \(\operatorname{Ext}_R^{k+n-m}(H^nC,H^mD)\). The different-weight terms vanish. The same-weight terms can occur only for \(k=0,1\); their \(k=0\) terms give (C.23.13). This proves the lemma. In particular mapping homotopies of positive dimension, which are negative-degree maps, have no nonzero homotopy classes here. Same-weight \(\operatorname{Ext}^1_R\) may still encode a nontrivial Jordan extension. ∎ #### C.23.4. The actual Weil diagram for convolution words Let \(\mathcal P_{\rm word}\) contain the normalized IC objects, the unit, their finite direct sums, and every finite ordered convolution word, with all parenthesizations retained. The Weil map on a word is the one induced by the actual descended convolution correspondence. It is not assigned after choosing a decomposition into its simple constituents. **Theorem C.23.4.1.** The enhanced geometric mapping objects of \(\mathcal P_{\rm word}\), together with their composition, convolution maps, unit, corrected symmetry and all coherence maps, form a Frobenius-equivariant diagram of diagonally pure objects of \(D^+(R)\). Applying Theorem C.23.3.1 gives an equivalence of the complete symmetric monoidal enhanced generator diagram with its actual graded Ext diagram. **Proof.** We first construct an action on mapping objects, not merely a list of actions on Ext groups. On a finite jet and action-nerve coefficient site, write \(\sigma\) for arithmetic pullback. It is an exact equivalence. A Weil coefficient object is an object \(P\) with an isomorphism \(\alpha_P:\sigma P\to P\). Work in the ambient sheaf or compatible coefficient-diagram categories used for the resolution enhancement in A.4; injectives are not required to be constructible. The forgetful functor from Weil objects has exact left adjoint \[ P\longmapsto\bigoplus_{j\in\mathbb Z}\sigma^jP, \tag{C.23.17} \] where the Weil map shifts the summand labels. Its right adjoint is the corresponding product. The adjunctions follow by evaluating at the summand or coordinate with label zero; the other coordinates of a Weil map are then determined by its structural isomorphism. The left adjoint is exact because \(\sigma\) and direct sums are exact. Consequently forgetting a Weil injective gives an underlying injective: its Hom functor is the Hom functor from the exact left adjoint. There are enough Weil injectives. Embed the underlying object in an injective \(I\), and map the Weil object into the coinduced product of the \(\sigma^j I\). The coordinate at zero makes this map a monomorphism. Coinduction sends an injective to an injective, since forgetting is exact. Repeating on cokernels constructs Weil injective resolutions. The bounded-below mapping-model argument in C.23.2 works both before and after forgetting. Thus the Hom complex between chosen resolved Weil objects computes the actual geometric derived mapping object. For a homogeneous map \(f\) its strict chain automorphism is \[ F(f)=\alpha_Q\,\sigma(f)\,\alpha_P^{-1}. \tag{C.23.18} \] It commutes with the differential and with composition. We have not taken invariant vectors in this Hom complex. Its cohomology is geometric Ext equipped with Frobenius, rather than Ext in the category of Weil objects. At finite coefficient levels (C.23.17) and the coinduced resolution are carried out in the ambient categories, retaining their coefficient reductions and action faces as a derived coefficient diagram. Resolving such a diagram retains its reduction comparisons as maps and their coherent homotopies; we do not claim that a nonflat coefficient reduction preserves an arbitrary injective term. The derived inverse limits in the rational-adic enhancement retain the Weil action: their product and mapping-cone models apply (C.23.18) to each coordinate and each transition. Inversion of \(\ell\) retains it as well. The actual mapping and composition comparisons are Lesson 5, Lemma P.1.3.1 and the enhanced construction in P.1.2; actual rational internal Hom is Theorem O.3.2.2 and rationalization Theorem N.4.3.1. Their underlying Hom complex is consequently the same actual rational-adic geometric mapping object. No locally finite assertion about these product cochain terms is needed. Next retain the operations. The pullbacks, tensor products, descent and proper images that define convolution are performed over the finite field. They act on Weil objects using their structural isomorphisms. Derive these functors in that category. Underlying injective resolutions compute the underlying derived proper images by the forgetful argument above; tensor products use flat resolutions, for which the exact free construction (C.23.17) retains flatness. Hence forgetting recovers the actual geometric derived convolution correspondence. This also proves compatibility at the enhanced level: the correspondence comparisons, adjunction units and counits, and their compositions are maps of Weil coefficient diagrams. On resolution models they give the corresponding maps and comparison homotopies, still with their Weil action. Resolving the entire coefficient diagram retains its faces, rather than choosing independent resolutions for its separate arrows. Derived passage retains the identities and their higher comparison homotopies, because a resolution equivalence induces the same mapping-complex equivalence at every face. This is the full action-nerve and fusion construction of Lesson 8, A–C, with the affine-parameter counits and all block maps proved in C.17. The sign correcting component symmetry is a constant scalar, so it too commutes with Frobenius. All finite ordered fusion coherence diagrams are therefore present in the Weil enhancement. Theorem C.22.3.1 now verifies purity for every member of this diagram: actual mapping cohomology between convolution words in degree \(n\) has sole eigenvalue \(a^n\), with Jordan parts permitted. The earlier parity and finiteness calculations make it finite-dimensional in each degree and zero below degree zero. Although an injective Hom complex may have nonzero cochain terms arbitrarily far below zero, it is a bounded-below *derived* object. Its tensor products remain bounded below and diagonally pure by the field Künneth calculation preceding C.23.3. Thus every required hypothesis of Theorem C.23.3.1 has been verified for this entire diagram. Apply its symmetric monoidal formal functor to the mapping objects and all the operations. Their zero-differential output is the actual graded Ext, with actual composition and convolution. Its associator, unit and corrected permutation maps are their degree-zero cohomology maps. Naturality retains all higher coherent identities. This proves the asserted equivalence of enhanced generator diagrams, not merely a monoidal structure transported abstractly across an unrelated category equivalence. ∎ #### C.23.5. The arithmetic monoidal Satake envelope Recall \(A=\operatorname{Sym}_E(\mathfrak h[-2])\) with zero differential and the full dual group \(H\), including its centre. A free generator is \(F_V=A\otimes_E V\), and \(\operatorname{Perf}_H(A)\) means its idempotent-complete finite-cone envelope, as in C.22.2. Its tensor product is the derived tensor product over \(A\). **Theorem C.23.5.1.** Over the arithmetic geometric fibre with the rational-adic coefficient field fixed above, there is an enhanced symmetric monoidal equivalence \[ \bigl(D^b_{L^+G,\mathrm{fin}}(\mathrm{Gr}_G,E),*\bigr) \ \simeq\ \bigl(\operatorname{Perf}_H(A),\otimes_A^{\mathbf L}\bigr), \qquad A_\lambda\longmapsto A\otimes_E V_\lambda. \tag{C.23.19} \] Its graded generator morphism comparison is the actual cup-and-Kostant map of C.21–C.22. The corrected component symmetry is used on the geometric side. This theorem does not assert a comparison with complex constructible coefficients. **Proof.** Theorem C.23.4.1 formalizes the full enhanced generator diagram, with convolution and every structural map retained. Theorem C.22.1.1 identifies that actual graded symmetric monoidal diagram with the free dual polynomial generators. This identification preserves the actual Yoneda composition and convolution of every homogeneous morphism; its cohomology degree and grading are the ones already checked in C.21. It sends the unit to \(A\) and sends \(A_\lambda\) to \(F_{V_\lambda}\). Composing the two comparisons gives an enhanced symmetric monoidal comparison on all generator objects and morphisms. We spell out extension to the finite envelope. A finite cone construction can be represented by a finite sum of shifted generators with a matrix differential of degree one, whose square is zero; successive cone matrices are upper triangular. Enhanced Yoneda identifies mapping complexes on generators, and its comparison on two such cone constructions is the corresponding finite total matrix mapping complex. It is a quasi-isomorphism by filtering the matrices by the number of cone steps, or by the two-variable mapping-cone induction of A.4 and C.22.2. In the idempotent completion a retract is its coherent projector and splitting data. A mapping-complex equivalence carries that entire data and its retraction identities, so extends to the retracts as well. No image of an arbitrary merely homotopy-idempotent cochain map is being taken as a new cochain subcomplex. The tensor comparison extends with these constructions. On a pair of shifted generators use the generator comparison with its shift signs. On their finite cone matrices the tensor differential is \[ d_{X\otimes Y}=d_X\otimes1+1\otimes d_Y, \tag{C.23.20} \] with the Koszul sign in the second term. The cross terms cancel, so its square is zero. Exactness of actual derived convolution and of derived \(A\)-tensor says that these are their actual total cone constructions. The comparison on the graded entries and on the degree-one cone maps identifies (C.23.20) on the two sides. For retracts tensor their splitting maps and their coherent projectors; the two retraction identities remain identities after tensoring. This supplies a natural tensor equivalence on every finite-envelope object and its full mapping complex. The unit and every associator and permutation comparison extend by the same finite matrix construction; all their coherent identities already hold on the enhanced generator diagram by C.23.4. Thus they hold on these total complexes and on their retracts. Equivalently each finite cone extends the corresponding coherent generator diagram by its actual cone functor, and its comparison is a mapping-complex equivalence in both variables. No new coherence class is introduced by that operation. Every bounded finite-support spherical object belongs to this envelope, by bounded perverse truncation and the semisimple spherical heart, proved in A.4. Every object of \(\operatorname{Perf}_H(A)\) belongs to its specified free envelope by definition. The free-generator mapping complexes are actual derived maps, as the injective-invariant argument of C.22.2 proved; ordinary equivariant coherent Hom has not been substituted for them. Hence the fully faithful comparison on the finite envelopes is essentially surjective and has the tensor structure just constructed. This proves (C.23.19). The entire construction retains the full group \(H\), its central directions and every connected component of \(\mathrm{Gr}_G\). ∎ ![Resolve the Laurent complex before taking primary vectors; transport its entire coherent Weil diagram; obtain actual arithmetic monoidal Satake](assets/derived-primary-monoidal-formality.png) The first panel is the actual complex and its explicit primary model in Exercise C.23.6.1; the basis is infinite, and a finite cutoff does not have the displayed cohomology. The middle panel displays the derived equivalence and simultaneous weight truncation of C.23.2.1–C.23.3.1, applied to the actual Weil mapping objects of C.23.4.1. Its tensor product is over \(E\) with diagonal Frobenius. The last panel is C.23.5.1, whose target tensor is over \(A\); its IC labels mean the normalized objects \(A_\lambda\). This is the arithmetic geometric fibre, with the full dual group and corrected component symmetry. Editable SVG source. #### C.23.6. Two derived-action exercises **Exercise C.23.6.1 (intermediate).** Put \(C=R\xrightarrow{t-a}R\) in degrees zero and one. Use the derived replacement theorem to compute \(L(C)\) up to quasi-isomorphism. What is the formal underlying \(E\)-complex, and why does the zero termwise locally finite subcomplex not compute it? **Solution.** The injective differential in \(C\) has cokernel \(R/(t-a)\), so \(H^1(C)=E\) with action \(a\), and all other cohomology is zero. Equations (C.23.3) and (C.23.8) give \(L(C)\simeq E[-1]\) as a derived \(R\)-object. An explicit locally finite model is obtained by setting \(p=t-a\) and \(P_a=R[p^{-1}]/R\), and placing \[ P_a\xrightarrow{p}P_a \quad\text{in degrees one and two}. \tag{C.23.21} \] Its basis \(e_j=p^{-j}\bmod R\), \(j\ge1\), satisfies \(p e_j=e_{j-1}\), where \(e_0=0\), and \(t e_j=a e_j+e_{j-1}\). To obtain that basis, reduce a numerator modulo \(p^j\); since \(a\ne0\), the Laurent variable is invertible in this quotient, and its residue has a unique polynomial representative of degree less than \(j\). Thus every vector in \(P_a\) lies in a finite Jordan block of eigenvalue \(a\). Multiplication by \(p\) is surjective with kernel \(Ee_1\). The cochain map from \(C\) to (C.23.21) is zero in degree zero and sends \(r\) in degree one to \(r e_1\). It is \(R\)-linear, commutes with the differential, and induces the isomorphism on \(H^1\). This also checks the recovery of cohomology without an abstract resolution choice. The underlying formal complex is the one-dimensional space in degree one. The original terms \(R\) are torsion free, so their termwise locally finite submodules are zero. The resolution adds the torsion injective terms needed before taking primary submodules. The computation distinguishes the derived replacement from that invalid zero complex. A finite cutoff of \(P_a\) would create a spurious cokernel at its last basis vector; the surjectivity just used depends on retaining all \(e_j\). **Exercise C.23.6.2 (advanced).** Let \(M=R/(t-a)^2\), with the two-dimensional Jordan action of eigenvalue \(a\), placed in cohomological degree one. Compute all \(H^kR\operatorname{Hom}_R(M[-1],M[-1])\), and compute the derived maps to \(E[-2]\) with eigenvalue \(a^2\). Explain which result is used for coherent formality. **Solution.** The resolution \(R\xrightarrow{(t-a)^2}R\) of \(M\) gives a zero differential after applying \(\operatorname{Hom}_R(-,M)\). Thus the self-mapping complex has \(M\) in degrees zero and one, and zero cohomology elsewhere. In particular its negative-degree cohomology is zero, but its degree-one group is two dimensional. For the target with eigenvalue \(a^2\), the same differential is multiplication by \((a^2-a)^2\), a nonzero scalar. Its Hom complex is acyclic; the shift of the target does not change that fact. The vanished cross-weight maps and negative-degree maps agree with C.23.3.2. The retained self-\(\operatorname{Ext}^1_R\) records possible Weil extensions; it is not an extra geometric IC Ext group. Coherent formality uses the derived primary equivalence and diagonal truncation, retaining the given Jordan action throughout. ### C.24. Descending composition formality through coefficient fields The following coefficient argument is independent of a Betti–étale comparison. Its conclusion supplies descent of the enhanced *composition* model. The actual geometric mapping diagrams are compared in C.27–C.29. Their complete rational symmetric monoidal comparison is [Theorem C.80.1.1](#c801-the-rational-represented-tensor-comparison). The lemmas here concern the enhanced composition model. The elementary algebraic inputs are proved in the earlier programme: The Nullstellensatz and Jacobson rings, Theorems 2.1–2.2, Formally smooth, unramified and étale ring maps, Theorems 1.2 and 4.1, and Krull dimension and Noether normalization, Lemmas 1.3 and 4.1 and Theorem 4.2. #### C.24.1. Rational points on triangular torsors **Lemma C.24.1.1.** Let \(k\) be a field of characteristic zero, \(U\) a reduced closed subgroup of a finite unitriangular matrix group over \(k\), and \(T\) a right \(U\)-torsor of finite type over \(k\). If \(T\) is geometrically nonempty, it has a \(k\)-point. Images and kernels of homomorphisms between such groups are reduced closed connected unipotent groups, and a surjection between them is surjective on points over any characteristic-zero extension of \(k\). **Proof.** Write \(N\) for the strictly upper triangular matrix algebra, with \(N^r=0\) for a sufficiently large \(r\). The matrix polynomials \[ \log(1+x)=\sum_{j=1}^{r-1}(-1)^{j+1}x^j/j, \qquad \exp(x)=\sum_{j=0}^{r-1}x^j/j! \tag{C.24.1} \] are inverse on \(N\) and \(1+N\). We verify their restriction to \(U\). Over an algebraic closure, a reduced algebraic group in characteristic zero is smooth. For the generic-point calculation, the function field of an irreducible component has a separating transcendence basis: choose any transcendence basis, and its remaining finite algebraic extension is separable in characteristic zero. Choose a finite generating tower for that extension. Each successive minimal polynomial has nonzero derivative, since an irreducible polynomial in characteristic zero is relatively prime to its derivative. Clearing the finitely many denominators of the equations and of the coordinate-ring generators, and inverting those derivatives, identifies a nonempty open with a successive tower of standard étale hypersurfaces over an open in affine space. The standard smooth presentation theorem and composition give a smooth nonempty open. Translation moves one such smooth point to any group point. Thus every point is smooth. For \(u\in U\), every integer power \(u^j\) belongs to \(U\). Each defining equation of \(U\) therefore vanishes on \(\exp(s\log u)\) at every integer \(s\). It is a polynomial in \(s\), so vanishes identically. Its derivative at zero shows \(\log u\in\mathfrak u=\operatorname{Lie}U\). Conversely let \(x\in\mathfrak u\). The left invariant vector field defined by \(x\) is tangent to \(U\) at each point, by translation. For a polynomial \(f\) in the radical defining ideal of \(U\), its derivative under that field vanishes on \(U\), so remains in that ideal. The same is true of all iterated derivatives. The derivative of \(f(\exp(sx))\) is that vector-field derivative along the curve, since \((\exp(sx))'=\exp(sx)x\). All Taylor coefficients at zero are therefore zero. As the expression is polynomial, \(f(\exp(sx))=0\). Hence \(\exp(x)\in U\). Thus (C.24.1) restricts to inverse polynomial isomorphisms between the variety \(U\) and the vector space \(\mathfrak u\). The argument is over the algebraic closure and all formulas are defined over \(k\), so the isomorphisms descend to \(k\). In particular \(U\) is connected. For a homomorphism \(\phi:U\to V\), \[ \phi(\exp(sx))=\exp(s\,d\phi(x)). \tag{C.24.2} \] Indeed its left side is a polynomial one-parameter group; differentiating \(h(s+t)=h(s)h(t)\) at \(t=0\) gives \(h'=h\,d\phi(x)\), \(h(0)=1\). Coefficient comparison in that polynomial differential equation proves the stated exponential. Its image is therefore \(\exp(d\phi(\mathfrak u))\), a closed connected subvariety and subgroup of \(V\); its kernel is \(\exp(\ker d\phi)\). Over any extension field, a point of the image has a logarithm in that linear image, and a linear lift gives a preimage. This proves the image, kernel and point-surjectivity assertions. Choose a point \(z\) of \(T\) over an algebraic closure. Because \(T\) is of finite type and nonempty, such a point exists and its finitely many coordinates lie in a finite extension; this is the weak Nullstellensatz applied after scalar extension. Enlarge it to a finite Galois extension \(L/k\) containing those coordinates. Write \[ \sigma z=z\,u_\sigma,\qquad u_{\sigma\tau}=u_\sigma\,\sigma(u_\tau). \tag{C.24.3} \] The \(u_\sigma\) lie in \(U(L)\), since the torsor action identifies each fibre with \(U\). Put \(U_i=U\cap(1+N^i)\). The degree-\(i\) matrix entries define a homomorphism from \(U_i\) into the additive vector space \(N^i/N^{i+1}\), with kernel \(U_{i+1}\). Its image is the \(k\)-linear image of \(\mathfrak u\cap N^i\): exponential reduces to \(1+x\) modulo \(N^{i+1}\). Therefore the map is onto that vector space on \(L\)-points, by a linear lift followed by exponential. Also \([U,U_i]\subset U_{i+1}\), since multiplying matrices of degrees one and \(i\) gives degree at least \(i+1\). Suppose the cocycle values have already been put in \(U_i\). Their degree-\(i\) components \(c_\sigma\) satisfy the additive cocycle identity \(c_{\sigma\tau}=c_\sigma+\sigma c_\tau\). Let \[ b=\frac1{|\operatorname{Gal}(L/k)|}\sum_\tau c_\tau. \tag{C.24.4} \] Then \(\sigma b=b-c_\sigma\). Choose \(v\in U_i(L)\) with degree-\(i\) component \(b\). Replacing \(z\) by \(zv\) replaces the cocycle by \(v^{-1}u_\sigma\sigma v\); its degree-\(i\) component is \(-b+c_\sigma+\sigma b=0\). Its values now lie in \(U_{i+1}\). Beginning with \(U_1=U\) and iterating through the finite matrix filtration kills the whole cocycle. The resulting point is fixed by \(\operatorname{Gal}(L/k)\), hence has coordinates in \(k\). It is the required \(k\)-point. ∎ #### C.24.2. Compatible points on a tower of formality torsors **Theorem C.24.2.1.** Suppose \(T_1\leftarrow T_2\leftarrow\cdots\) is a tower of geometrically nonempty finite-type torsors under finite-dimensional reduced closed unitriangular groups \(G_i\), with transition maps equivariant for group homomorphisms \(G_{i+1}\to G_i\). Then there is a compatible system of \(k\)-points. Surjectivity of the original transition maps is not required. **Proof.** For \(i\ge j\), the geometric image of \(T_i\to T_j\) is a coset of the image \(G_i\to G_j\). The latter is closed and connected by C.24.1. Thus this image is a reduced closed subvariety of \(T_j\); its defining ideal is the scheme-theoretic image ideal, and it descends to \(k\) because the transition map is defined over \(k\). It is a torsor under that image group. For a fixed \(j\), these closed image groups form a descending chain in \(G_j\). Their dimensions can decrease only finitely often. Once the dimension is constant, a closed connected subgroup contained in the previous connected group of the same dimension equals it: a proper closed subset of an irreducible variety has smaller dimension, and a connected unipotent group is irreducible by its exponential vector-space parametrization. The groups therefore stabilize. Their nested nonempty cosets stabilize too. Denote the resulting image torsor in \(T_j\) by \(S_j\), and its group by \(H_j\). The map \(S_{j+1}\to S_j\) is geometrically onto. To check this, choose \(i\) large enough that the images in both \(T_j\) and \(T_{j+1}\) have stabilized. The image of \(T_i\) in \(T_{j+1}\) is \(S_{j+1}\), and its image in \(T_j\) is \(S_j\); composition gives the assertion. The corresponding homomorphism \(H_{j+1}\to H_j\) is onto as well, by the same stabilized image calculation. By C.24.1, \(S_1\) has a \(k\)-point. Over any chosen \(k\)-point of \(S_j\), the fibre of \(S_{j+1}\to S_j\) is a geometrically nonempty torsor under the kernel \(H_{j+1}\to H_j\). Indeed the map is equivariant and geometrically surjective, and two lifts differ by precisely that kernel. The kernel is a reduced closed connected unipotent group by C.24.1. The same lemma supplies a \(k\)-point in the fibre. Induction chooses compatible \(k\)-points for all \(j\), giving a compatible system in the original \(T_j\). This proves the assertion without choosing points independently in a nonsurjective tower. ∎ #### C.24.3. A composition-formality descent theorem **Theorem C.24.3.1.** Let \(\mathcal C\) be a strictly unital minimal \(A_\infty\) category over \(k\) with countably many objects. Suppose each Hom space is zero in negative degrees and degree one, is finite-dimensional in each degree, and has only the scalar identities in degree zero: \(\operatorname{Hom}^0(X,Y)=0\) for \(X\ne Y\), and \(\operatorname{Hom}^0(X,X)=k\,1_X\). If a characteristic-zero field extension makes \(\mathcal C\) strictly unital \(A_\infty\)-equivalent to its ordinary graded category, then such an equivalence already exists over \(k\), with first component the identity on the graded Hom spaces. **Proof.** Normalize all higher operations at identity arguments; strict unitality means they vanish there. Use the reduced suspended Hom spaces \(s\overline{\operatorname{Hom}}\), where \(s\) lowers degree by one and the identities have been omitted. They have degrees at least one. The reduced bar coalgebra \(B\) is the direct sum of their finite composable words. Its degree-one coderivation \(b\) records the given operations; \(b_2\) records just the ordinary product, with \(b_2^2=0\). An equivalence to the formal model is a degree-zero coalgebra automorphism \(g\) with \[ g b=b_2g. \tag{C.24.5} \] If its first component is an invertible graded map, the arity-two identity says that component preserves the ordinary product. Composing on the target by its graded inverse consequently makes the first component the identity. Thus the hypothesis gives a solution of (C.24.5) with that first component over the extension field. Enumerate the objects. For stage \(D\), retain the first \(D\) objects and the reduced bar subcoalgebra \(B_{\le D}\), relative to their scalar identity algebra, of total suspended degree at most \(D\). It is finite-dimensional: a word has at most \(D\) letters, all their degrees are between one and \(D\), the object set is finite, and each Hom degree is finite-dimensional. Its coproduct remains inside it because each subword has no greater degree. Coordinate changes fix the objects, respect every Hom source and target, preserve degree, and have first component the identity. Every other component lowers word length. They form a reduced closed algebraic subgroup \(U_D\) of a finite unitriangular group: the coalgebra, object-label and identity-component equations are polynomial, and inversion is the finite sum for a word-length-lowering matrix. Impose (C.24.5) on inputs of total degree at most \(D-1\), so every required output has degree at most \(D\). No claim that \(b\) preserves the bar subcoalgebra in its top degree is needed. These are finite polynomial equations in \(U_D\). Over the extension field they have a solution. If \(g,h\) are two solutions, then \(h g^{-1}\) is precisely a coordinate change commuting with \(b_2\) in the same output degree range; conversely any such change sends a solution to a solution. Thus the reduced solution variety \(T_D\) is a geometrically nonempty torsor under the reduced closed stabilizer \(G_D\subset U_D\). Restriction of words and objects gives the equivariant transition maps \(T_{D+1}\to T_D\) and \(G_{D+1}\to G_D\). Theorem C.24.2.1 gives compatible solutions over \(k\). Each fixed finite word, output coefficient and object tuple occurs at some finite stage and its coordinate stabilizes thereafter. The compatible solutions therefore define a coalgebra automorphism on all of \(B\). Its inverse is still determined by the finite word-length recursion. Every instance of (C.24.5) has finite input degree and finitely many object labels, so is checked at a sufficiently large stage. The resulting \(g\) proves the theorem. For a dg category with these cohomological properties, A.3.1.1 supplies the minimal model over \(k\). Its chosen contraction remains a contraction after field extension, so the minimal-model operations base-change to that same model. The theorem then descends composition formality of the dg generator category and its finite enhanced Yoneda envelope. This last passage is the two-variable cone and coherent-retract argument of A.4. It does not descend extra tensor coherence that has not been included in the coordinate equations, and does not supply a geometric Betti–étale comparison. ∎ ### C.25. Actual Betti comparison on rootwise resolution generators Fix a finite extension \(E/\mathbf Q_\ell\), its valuation ring \(O\), and \(O_n=O/\varpi^n\). Analytification is first applied to the finite coefficient diagrams, before derived completion and rationalization. On the Betti side the resulting comparison with ordinary constructible \(E\)-complexes will be proved on the generator category in question. In particular, a discrete constant étale \(\mathbf Q_\ell\)-sheaf is not being substituted for the complete coefficient theory. Trivialize the Tate lines on the complex side with compatible positive complex roots of unity. Only the comparison maps, rather than the general comparison theorem, are imported from *Comparison with singular cohomology*, Section 3. Its maps are transposed inverse-image counits, preserve the mixed compact/ordinary cup product, and commute with localization and proper base change. The proof of its Theorem 7.2 uses a proper-constructibility input. None of C.25.1–C.25.5 below consumes that theorem. #### C.25.1. Compact comparison for a finite toric-cell filtration Call a finite-type complex scheme toric-cell filtered if its reduced scheme has a finite closed filtration whose differences are finite disjoint unions of \(\mathbf A^a\times\mathbf G_m^b\). The filtration is part of the hypothesis; a partition without closed initial unions is insufficient. **Theorem C.25.1.1.** For such a scheme \(C\), the actual compact analytification map \[ R\Gamma_{c,\mathrm{et}}(C,O_n) \longrightarrow R\Gamma_c(C^{\mathrm{an}},O_n) \tag{C.25.1} \] is a quasi-isomorphism. These maps preserve coefficient reductions, the compact/ordinary cup product, and all the localization triangles defining the filtration. Both sides are perfect coefficient complexes. Derived completion and rationalization give the corresponding comparison for \(E\), with its actual cup and localization maps. **Proof.** Start with \(\mathbf P^1\). The unit and the trace-positive \(c_1\mathcal O(1)\) give its two free groups in degrees zero and two, by Lesson 6, Section 5.13(c). Section 5.13(d) checks the actual analytification map on those classes: for transition functions \(g_{ij}=\exp(2\pi i u_{ij})\), the integral cocycle \(u_{ij}+u_{jk}-u_{ik}\) reduces to the Kummer boundary of \(\exp(2\pi i u_{ij}/\ell^m)\). The positive-root trivialization therefore sends that Chern class to the positive classical class. This proves the projective-line comparison itself, not just equality of its two ranks. Products of projective lines have the tensor product of these two-term projective complexes. The actual external cup and iterated proper projection identify the monomial bases, as in (5.45) there. Coordinate boundary intersections are again products of projective lines or points, and their restriction maps retain the units and Chern classes. For \(\mathbf A^a\times\mathbf G_m^b\), take the product compactification \((\mathbf P^1)^{a+b}\), removing infinity in the first \(a\) factors and zero and infinity in the last \(b\). The finite boundary Čech resolution is the tensor product of the one-factor localization triangles. Comparison is an isomorphism on every compactification and boundary term, and commutes with their restriction maps. It is consequently an isomorphism on their total compact-support complex. This also proves that this complex is perfect; the individual boundary terms are finite projective complexes. Induct along the stated closed filtration. For a closed previous stage and its open difference, extension by zero gives a compact localization triangle on each side. The maps between these triangles are the actual maps of Section 3 of the comparison lesson. The open term compares by the preceding computation and the previous closed term by induction. Two out of three gives the new stage, with its original map. Disjoint finite unions cause no additional issue. Reduction to \(C_{\mathrm{red}}\) is legitimate by nilpotent invariance of the étale site and of the analytic underlying space. Use the same finite compactifications and localization diagrams with \(O\) coefficients on the classical side. Their cell complexes are finite free, and the resulting complex is perfect by closure under finitely many cones. Its reductions are the preceding finite complexes. A bounded finite projective \(O\)-complex maps isomorphically to the derived inverse limit of its reductions: termwise the limit is the original finite projective module, and the surjective reductions have no positive derived limit. The complete étale comparison is the derived limit of (C.25.1), so is an isomorphism as well. Tensoring by \(E\) is exact. Since each step used the same finite diagrams, all the asserted maps, rather than independently chosen cohomology identifications, survive completion and rationalization. ∎ #### C.25.2. The trace in a smooth toric-cell filtered scheme **Lemma C.25.2.1.** If \(Z\) is smooth of pure complex dimension \(d\) and toric-cell filtered, compact comparison commutes with the structural trace \[ R\Gamma_{c,\mathrm{et}}(Z,E(d))[2d]\longrightarrow E \quad\text{and}\quad R\Gamma_c(Z^{\mathrm{an}},E)[2d]\longrightarrow E. \tag{C.25.2} \] The second trace uses the complex orientation. The compatibility is retained for finite coefficients and their complete diagrams when the input has finite Tor amplitude. **Proof.** The smooth trace and supported point classes on the étale side are those of Lesson 5, P.2.4 and Q.1.2. The classical compact-support adjoint and its orientation are constructed in GL-PERV, Appendices F.4–F.5 and H.5. Both point classes have trace one. Their local compatibility can be checked without a comparison theorem for a general smooth variety. At the origin of the affine line, the trace-positive class is the negative of the raw supported Kummer boundary. The explicit singular-cochain calculation in *Sheaf cohomology and singular cohomology on manifolds*, Proposition 6.3 and Corollary 6.4, evaluates that boundary as the inverse positive root on the oriented relative disc. Its negative is the positive oriented class. Taking external products gives the same statement for the origin of \(\mathbf A^d\); there is no permutation sign because the point classes have even degree. An étale coordinate chart at a smooth complex point is an analytic local biholomorphism. Restriction, étale purity and excision preserve these supported classes, so this local assertion holds at every smooth point of \(Z\). For each connected component, the étale top compact group is \(E(-d)\), with the point class a generator and trace one, by the trace-normalized smooth formula of Lesson 5. The classical group is \(E\), with the same property, by oriented compact-support duality and GL-PERV F.5. The actual compact comparison sends the point class to that class: this is naturality for its closed support and the local calculation just made. Thus the two traces agree on a generator, and on the finite sum over components. This is an equality of derived maps as well. Smooth trace and classical orientation vanish outside the top compact degree. More explicitly, over \(E\), maps from a bounded complex to \(E[-2d]\) are its top-degree linear dual; no higher coefficient Ext remains. At finite level \(O_n\) the coefficient ring is injective over itself: a map from its ideal \((\varpi^a)\) to \(O_n\) takes the generator to an element annihilated by \(\varpi^{n-a}\), hence to \(\varpi^a y\), and extends by multiplication by \(y\). The ideal extension argument of C.23.1.1 then gives injectivity. Consequently the same assertion about maps into the top coefficient holds there. The classical adjoint construction can be computed on bounded flat models of finite-Tor inputs; replacing tensor by derived tensor in F.4–F.5 preserves the adjunction calculation. The point calculation and traces commute with reduction, so their equalities hold in the whole complete diagram. ∎ **Lemma C.25.2.2 (the classical adjoint with these coefficient rings).** The classical compact-support adjoint and complex-manifold orientation used above exist over \(O\) and \(O_n\), with derived coefficient Hom. On a smooth complex \(d\)-manifold their structural dualizing object is respectively \(O[2d]\) and \(O_n[2d]\), with its orientation trace. Thus the classical adjunction in C.25.3 may be computed integrally before completion. **Proof.** We give the coefficient repair to GL-PERV F.1–F.5 explicitly, because their stated coefficient scope is a field. The compact closed-section extension, c-soft quotient argument and compact continuity in F.1–F.2 use only finite gluing and exact filtered colimits. They apply to modules over either ring. The dimension bound in F.2 uses the minimal closed subset of the real line on which a nonzero class survives, and the exact closed-cover sequence splitting that subset into two half-lines. That proof likewise uses no tensor operation or field property. Iterating it and the local c-soft gluing proves a finite uniform compact cohomological bound on the finite Euclidean atlases in use. Products and filtered colimits of flat modules are flat over either coefficient ring. Over \(O\), a product of flat modules is torsion free, and the finite principal-ideal tensor test proves flatness; filtered colimits preserve that test. For \(O_n\), the test is \[ \ker(\varpi^a:M\to M)=\varpi^{n-a}M,\qquad 1\leq a\leq n. \tag{C.25.2a} \] Indeed the alternating free resolution of \(O_n/(\varpi^a)\), with maps \(\varpi^a\) and \(\varpi^{n-a}\), identifies its first Tor with this kernel quotient. Every finitely presented module is a finite sum of cyclic modules: diagonalize its finite presentation matrix by choosing an entry of least valuation, using divisibility to clear its row and column, and repeating on the smaller matrix. Testing the resulting cyclic summands proves the flatness criterion. The passage from finitely presented tests to arbitrary modules is by their finite presentation diagrams and exact filtered colimits, equivalently the finite-relation tensor test. Equation (C.25.2a) holds for a product of flat modules componentwise; a family in the right-hand image has a preimage chosen in each component. It also passes to filtered colimits, since the kernel is a finite linear equation. This proves the assertion. For a sheaf with flat stalks, its Godement sheaf has flat stalks: sections of that sheaf are products of the original stalk modules, and a new stalk is the filtered colimit of such products. The original sheaf embeds with a stalkwise split map, since evaluation of a germ of a Godement section at that very point is a retraction. Its quotient therefore has flat stalks too. Iterate. The Godement terms are flasque and hence c-soft here. Truncate at the compact cohomological bound \(N\). Dimension shifting and the c-soft criterion of F.1 show that the last quotient is c-soft, and its stalks are still flat. We obtain a bounded flat c-soft resolution of the constant coefficient sheaf. Its augmented complex is stalkwise split exact, by the successive evaluation splittings, so tensoring it with any sheaf preserves the augmentation quasi-isomorphism. If \(M\) is flat and c-soft, the proof of F.3 shows that \(F\otimes M\) is c-soft for every sheaf \(F\). Its resolution by sums of the free open-extension sheaves stays exact after tensoring, because \(M\) is flat. The tensor terms are sums of open extensions of \(M\), and are c-soft. Dimension shifting through \(N\) of these terms gives vanishing of \(H_c^1\) for the tensor; the criterion in F.1 proves c-softness. Thus \(T_M(F)=\Gamma_c(F\otimes M)\) is exact and preserves colimits. The explicit adjoint \[ \mathcal D_M(J)(U)= \operatorname{Hom}_R\!\left(\Gamma_c(U,M|_U),J\right), \qquad R=O\text{ or }O_n, \tag{C.25.2b} \] is proved by the same free-open section presentation and its relation presentation in F.3. For injective \(J\), it is an injective sheaf, since \(T_M\) is exact. Use the bounded flat c-soft resolution just constructed in formula F.12, and an injective coefficient resolution of \(R\). For \(O_n\), \(R\) itself is injective by C.25.2.1. For \(O\), use \(O\to\operatorname{Frac}(O)\to\operatorname{Frac}(O)/O\); the last two modules are divisible, and the ideal extension proof of C.23.1.1, applied to this principal ideal domain, makes them injective. The same finite-total-complex currying and its differential in F.13 give \[ R\operatorname{Hom}_R(R\Gamma_c(U,F),R) \simeq R\operatorname{Hom}_U(F,a_U^!R). \tag{C.25.2c} \] All target terms are injective by (C.25.2b). This verifies derived Hom, not only the underived adjunction. The proper-support composition argument of H.3 uses sections and their supports, so retains its proof with these coefficients as well. On coordinate balls of real dimension \(m\), compact singular cochains have the sole group \(R\) in degree \(m\), by the relative sphere calculation of GL-PERV C.5. The local form of (C.25.2c) therefore gives the orientation sheaf in degree \(-m\), just as in F.5. Smaller-ball extension preserves its oriented generator, and complex-coordinate changes have positive real determinant. Hence on a complex \(d\)-manifold it is \(R[2d]\). Evaluation identifies its counit with the orientation trace. This proves the coefficient extension used in the preceding lemma and in the following correspondence formula. ∎ #### C.25.3. A correspondence formula for the actual mapping complex Let \(f:Y\to X\) and \(g:Z\to X\) be proper maps of separated finite-type complex schemes, with \(Y,Z\) smooth of dimensions \(d_Y,d_Z\). Put \(C=Y\times_XZ\), with projections \(k:C\to Y\) and \(h:C\to Z\). Suppose \(Z\) and \(C\) are toric-cell filtered, and all geometric fibres of \(f\) and \(g\) have such filtrations. Let \[ B_f=Rf_*E_Y[d_Y],\qquad B_g=Rg_*E_Z[d_Z]. \tag{C.25.3} \] Their Betti versions use the same proper maps and shifts. **Theorem C.25.3.1.** Completed analytification identifies these objects and induces a quasi-isomorphism on their actual enhanced mapping complexes. Under the trace-normalized identifications, the mapping complex is \[ R\operatorname{Hom}_X(B_f,B_g) \simeq R\operatorname{Hom}_E\!\left(R\Gamma_c(C,E),E\right) [-d_Y-d_Z](-d_Z). \tag{C.25.4} \] The analogous Betti formula uses the complex orientation and the trivialized Tate line. If \(c\) is the compact comparison from the étale complex to the Betti complex, the map from the étale Hom complex to the Betti Hom complex is \(D_E(c^{-1})\), with the displayed shift and twist. The inverse matters because linear duality reverses arrows. This is the map induced by analytification of morphisms, rather than an unrelated isomorphism having the same dimensions. **Proof.** Proper base change computes every stalk of the object comparison for \(Rf_*O_n\) as the ordinary comparison of its proper fibre. That fibre is toric-cell filtered by hypothesis, and ordinary and compact cohomology coincide on it. C.25.1.1 therefore proves that the original proper-pushforward comparison is a stalkwise quasi-isomorphism. The same holds for \(g\). To compute the map on Hom, apply ordinary adjunction for \(g^*,Rg_*\), then actual proper base change for \(f\): \[ \begin{aligned} R\operatorname{Hom}_X(Rf_*E_Y,Rg_*E_Z) &\simeq R\operatorname{Hom}_Z(g^*Rf_*E_Y,E_Z)\\ &\simeq R\operatorname{Hom}_Z(Rh_*E_C,E_Z). \end{aligned} \tag{C.25.5} \] These equalities are equalities of enhanced mapping objects. They do not take invariants or discard chain homotopies. If \(a_Z\) is the structural map, its smooth dualizing object is \(a_Z^!E=E_Z(d_Z)[2d_Z]\). Properness of \(h\) and proper-support adjunction give \[ \begin{aligned} R\operatorname{Hom}_Z(Rh_*E_C,a_Z^!E) &\simeq R\operatorname{Hom}_E(Ra_{Z!}Rh_*E_C,E)\\ &\simeq R\operatorname{Hom}_E(R\Gamma_c(C,E),E). \end{aligned} \tag{C.25.6} \] Here \(Z\) need not be proper over the point. The first equality uses \(a_{Z!}\), and \(Rh_*=Rh_!\) only because \(h\) is proper. Removing the smooth dualizing shift and then inserting the two shifts in (C.25.3) gives \(-2d_Z+d_Z-d_Y=-d_Y-d_Z\), with twist \(-d_Z\). This proves (C.25.4). The algebraic adjunction and smooth normalization are precisely Lesson 5, P.2.3–P.2.4; the classical adjunction and orientation are GL-PERV F.4–F.5 and H.3–H.5. There is a useful way to verify which Hom map this computes. The pairing of a morphism in (C.25.5) with compact cohomology of \(C\) is pullback and proper base change to \(Z\), followed by evaluation and the structural trace of \(Z\). The analytification maps commute with the first two operations and evaluation, by the transposed-counit calculation in the comparison lesson, Lemma 3.1. They commute with the last trace by C.25.2.1. The resulting square of pairings commutes. Since (C.25.6) is exactly the derived-dual adjunction representing this pairing, that square gives \(\langle a(H),c(v)\rangle_{\mathrm{an}}=\langle H,v\rangle_{\mathrm{et}}\). Thus \(D_E(c)\,a=1\) under the displayed identifications, and \(a=D_E(c^{-1})\), with the stated shift and twist. A degree-by-degree dimension argument would not establish this assertion. For clarity about coefficients, do this calculation first on finite-Tor \(O_n\)-models. Classical smooth duality on these models is the explicit flat c-soft and injective coefficient construction of C.25.2.2; algebraic finite reduction and mapping adjunction are Lesson 5, P.1.2 and P.2.2. All the compact complexes of \(C\) are the reductions of the perfect \(O\)-complex constructed in C.25.1.1. Their duals therefore have the same property. Taking derived limits computes the complete Hom comparison, and localization computes the rational one. On the ordinary Betti side one obtains the same perfect integral mapping model. One can use an injective coefficient resolution \(O\to\operatorname{Frac}(O)\to\operatorname{Frac}(O)/O\) in the classical compact-adjoint construction; both latter modules are injective by the divisible-module ideal extension proof, and flat resolutions supply derived tensor. A coordinate ball has free compact cohomology in its top dimension, so that construction gives \(O_Z[2d_Z]\) with its trace. Equations (C.25.5)–(C.25.6) then give the dual of the perfect \(O\)-complex of \(C\). Its completion and rationalization are exactly its ordinary \(E\)-version. This proves the required identification of the completed Betti generator enhancement with the ordinary constructible \(E\)-enhancement, without a claim about naive inverse limits of arbitrary analytic sheaves. ∎ #### C.25.4. Retaining the entire group action nerve **Theorem C.25.4.1.** Let \(Q\) be a smooth complex algebraic group with a finite toric-cell filtration, and let \(f_i:Y_i\to X\) be proper \(Q\)-equivariant maps with smooth toric-cell filtered sources. Suppose every \(Y_i\times_XY_j\) and every fibre of \(f_i\) is toric-cell filtered. Completed analytification is fully faithful on the enhanced generator category of the actual \(Rf_{i*}E[d_i]\), both underlying and genuinely \(Q\)-equivariant, and on its finite cone and coherent-retract envelope. It retains all compositions, units, and homotopies in these diagrams. **Proof.** On action-nerve level \(m\), use \(X_m=Q^m\times X\), \(Y_{i,m}=Q^m\times Y_i\), and the proper map \(f_{i,m}\). Its fibres are the original proper fibres. Its pair correspondence is \(Q^m\times(Y_i\times_XY_j)\). Products of two finite toric-cell filtrations have a finite closed toric-cell filtration: use the product closed-open triangles, or order pairs of closed stages lexicographically and insert their open product filtrations. Its source is smooth of dimension \(m\dim Q+d_i\). Thus C.25.3.1 applies at every nerve level. The objects at that level retain shift \(d_i\), rather than the dimension of \(Y_{i,m}\). Repeating (C.25.5)–(C.25.6) with these actual shifts gives \[ \begin{aligned} R\operatorname{Hom}_{X_m}(B_{i,m},B_{j,m}) \simeq{}& R\operatorname{Hom}_E\!\left( R\Gamma_c(Q^m\times(Y_i\times_XY_j),E),E\right)\\ &[-2m\dim Q-d_i-d_j](-m\dim Q-d_j). \end{aligned} \tag{C.25.7} \] This is the levelwise formula, not the equivariant answer at level zero. The object comparison was defined by the same inverse-image counits on every level. It is therefore natural for the action, projection, multiplication, identity and all their iterated pullback comparisons. The Hom comparison is induced by that functor on morphisms; C.25.3.1 proves that it is an equivalence, without replacing it by an independently chosen map. It follows that the whole cosimplicial mapping diagram compares. The genuine equivariant mapping object is its derived totalization, by Lesson 5 N.4 and the complete action-nerve mapping construction of P.1.3.1. A levelwise equivalence induces an equivalence on the derived limit: this follows directly by computing a homotopy limit with its product and cone model, or by the defining mapping property of a limit. No invariants in a single ordinary group representation, nor an underived product totalization, is substituted. Composition, identity and their higher coherence are already preserved by completed analytification. The totalized maps inherit them from that same diagram. Finally extend to finite shifts and cones in two variables, using the actual long exact mapping triangles. This is the fully faithful finite-envelope argument of A.4. Extend to coherent retracts using their actual projector and splitting diagrams; apply the functor to those diagrams. Thus no image of an arbitrary homotopy-idempotent cochain is taken. This proves the assertion on the full stated envelope. ∎ #### C.25.5. Applying the comparison to the rootwise Grassmannian objects **Theorem C.25.5.1.** For a split complex reductive \(G\), the preceding comparison applies to every finite collection of rootwise resolution images on finite Grassmannian supports, including their full finite-jet Iwahori action diagrams. It identifies the corresponding underlying and genuinely Iwahori-equivariant \(E\)-generator envelopes with their ordinary Betti versions. The actual IC objects in those envelopes are sent to the Betti IC objects, with their stalk and costalk tests. Ordered wall-operation diagrams in this scope also compare. **Proof.** Use the wall resolutions of Lesson 11, C.2.1.1. Its proof establishes a smooth projective source, an actual \(I\)-equivariant proper map, and affine-paved reduced geometric fibres. These are all-characteristic statements, so apply over \(\mathbf C\). The finite-word root charts bound every loop coefficient needed for the source action. For a given finite collection, take a common jet order larger than those finitely many bounds and the target-support bounds. This gives an actual smooth finite-jet Iwahori quotient \(I_N\), not just its action on tangent vectors. Its scheme is \(T\times\mathbf A^u\) by the ordered root and congruence coordinates of Lesson 1, Theorem 2.C, and C.13. Consequently it has a toric-cell filtration. The source of each wall word has a finite affine paving. To see it in the same charts, select successively the closed point and the open affine-line chart in each wall factor. Over an already selected root chart the torsor has its explicit root representative, so the next open chart is its product with \(\mathbf A^1\); the next closed chart is that previous chart. Ordering the closed chart before the open chart recursively gives closed initial unions. This is the wall-chart construction of Lesson 2, Section 7.4 and Lesson 6, Section 5.3. It remains to check the pair correspondence, since fibre pavings alone would not establish that hypothesis of C.25.4.1. Choose a finite union of Iwahori cells containing both images and a finite closed cell ordering. Over one such cell \(S\simeq\mathbf A^e\), its ordered root coordinates give a section \(S\to I_N\) taking a fixed normalizer point \(x\) to the varying point. Acting with this section on each resolution gives actual isomorphisms \[ f_i^{-1}(S)\simeq S\times f_i^{-1}(x),\qquad (Y_i\times_XY_j)|_S \simeq S\times f_i^{-1}(x)\times f_j^{-1}(x). \tag{C.25.8} \] The inverses act with the inverse section. These are scheme isomorphisms coming from the action, not merely pointwise homeomorphisms. The fibres have the closed affine pavings of Lesson 11, C.2.1.1: at a flag endpoint its closed/open induction is \(F_{a-1}(v)\) and \(\mathbf A^1\times F_{a-1}(vs_a)\); at a Grassmannian endpoint the auxiliary finite-flag root sections give the products in (C.2.1.4) there. Product pavings and the pullback of the closed base-cell ordering now give a finite closed affine paving of the reduced pair correspondence. To insert a closed stage of an open product cell, adjoin the whole preceding closed base part; its complement is open in the total stage. This verifies closedness at every step. Since the maps are proper and the supports proper, the pair correspondence itself is proper. All hypotheses of C.25.4.1 are now verified for \(Q=I_N\). Increasing the common jet level retains the full group nerve; the omitted congruence kernel is handled by the actual affine-space unit comparison on that nerve in C.13. It is not discarded on the basis of one fibre computation. For completeness, an IC retract can be chosen in the genuine Iwahori envelope, as well as in the underlying envelope. At an Iwahori cell, the finite-jet stabilizer of its normalizer point has its split torus and the explicitly ordered affine unipotent root coordinates. It is connected. Its classifying-space cohomology is therefore the even polynomial torus cohomology: the full unipotent nerve comparison is C.13, and the torus calculation is C.14. A genuine equivariant constant cohomology sheaf there has the trivial stabilizer coefficient action, because its isomorphism matrix is locally constant on this connected group and equals the identity at the unit. Successive Postnikov attachments between cohomology degrees of the same parity vanish: their degree differences plus one are odd, whereas this actual stabilizer Hom has only even degrees. Thus these restrictions split into shifted constant coefficients in the stabilizer category. The closed-open Hom induction of Lesson 11, C.2.2.1 consequently works equivariantly too. Its open-cell term is computed with that stabilizer polynomial cohomology; its closed-cell term uses ordinary restriction of the source and exceptional restriction of the target. Matching parities make both odd Hom groups zero. It follows that the dense-cell identities lift to equivariant maps \(u:I_\lambda\to B_\lambda\) and \(v:B_\lambda\to I_\lambda\). Their underlying composite is the identity by intermediate-extension uniqueness, exactly as in Lesson 11, C.2.3.1. Their equivariant composite is the identity as well, since degree-zero morphisms between perverse sheaves are their actual sheaf morphisms with a compatibility condition; the forgetful map in degree zero is injective. The split triangle for \(u\) therefore gives \(B_\lambda\simeq I_\lambda\oplus\operatorname{Cone}(u)\). Transport the literal direct-sum projection and inclusion along this equivalence. This supplies the coherent projector and splitting diagram needed in the finite envelope, rather than the image of a homotopy-idempotent cochain. Analytification carries this coherent splitting to a Betti retract \(P\). Its ordinary restrictions are the analytifications of those of \(I_\lambda\), by the proper object comparison and the same projector. Duality on the resolution image is its smooth-source orientation followed by proper duality. The trace compatibility of C.25.2.1 and the adjunction calculation of C.25.3.1 identify this duality with Betti duality, including its evaluation on morphisms. It extends over the finite cone and coherent-retract envelope by applying these actual dual diagrams. Therefore exceptional restriction of \(P\), computed as stratum dual of ordinary restriction of its dual, is also the comparison of the original exceptional restriction. All the perverse inequalities and strict boundary inequalities of the IC are retained. The dense restriction is the constant line in degree \(-d_\lambda\). The intermediate-extension characterization identifies \(P\) with the Betti \(I_\lambda\). Finally, the proper wall operation is built from a wall contracted product, exact inverse image, tensor, and its proper image. The finite root frames and their action nerve are the same ones used above. Analytification preserves the first two operations and their units, and its proper comparison on these wall-word resolution objects is the already proved one. The comparisons are natural for concatenation and for the actual descended maps. Thus the ordered wall-operation diagrams, their compositions and homotopies compare as asserted. This statement concerns these Iwahori generators and wall operations; spherical induction, geometric specialization from a finite-field fibre, and ordered moving fusion still require their own comparisons. ∎ ![The actual correspondence Hom formula retains dimensions and twists; its forward Betti map is the dual of the inverse compact comparison; every Iwahori nerve level is retained](assets/complex-generator-correspondence.png) Theorem C.25.3.1 proves the displayed mapping formula and identifies the actual arrow by evaluation and trace. Theorems C.25.4.1-C.25.5.1 retain the full action nerve and verify the rootwise pair-correspondence pavings and both IC perverse tests. Editable SVG source. #### C.25.6. Two shift and coefficient checks **Exercise C.25.6.1.** Apply (C.25.4) to \(f=g=1_{\mathbf P^1}\). What are the two Hom degrees, including the Tate coefficient? **Solution.** Here \(d_Y=d_Z=1\) and \(C=\mathbf P^1\). Its compact complex is \(E\oplus E(-1)[-2]\). Formula (C.25.4) is \[ (E\oplus E(1)[2])[-2](-1) =E(-1)[-2]\oplus E. \tag{C.25.9} \] It has \(E\) in degree zero and \(E(-1)\) in degree two, as \(R\Gamma(\mathbf P^1,E)\) must. Using \(+d_Y+d_Z\), or omitting the twist, would fail this check. **Exercise C.25.6.2.** Let both maps be \(\mathbf P^1\to\operatorname{Spec}\mathbf C\). Can the resolution generator have negative self-Hom, and what does its pair correspondence predict? **Solution.** The shifted image is \(B=E[1]\oplus E(-1)[-1]\). Its pair correspondence is \(\mathbf P^1\times\mathbf P^1\), with compact groups in degrees zero, two and four of ranks \(1,2,1\). Formula (C.25.4), with the same shift \([-2](-1)\), gives self-Hom in degrees \(-2,0,2\), of those reversed ranks. Explicitly the negative-degree entry maps the degree-one coefficient of \(B\) to its degree-minus-one coefficient; the positive-degree entry reverses them. Their two compositions are the diagonal projectors in degree zero. Thus the formula retains the full matrix algebra and its composition. A nonnegative-Hom claim about arbitrary resolution images would be false; the nonnegative IC-generator hypothesis in C.23–C.24 is a separate statement. ### C.26. Actual specialization of the rootwise enhanced diagram Let \(R\) be a strictly henselian discrete valuation ring, let \(S=\operatorname{Spec}R\), and let \(\ell\) be invertible on \(S\). Retain \(O,O_n,E\) from C.25. A relative toric-cell filtration means a finite closed filtration whose open differences have étale topoi identified with finite disjoint unions of \(\mathbf A^a_S\times\mathbf G_{m,S}^b\) by closed nilpotent immersions. An actual isomorphism is allowed as the simplest case. Thus the definition concerns actual scheme maps and topological invariance, rather than a list of fibre dimensions. Choose compatible Tate generators on \(S\), once, and use their pullbacks on every fibre. The precise arbitrary-base inputs are the programme lesson *Cohomology with compact support*, Theorems 8.1, 9.1 and 10.3, and *Poincaré duality for smooth varieties*, Sections 2, 4–6, 8 and Theorem 9.1. The latter proves \(q^!A=q^*A(d)[2d]\), with the trace counit, for a separated smooth finite-presentation \(q\) over a quasi-compact quasi-separated base. Its proof uses a uniform derived trace system, including constructibility and bounded ghosts, rather than a field-only smooth normalization. Theorem 12.3 of the compact-support lesson contains its exact affine-line constructibility proof reduction; this is the premise used in that trace-system proof. C.25's independent toric-cell argument remains sufficient for the complex comparison previously proved there. #### C.26.1. Relative compact objects are perfect constants **Theorem C.26.1.1.** Suppose \(a:C\to S\) is separated of finite presentation and has a relative toric-cell filtration. Then \(Ra_!O_n\) is a perfect constant coefficient object on \(S_{\mathrm{et}}\). The actual base-change maps, followed by sections, give quasi-isomorphisms \[ K_{C,n}:=R\Gamma(S,Ra_!O_n) \longrightarrow R\Gamma_c(C_b,O_n) \tag{C.26.1} \] for every algebraically closed geometric fibre \(b\to S\), including an algebraically closed extension of the generic field. These comparisons retain the entire finite localization diagram, coefficient reductions and compact/ordinary cup products. The compatible complete and rational comparisons have the same properties. **Proof.** Global sections on the strictly henselian local étale site are exact. Indeed, lift a given section under an epimorphism on an étale cover. One member contains a point over the closed geometric point. The henselian property and separably closed residue field give a section of that pointed étale neighbourhood over \(S\); pulling the lift back gives the required global lift. The same argument shows that restriction of sections to the closed geometric stalk is bijective. For a constant module its sections are that module. Hence the constant functor sends a bounded finite projective coefficient complex \(K\) to a K-projective sheaf complex, with \[ R\operatorname{Hom}_{S_{\mathrm{et}}}(K_S,L_S) \simeq R\operatorname{Hom}_{O_n}(K,L). \tag{C.26.2} \] For finite projective terms this is the exact sections calculation; bounded totalization proves it for perfect complexes. In particular the image of the perfect constant functor is closed under finite cones and coherent retracts. All connecting maps between its objects lift through the same fully faithful enhanced functor. For \(\mathbf P^1_S\), the actual unit and relative \(c_1\mathcal O(1)\) give \[ O_{n,S}\oplus O_{n,S}(-1)[-2] \xrightarrow{\sim}R\pi_*O_n. \tag{C.26.3} \] This is The Satake category, Section 5.13(c)'s relative projective-line computation. It may also be checked on every geometric stalk using proper base change and the two trace-normalized generators. Its maps are the same relative unit and Kummer class before and after base change. Use the product compactification by projective lines for a relative toric cell. Remove infinity in an affine-line factor, and zero and infinity in a multiplicative-line factor. Products of (C.26.3) and their actual boundary restrictions give a finite Čech compact-support complex of perfect constant objects. Consequently the compact object of the cell is a perfect constant. This construction uses a proper compactification over \(S\), so its base-change map is precisely the compact-support base-change map, with its units and localization boundaries. Induct along the relative closed filtration. The compact localization triangle has perfect constant open term and perfect constant previous-stage term. Its connecting morphism lifts by (C.26.2), and its middle term is the corresponding coefficient cone, pulled back to \(S\). This proves the assertion about the whole object. Closed nilpotent immersions do not change the étale site or these coefficient functors, so the permitted thickened strata give the same calculation. The actual counit \[ (K_{C,n})_S\longrightarrow Ra_!O_n \tag{C.26.4} \] is an isomorphism because the target is in this constant image. Pulling it back to \(b\) and applying compact base change identifies it with (C.26.1). This proves the identity of the comparison map, not only its invertibility. Do the finite compactification construction with the compatible \(O_n\)-diagram. The cell objects come from bounded finite free \(O\)-complexes. At a filtration step, take the connecting map in the complete constant category and its finite cone. The complete constant category on perfect inputs is again the coefficient category: the finite constant mapping complexes in (C.26.2) compute their compatible derived limit, equal to the mapping complex of the finite projective \(O\)-models. Thus each step gives a perfect \(O\)-model with its actual derived reductions. The inverse-limit argument of C.25.1.1 proves the complete comparison, and exact localization proves the \(E\)-comparison. No cohomology-before-reduction interchange is used. ∎ #### C.26.2. The relative trace identifies specialization on Hom Let \(f:Y\to X\) and \(g:Z\to X\) be proper \(S\)-maps of separated finite-presentation schemes. Suppose \(Y,Z\) are smooth over \(S\), of relative dimensions \(r,s\), and \(C=Y\times_XZ\) has a relative toric-cell filtration. Set \[ B_f=Rf_*O_n[r],\qquad B_g=Rg_*O_n[s]. \tag{C.26.5} \] **Theorem C.26.2.1.** Actual geometric restriction gives a quasi-isomorphism on the enhanced Hom between (C.26.5) and its corresponding Hom on every algebraically closed fibre. With the compatible Tate generators, its relative formula is \[ R\operatorname{Hom}_{X}(B_f,B_g) \simeq R\operatorname{Hom}_{O_n}(K_{C,n},O_n) [-r-s](-s). \tag{C.26.6} \] If \(\kappa_b\) is (C.26.1), restriction on Hom is \(D_{O_n}(\kappa_b^{-1})\), with these same shifts and twist. The complete and rational formulas hold as well. **Proof.** The arbitrary-base relative smooth formula quoted above is stated for cyclic coefficients. Here is the finite scalar extension needed for \(O_n\). Choose \(\mathbf Z/\ell^m\to O_n\). A derived \(O_n\)-module object is resolved by its free-coefficient bar resolution, whose terms are \(O_n\otimes^L_{\mathbf Z/\ell^m}F\). Its augmentation becomes split after forgetting the coefficient action, by the extra degeneracy given by that action and the unit. The realization is therefore the original module object. On a free term, proper-support projection and scalar adjunction identify maps into a proposed \(O_n\)-right adjoint with the cyclic-coefficient smooth adjunction. Its formula \(q^*A(d)[2d]\) retains the \(O_n\)-action of \(A\). Passing through the bar resolution proves the adjunction for all \(O_n\)-module inputs; its counit is the derived scalar extension of the same trace. Equivalently one can use a K-flat dg algebra resolution of the coefficient ring to implement this free bar resolution. This argument does not require flatness of \(O_n\) over the cyclic coefficient ring. Let \(h:C\to Z\). Ordinary adjunction and actual proper base change give \[ \begin{aligned} R\operatorname{Hom}_{X}(Rf_*O_n,Rg_*O_n) &\simeq R\operatorname{Hom}_{Z}(g^*Rf_*O_n,O_{n,Z})\\ &\simeq R\operatorname{Hom}_{Z}(Rh_*O_{n,C},O_{n,Z}). \end{aligned} \tag{C.26.7} \] Writing \(q:Z\to S\), relative smooth duality says \(q^!O_{n,S}=O_{n,Z}(s)[2s]\), with the trace counit. Properness of \(h\) and proper-support composition then give \[ R\operatorname{Hom}_{Z}(Rh_*O_{n,C},q^!O_{n,S}) \simeq R\operatorname{Hom}_{S}(Ra_!O_{n,C},O_{n,S}). \tag{C.26.8} \] Use C.26.1.1 and (C.26.2) on the right. Removing \([2s](s)\) and inserting the two shifts in (C.26.5) yields \(-2s+s-r=-r-s\) and twist \(-s\), proving (C.26.6). These are relative dimensions: the dimension of the trait has not been added. The pairing represented by (C.26.8) is evaluation followed by the relative trace of \(q\). Proper base change, evaluation, and this trace commute with the given geometric restriction; the trace compatibility is Theorem 2.1(T1) and Theorem 9.1 of the relative-duality lesson. Thus, if \(a_b\) is the actual map on Hom, \[ \langle a_b(H),\kappa_b(v)\rangle_b=\langle H,v\rangle_S, \qquad D_{O_n}(\kappa_b)\,a_b=1. \tag{C.26.9} \] Since \(\kappa_b\) is an equivalence, this proves \(a_b=D_{O_n}(\kappa_b^{-1})\). It also proves that the original restriction functor supplies the quasi-isomorphism. The same functor therefore preserves compositions and homotopies. The compact \(O_n\)-complexes are derived reductions of the perfect \(O\)-object in C.26.1.1, so their duals are reductions of its perfect dual. The compatible adjunctions and traces give the same inverse-limit mapping diagram. Taking its limit, then localizing, proves the complete and rational assertions, retaining (C.26.9). This is a comparison on the family generator diagram, rather than a claim that arbitrary coefficient sheaves on the trait are determined by one geometric fibre. ∎ #### C.26.3. Specialization on the genuine action nerve **Theorem C.26.3.1.** Let \(Q/S\) be a smooth affine group of relative dimension \(D\), with a relative toric-cell filtration. Suppose \(f_i:Y_i\to X\) are proper \(Q\)-equivariant maps with \(Y_i/S\) smooth of relative dimensions \(r_i\), and every pair correspondence \(Y_i\times_XY_j\) has a relative toric-cell filtration. Then geometric restriction is an enhanced equivalence from the finite shift, cone and coherent-retract envelope of these family generators onto the corresponding generator envelope of any algebraically closed fibre, both underlying and genuinely \(Q\)-equivariant. The assertion holds with the compatible complete and rational coefficients. **Proof.** On action-nerve level \(m\), the corresponding source is \(Q^m\times_SY_i\), smooth over \(S\) of relative dimension \(mD+r_i\). The proper map to \(Q^m\times_SX\) has pair correspondence \(Q^m\times_S(Y_i\times_XY_j)\). Product closed filtrations give a relative toric-cell filtration, just as in C.25.4.1. Its object retains shift \(r_i\). C.26.2.1 therefore gives the levelwise formula \[ \begin{aligned} R\operatorname{Hom}_{X_m}(B_{i,m},B_{j,m}) \simeq{}& D_{O_n}K_{\,Q^m\times_S(Y_i\times_XY_j),n}\\ &[-2mD-r_i-r_j](-mD-r_j). \end{aligned} \tag{C.26.10} \] It proves that the actual geometric-restriction map is an equivalence on each level. Inverse image commutes with every action, identity and multiplication square. Its proper comparisons are their original adjunction mates. Hence the whole cosimplicial mapping diagram, with all its faces and degeneracies, compares. Taking its derived totalization proves fully faithful genuine equivariance. This is the complete action-nerve construction of Semi-infinite orbits and weight functors, N.4 and P.1.3.1, applied to these actual finite diagrams. There is a coefficient point when rationalizing that totalization. Put \(d_{ij}=\dim((Y_i\times_XY_j)_{\bar b})\), or any fixed uniform bound for these relative fibre dimensions. Compact cohomology on level \(m\) vanishes above \(2mD+2d_{ij}\), by the relative compact-support bound. The dual formula (C.26.10) therefore has no cohomology below \[ L_{ij}=r_i+r_j-2d_{ij}, \tag{C.26.11} \] independently of \(m\). Each level is a perfect complete coefficient complex. The finite coefficient cohomology groups are finite, so their inverse systems satisfy the Mittag–Leffler condition and completion retains this lower bound. For a cohomological window through degree \(N\), good truncation at the uniform lower bound uses only finitely many cosimplicial degrees, including those in degree \(N+1\) needed to compute cycles: \(m\leq N+1-L_{ij}\), since the total degree is \(m+q\) with \(q\geq L_{ij}\). Finite cohomological windows and their finite product-and-cone limit models consequently commute with exact coefficient localization. Equivalently apply the finite-filtration cosimplicial spectral sequence in each such window. Completion is already the limit of the compatible diagrams. This justifies the rational mapping comparison; it does not exchange localization with an arbitrary infinite product of coefficient modules. Extend full faithfulness to finite shifts and cones in two variables by the original mapping triangles, and to coherent retracts by applying restriction to their full splitting diagrams. Every fibre generator is the actual restriction of its family generator. A morphism or coherent projector used to build a fibre envelope object lifts through the fully faithful enhanced functor; its entire coherence diagram lifts with contractible comparison fibres, as in C.23.2.1 and A.4. Induction through the finite construction proves essential surjectivity. This proves the enhanced equivalences, including all the indicated diagrams. ∎ #### C.26.4. Rootwise families and the actual IC identification **Theorem C.26.4.1.** Fix a pinned split reductive datum. There is a strictly henselian arithmetic trait with geometric closed fibre \(\overline{\mathbf F}_p\) and a complex geometric generic fibre for which C.26.3.1 applies to every finite collection of rootwise resolution words and their finite-jet Iwahori diagrams. It identifies their full generator envelopes and carries the corresponding actual IC objects to one another. It preserves the ordered wall-operation diagrams. Combining it with C.25 gives the actual comparison of these arithmetic geometric and ordinary complex Betti generator diagrams over finite \(E/\mathbf Q_\ell\). **Proof.** Take the integral pinned group of Pinnings and the classification of split reductive groups, Theorem 10.1. Choose \(p\ne\ell\), and take the strict henselization of \(\mathbf Z_{(p)}\) at a geometric closed point. Its pointed étale construction, localness and residue field are proved in *Henselian local rings and henselization*, Theorem 5.1. We verify the needed discrete valuation assertion. Localize each finite-presentation pointed étale stage at its selected closed prime, obtaining a noetherian local flat algebra \(A\). Its closed fibre is a finite separable field, so its maximal ideal is \((p)\); flatness makes multiplication by \(p\) injective. By *Noetherian and Artinian rings*, Theorem 6.1, \(\bigcap_a p^aA=0\). Every nonzero element therefore has a greatest exponent \(a\) for which it is in \(p^aA\), and is \(p^a\) times a unit. Products of two such elements are nonzero, because powers of \(p\) are nonzero divisors. Thus \(A\) is a domain and a discrete valuation ring with uniformizer \(p\). The local transition maps preserve \(p\) and units and are injective: a nonzero element \(p^a u\) cannot become zero at a later stage. Localization at a selected prime gives the same colimit as the pointed construction, since every element outside that prime becomes a unit in a further pointed neighbourhood. A nonzero ideal in this union has a least occurring exponent \(a\), and is exactly \((p^a)\). The union is consequently a noetherian discrete valuation ring. Its residue field is \(\overline{\mathbf F}_p\). Each local stage's fraction field is a finite extension of \(\mathbf Q\): its generic étale finite-presentation algebra is a finite product of finite separable fields, and localization selects one factor. The union's fraction field is algebraic over \(\mathbf Q\). Its algebraic closure embeds in \(\mathbf C\), supplying the required complex geometric generic fibre. Use the integral wall quotients and contracted products of Loop groups and the affine Grassmannian, Section 7.4, with the initial length-zero torsor. They give the family maps \(f_i:\mathcal B_i\to X\), with smooth projective sources of the word lengths as relative dimensions. All root exponents, normalizer representatives and action bounds are the same integral ones. A finite collection admits one common finite-jet Iwahori quotient. Its underlying scheme is \(T_S\times\mathbf A_S^u\), by the full ordered root and congruence coordinates of C.13. In particular it is smooth and relative toric-cell filtered. We check the relative fibre paving, not merely its two geometric-fibre versions. For a flag endpoint section \(yI\), forgetting the final wall identifies its fibre with the preceding endpoint's inverse image of \(yP_s/I\). If \(v\) is the shorter of \(y,ys\), the latter relative projective line has closed section \(vI\) and open root chart \[ z\longmapsto x_{v(a_s)}(z)\,vsI,\qquad z\in\mathbf A^1_S. \tag{C.26.12} \] The root is positive and lies in the Iwahori. Its left action gives a scheme isomorphism of the open inverse image with \(\mathbf A^1_S\times F_{\mathrm{previous}}(vs)\); the closed part is \(F_{\mathrm{previous}}(v)\). Both isomorphisms are those determinant-one wall-chart formulas on all rings. At length zero the fibre is the section \(S\), or empty. Insert the recursively closed filtrations, adjoining the whole preceding closed part to each open-filtration stage. This gives the relative affine paving by induction. No root structure constant has been inverted. For a Grassmannian endpoint, insert the auxiliary flag fibre \(G/B\) and the Borel selected by the signs of \(\nu+\epsilon\gamma\), exactly as in The Mirković–Vilonen theorem: identifying the dual group, C.2.1.1. Automorphisms, forms and parabolic subgroups, Theorem 7.1 gives its relative root sections. Their conjugates by \(t^\nu\) are in the Iwahori, by the same integer root exponents. They trivialize the inverse image over a flag cell as its affine-space product with the flag-endpoint fibre just treated. The closed flag and Schubert stage orderings can be taken from the finite proper wall-word images. On every geometric fibre these images contain exactly the downward Bruhat labels established by the integral root-coordinate calculation. The relative root cells are locally closed immersions with those same labels. Hence on an open difference of two such closed stages, its root-cell immersion covers the whole underlying space. It is then a closed immersion with nilpotent defining ideal: a locally closed immersion with full underlying image is closed; every element of its ideal lies in the ambient nilradical, and finite generation makes the ideal nilpotent. Thus these differences have the étale topoi of the stated relative affine cells. This explains why neither flatness of every closed stage nor an identity of the coproduct of strata with the full test-ring functor is being assumed. It suffices for the precise relative-filtration hypothesis. For a pair correspondence, apply the root section of an Iwahori cell simultaneously to the two resolutions. On the actual root-cell scheme \(A_S^e\) this gives \[ (\mathcal B_i\times_X\mathcal B_j)|_{A_S^e} \simeq A_S^e\times_S F_i(x)\times_S F_j(x). \tag{C.26.13} \] Its inverse uses the inverse root section. Products of the relative fibre pavings, inserted into the pulled-back finite closed base-cell ordering, give the required relative affine paving of the pair correspondence. Pullback of the possible nilpotent stage immersions is still nilpotent. This verifies every geometric hypothesis of C.26.3.1 over the trait itself. An actual IC in the closed-fibre Iwahori category is a coherent retract of its rootwise resolution image. The equivariant lifting argument in C.25.5.1 applies in this characteristic as well: finite-jet stabilizers are connected split torus by affine unipotent coordinates; C.13–C.14 compute their full even classifying-space cohomology; matching IC and resolution parity kills the closed-open odd-Hom obstruction. Dense-cell identities give \(u,v\) with \(vu=1\), and the split triangle supplies the coherent splitting. General IC parity and the required actual dense identity are already Lesson 11, C.2–C.4. Lift this whole retract through the enhanced equivalence of C.26.3.1. To identify its generic fibre, retain both perverse tests. Relative duality on a resolution image is its smooth-source relative orientation followed by proper duality: \[ D_{X/S}B_i=B_i(r_i). \tag{C.26.14} \] Here is the proper-duality identity being used. For any test object \(A\), the projection formula and exceptional adjunction identify \[ \begin{aligned} \operatorname{RHom}_X(A,D_{X/S}Rf_*F) &\simeq\operatorname{RHom}_X(A\otimes^LRf_*F,\omega_{X/S})\\ &\simeq\operatorname{RHom}_Y(f^*A\otimes^LF,\omega_{Y/S})\\ &\simeq\operatorname{RHom}_X(A,Rf_*D_{Y/S}F). \end{aligned} \] Properness changes \(f_!\) into \(f_*\), and composition of exceptional right adjoints gives \(f^!\omega_{X/S}=\omega_{Y/S}\). These are the projection and composition identities already retained in C.26.2.1. Yoneda gives the asserted identity, with its evaluation map. Smooth-source orientation makes its bidual evaluation an isomorphism. This proves (C.26.14) and its biduality assertion. Finite mapping triangles and coherent splittings extend the assertion to finite cones and coherent retracts. Relative trace and the comparison of C.26.2.1 make these identities, evaluations and maps commute with restriction. On a relative Iwahori cell, ordinary restriction of a resolution image is the constant perfect relative compact object of its proper fibre, pulled to that cell by the root section. C.26.1.1 proves its actual comparison on either geometric fibre. The same is true of its dual and its coherent retracts. Write \(i:T\to X\) for the cell inclusion. The exact exceptional restriction identity is \[ i^!D_{X/S}P\simeq D_{T/S}i^*P. \] Indeed, test its left side against \(A\), move \(i_!A\) by adjunction, and use the projection formula \(i_!A\otimes^LP=i_!(A\otimes^Li^*P)\). The resulting pairing has target \(i^!\omega_{X/S}=\omega_{T/S}\) and is exactly the right side. Yoneda proves the identity. Apply it to \(D_{X/S}P\) and the bidual evaluation just proved to obtain \(i^!P=D_{T/S}i^*D_{X/S}P\). The cell's relative orientation adds exactly twice its relative dimension. Consequently the lifted IC's ordinary and exceptional restrictions have the same coefficient complexes, with the same degrees, on both geometric fibres. The strict boundary inequalities and dense constant line of the closed-fibre IC are therefore retained on the generic fibre. Intermediate-extension uniqueness identifies that fibre with the actual generic IC of the same label. This argument does not put an absolute perverse shift for the dimension of the trait into the IC. Finally, wall operations use the same integral contracted products, coefficient inverse images and proper images. Their frame actions and complete nerves commute with restriction, and the actual proper base-change maps identify their resulting words. Thus their ordered operation diagrams and homotopies compare as well. Increasing a jet level retains the affine congruence-kernel nerve comparison of C.13. Applying C.25 to the complex geometric fibre gives the stated Betti comparison. This theorem establishes the rootwise and Iwahori enhanced diagram; the spherical convolution and full fusion comparison are separate remaining assertions. ∎ ![Relative compact objects are perfect constants; specialization on Hom is the dual of the inverse compact comparison, with relative shifts and full action nerve](assets/relative-specialization.png) Theorems C.26.1.1–C.26.2.1 prove the compact object and trace identities in the diagram. Theorem C.26.3.1 retains the full nerve and the uniform degree bound before coefficient localization; C.26.4.1 transports the actual IC using both stalk and costalk tests. Editable SVG source. #### C.26.5. Coefficients and the growing nerve **Exercise C.26.5.1.** Let \(K=[O\xrightarrow{\varpi}O]\) in degrees zero and one. Compare \(H^0(K)\otimes_OO_1\) with \(H^0(K\otimes_O^LO_1)\). Which operation in C.26.1 preserves the correct coefficient diagram? **Solution.** The first group is zero. Since the two terms of \(K\) are free, its derived reduction is represented by the same two terms over \(O_1\) with zero differential. Its degree-zero and degree-one groups are both \(O_1\). The extra degree-zero group is the Tor contribution of \(H^1(K)=O/\varpi\). Therefore take the finite projective coefficient complex, its actual derived reductions and their cones, as in C.26.1.1. Reducing an already extracted cohomology group loses part of this diagram. After tensoring by \(E\), multiplication by \(\varpi\) is invertible and the whole complex is acyclic. **Exercise C.26.5.2.** Let \(Q=\mathbf G_m\times\mathbf A^u\) act trivially on \(X=\mathbf P^1\), and take both generator maps to be the identity. Compute the level-\(m\) Hom range and the eventual genuine equivariant self-Hom. Explain the role of the dimension cancellation in (C.26.10). **Solution.** Both relative generator dimensions are one. The level pair correspondence is \(Q^m\times\mathbf P^1\). Its compact complex has degrees from \(2mu+m\) through \(2m(u+1)+2\): an affine factor contributes only its top degree, a multiplicative factor contributes degrees one and two, and the projective line contributes zero and two. Formula (C.26.10) shifts its dual by \(-2m(u+1)-2\), giving Hom degrees zero through \(m+2\). The lower degree is independent of both \(m\) and \(u\). Hence a fixed total cohomological degree sees only finitely many nerve degrees. The affine unipotent nerve has the actual unit equivalence of C.13. The multiplicative-group classifying-space calculation of C.14 gives the degree-two class \(c\). The trivial-action product and the projective-line Chern class \(h\) therefore give \[ H^*\operatorname{RHom}_{Q}(E_{\mathbf P^1}[1],E_{\mathbf P^1}[1]) =E[c,h]/(h^2),\qquad |c|=|h|=2. \tag{C.26.15} \] Each class carries coefficient \(E(-1)\) in degree two before the chosen Tate trivialization. Level zero sees only \(E[h]/(h^2)\), and misses every positive power of \(c\). Thus agreement on the level-zero fibre is insufficient for genuine equivariant Hom. ### C.27. Spherical induction, full coordinates and ordered convolution Retain the arithmetic trait and coefficient field \(E\) of C.26. All objects below have a finite Schubert support, enlarged whenever a finite convolution word requires it. A root-word envelope contains all finite wall words, including nonreduced words and their length-zero translates, with finite shifts, cones and coherent retracts. Choose compatible finite loop and frame bounds for each finite diagram. Write \(H=J_NG\) and \(I_H\) for the inverse image of \(B\) under evaluation \(H\to G\). Thus \(H/I_H=G/B=:F\). An induced object means actual proper induction along \[ \kappa:[X/I_H]\longrightarrow[X/H], \qquad \operatorname{Ind}A=R\kappa_*A. \tag{C.27.1} \] These quotient presentations use the full action nerves. On the \(H\)-atlas its object is the proper endpoint image on \(H\times^{I_H}X\), with the coefficient descended from \(A\); no invariants of a graded Hom module define this induction. #### C.27.1. The unit projection and induced wall words **Theorem C.27.1.1.** For every genuinely \(H\)-equivariant bounded coefficient object \(P\), the actual flag unit is the first summand of an enhanced decomposition \[ \bigoplus_j P(-a_j)[-2a_j] \xrightarrow{\sim} \operatorname{Ind}\operatorname{Res}P, \qquad a_1=0,\quad b_1=1. \tag{C.27.2} \] Here the \(b_j\) are the homogeneous flag Chern basis of C.14.2–C.14.3, of polynomial degrees \(a_j\). Projection onto the first summand gives a coherent retraction of \(P\). Each induced root-word resolution object restricts to the Iwahori root-word envelope. The same assertions hold with the full finite coordinate group included. **Proof.** Character lines on the \(I_H\)-presentation give the actual classes \(c(b_j)\). The unit of \(P\), followed by cup with those classes, defines (C.27.2), just as in (C.14.8). We verify its relative assertion. On the action atlas it is the pullback of \[ \bigoplus_j E_S(-a_j)[-2a_j]\longrightarrow R\pi_*E_F, \qquad \pi:F\to S. \tag{C.27.3} \] On each geometric fibre the actual Chern basis is C.14.2.1, valid in that characteristic. Relative flag root cells are those of Automorphisms, forms and parabolic subgroups, Theorem 7.1; C.26.1.1 makes the target a perfect constant object. Alternatively proper base change checks (C.27.3) directly on every geometric stalk, using the same relative character lines. It is therefore an equivalence. Projection formula gives (C.27.2) for the possibly singular coefficient \(P\). Conservative descent on the complete action atlas proves it on the quotient presentation. No internal-Hom pullback identity for singular coefficients is used. Choose the inverse of this equivalence in the enhanced category; its space of inverse data is contractible. Its projection onto \(j=1\), together with the specified unit inclusion, is a coherent splitting. All maps are the relative Chern, unit and projection comparisons, so their entire diagrams commute with geometric restriction. Full coordinates fix evaluation at the origin and every character of \(T\), hence these same global line classes exist on the enlarged presentation; the same atlas proof applies there. For the assertion about induced words, use the longest finite Weyl word and its integral resolution \[ \rho:\mathcal B_{w_0}\longrightarrow F. \tag{C.27.4} \] Both source and target are smooth over \(S\), of relative dimension \(d_F\); \(\rho\) is proper and an isomorphism on the dense root cell. There are actual maps \[ E_F\xrightarrow{u}R\rho_*E_{\mathcal B_{w_0}} \xrightarrow{v}E_F. \tag{C.27.5} \] The first is the unit. The second is the proper counit after the smooth relative orientations identify \(\rho^!E_F=E_{\mathcal B_{w_0}}\): composition of exceptional adjoints gives \(\rho^!\omega_{F/S}=\omega_{\mathcal B_{w_0}/S}\), and the two equal-dimensional orientation shifts cancel. This is precisely the proper-duality calculation proved in C.26.4.1. On the dense cell the two maps are identities. Their composite lies in \(\operatorname{Hom}(E_F,E_F)=H^0(F,E)=E\), so it is the identity globally. Here \(F\) is connected and \(R\Gamma(F,E)\) is the perfect constant flag object just calculated. The maps retain the Iwahori and coordinate actions: the source, orientations and trace are their actual equivariant ones on every nerve level. Degree-zero endomorphisms on that presentation are again \(E\). Indeed each underlying nerve level is connected, has no negative constant-coefficient cohomology, and has \(H^0=E\); the degree-zero total equalizer is the coefficient line. Negative Hom of this constant unit is zero by the same bounded-below totalization. Thus the identity composite has its coherent splitting by the split-triangle construction used in C.25.5.1. Induction at an \(H\)-atlas is convolution of the constant coefficient on the finite flag \(F\) with the given genuinely Iwahori coefficient. The finite flag frame and its right \(I_H\)-action supply the associated kernel. Apply this actual exact convolution functor to (C.27.5), fixing its second factor. Effective coherent descent is the frame-chart Čech construction of The Mirković–Vilonen theorem: identifying the dual group, Lemma C.1.1.1; for a given coherent derived second factor its action is already specified, and the same bounded totalization constructs its kernel. This construction works over \(S\) as well: finite étale charts, their diagonal contraction and the actual ordinary étale comparison have the same proof, and compact constructibility is the arbitrary-base theorem used in C.26. On a root-word second factor, convolution with \(R\rho_*E\) is the endpoint image of the concatenated finite-wall word and that second word. This equality is checked on the final finite frame: pull the first proper image back, move the second coefficient through it by projection formula, and multiply the contracted products. These are the exact maps of The Mirković–Vilonen theorem: identifying the dual group, equation C.3.2.2. If the second word has length \(r\), its shift is \([r]\); the concatenated resolution generator has shift \([d_F+r]\), so the resulting object is that generator shifted by \([-d_F]\). A length-zero translate moves across a wall by conjugating its simple root, which still gives an allowed finite wall word. Consequently \(\operatorname{Res}\operatorname{Ind}B_i\) is a coherent retract of a finite root-word object with that shift. The same argument transports all first-factor maps and splittings, not only their cohomology. All contracted products, units and traces are integral and commute with geometric restriction. This proves the claims. ∎ #### C.27.2. Every coordinate coefficient and every spherical nerve level **Lemma C.27.2.1.** At suitable finite bounds, the full origin-preserving coordinate group acts on the integral root-word and induced-word families and their frame diagrams. Its underlying scheme is \(\mathbf G_{m,S}\times\mathbf A_S^{m-2}\) for coordinates modulo \(t^m\). Both the finite loop semidirect group and its Iwahori subgroup have relative toric-cell filtrations. These statements retain all substitution coefficients, rather than only the derivative at the origin. **Proof.** A truncated coordinate has the form \[ \phi(t)=z t+b_2t^2+\cdots+b_{m-1}t^{m-1}\pmod {t^m}, \qquad z\in A^\times \tag{C.27.6} \] on every \(S\)-algebra \(A\). Recursive comparison of coefficients constructs its inverse, dividing only by powers of \(z\). Composition and inverse therefore define the actual smooth group on the stated scheme. The precise finite-support substitution estimate and its inverse are Convolution and rigidity, Theorem C.1.2.1, and Fusion and the commutativity constraint, Theorem B.2.1.1. Their proof uses the integral binomial expansion for every integer Laurent exponent; it remains valid over the trait. More explicitly, let \(A_{\mathrm{fr}}\) bound the pole orders of each required faithful frame and its inverse, and let \(n_{\mathrm{fr}}\) be its right congruence depth. A kernel substitution \(t\pmod {t^m}\) gives \[ g^{-1}\phi(g)=1\pmod {t^{m-1-2A_{\mathrm{fr}}}}. \tag{C.27.6a} \] For an entry \(t^j\), its change starts in order \(j+m-1\); multiplication by \(g^{-1}\) loses at most the second pole bound. This proves (C.27.6a), also for the inverse frame. Choose \(m\geq\max(2,N,n_{\mathrm{fr}}+2A_{\mathrm{fr}}+1)\) over the finite diagram. It kills the kernel on every required frame quotient and support, and also allows substitution on the chosen loop jet group. Substitution preserves every root ideal \(t^aA[[t]]\), because \(\phi(t)=t\,v(t)\) with \(v(t)\) a unit, also for negative integer \(a\). It fixes the constant group and evaluation at the origin. Hence it preserves the Iwahori and each wall parahoric: their ordered root ideals and their torus are preserved. For a length-zero representative \(n_\omega=t^\nu n_w\), one has \(\phi(n_\omega)=t^\nu\nu(v)n_w\in n_\omega I\). Thus it acts on the initial coset and on each contracted wall product. The action is actual substitution, so its multiplication, unit and inverse identities hold on all parameter rings and all higher frame diagrams. It also acts on the induced product by \([h,y]\mapsto[\phi(h),\phi(y)]\); preservation of \(I\) makes this well defined. Bounded congruence coordinates make these actions factor through the selected finite groups. The loop jet scheme is \(G_S\) times an affine congruence kernel by C.13. Pull the finite closed root-cell filtration of \(G/B\) back to \(G\). Its root sections trivialize every difference as an affine root cell times \(B=T\times U\); ordered root coordinates make \(U\) an affine space. This is a relative toric-cell filtration, with the possible nilpotent stage immersions allowed and proved in C.26.4.1. Product with the affine congruence kernel proves the assertion for loop jets. The Iwahori jet scheme is \(T_S\times\mathbf A_S^u\), already C.13. The underlying semidirect scheme is the product with (C.27.6), so the product closed filtrations give the claimed filtrations. The group multiplication need not be a product multiplication; only this actual scheme filtration is used in the Hom calculation. ∎ **Theorem C.27.2.2.** Geometric restriction is an enhanced equivalence between the finite induced-generator envelope on the trait and the corresponding genuinely spherical finite-support category on either algebraically closed fibre. This holds for the complete loop action nerve, and for the loop action enlarged by full finite coordinates when that structure is retained. The restriction of each spherical IC lift to the complex fibre is the actual IC of its label. Combining with C.25 compares the corresponding genuinely equivariant complex étale and ordinary Betti envelopes. **Proof.** First compare mapping objects. By C.27.1.1 the underlying restriction of each induced generator is in the root-word envelope. At nerve level \(q\) for a chosen loop or coordinate semidirect group \(Q\), the underlying object is the ordinary pullback of that generator on \(Q^q\times_SX\). Its actual coherent action identifies all action pullbacks with this object. Product with \(Q^q\) sends the root-word expression, including its coherent retraction, to the same product envelope. C.26.2.1 applies to the original root-word generators on that product: their sources are \(Q^q\times_SY_i\), and their pairs are \(Q^q\times_S(Y_i\times_XY_j)\). Lemma C.27.2.1 and C.26.4.1 prove their relative toric-cell filtrations. Use their original generator shifts \(r_i\), not their enlarged dimensions, to get the exact formula (C.26.10). Finite mapping triangles in both variables and coherent splittings then prove mapping comparison for every pair of induced-envelope objects on that level. Restriction is an actual functor on the coherent presentation, so the comparisons commute with all action, multiplication, identity and coordinate faces, and with all degeneracies. Derived totalization gives fully faithful genuine equivariance. The coefficient justification also survives passage to these envelopes. Each fixed finite construction introduces only finitely many shifts into the uniform bounds (C.26.11); their minimum is still independent of \(q\). Thus the finite-window and completion proof of C.26.3.1 applies unchanged. No coefficient localization is exchanged with an unbounded product. Every induced fibre generator is the actual restriction of its family source and coefficient. Fully faithful enhanced mapping comparison lifts finite maps and coherent projectors, so it gives an equivalence of the induced-generator envelopes. We identify the fibre envelope. A spherical IC restricts to an Iwahori IC, with its canonical action, and is a coherent retract of a root-word image by C.25.5.1 and C.26.4.1. Induction carries its entire finite construction into the induced-generator envelope. The actual unit projection (C.27.2) then makes the original spherical IC a coherent retract there. This argument works with coordinates as well: the IC has its canonical full-coordinate action by Fusion and the commutativity constraint, Theorem B.2.1.1, and the root-word retraction lifts on the enlarged Iwahori nerve. We verify that lift on each root cell. Every substitution fixes the normalizer endpoint in the Grassmannian: the extra factor \(\nu(v)\) is integral torus and belongs to its right loop stabilizer. It also fixes the corresponding affine-flag endpoint, since the conjugated torus unit belongs to \(I\). The enlarged stabilizer is therefore the semidirect product of the original Iwahori endpoint stabilizer with the full coordinate group. The original stabilizer's torus is constant, and substitution fixes it. Quotienting by its ordered root and congruence kernel and then by derivative-one coordinates leaves exactly \(T\times\mathbf G_m\). The combined kernel has an underlying affine space: the original root/congruence coordinates and the derivative-one coordinates are independent factors, with the semidirect polynomial multiplication. Its powers on the group nerve are affine spaces as well. The actual affine-line units of C.13 remove this kernel on the complete nerve; this argument does not require its multiplication to be additive. C.14 then gives the even polynomial cohomology of \(T\times\mathbf G_m\). Connected stabilizer action makes the IC and resolution cohomology sheaves constant on the root cell. In its equivariant presentation their parity-matching Postnikov attaching maps have odd degree and vanish by this even stabilizer calculation. Apply the closed-open lifting triangles of C.25.5.1: both their odd-Hom obstructions vanish by finite cell induction. The dense identity lifts in both directions, its composite is the scalar identity of the IC, and the split triangle gives the whole coherent retraction with coordinates. This proves the claimed enlarged lift, not only a retraction after forgetting the coordinate action. Every bounded genuinely spherical constructible object with finite support is generated by its finite perverse truncation tower. Its perverse cohomology belongs to the finite-length spherical heart; semisimplicity and the classification of its simples were proved in The Satake category, Theorem B.2.3.1 and The Mirković–Vilonen theorem: identifying the dual group, Theorem B.4.1.1. Thus it is a finite direct sum of these ICs. The tower gives its finite cone construction in their enhanced envelope. With coordinates the canonical heart action is unique, by Fusion and the commutativity constraint, Theorem B.2.1.1, and the same truncation tower retains it. Conversely the induced objects have bounded constructible finite support and the indicated genuine actions. Therefore the fibre envelope is exactly the category stated in the theorem. The lift of a closed-fibre spherical IC, when restricted to the Iwahori presentation, is the lift in C.26's root-word envelope. Indeed C.27.1.1 puts it in that envelope, and full faithfulness identifies its whole closed-fibre splitting with the root-word IC splitting. C.26.4.1's ordinary and exceptional restriction comparison therefore applies. It retains the strict boundary degrees and the dense constant line, and identifies the generic object with the actual IC of the same label. Its genuine spherical and coordinate structures are the canonical ones: on the field heart their identity-normalized mapping components are contractible by negative perverse Hom vanishing, exactly the coherent-action argument of The Mirković–Vilonen theorem: identifying the dual group, Lemma C.1.1.1. Finally on the complex fibre apply C.25 to each product root-word mapping diagram just used. The group schemes are complex toric-cell filtered, and the proof needs their full nerve rather than their multiplication being a toric multiplication. Its trace and proper comparison maps are natural on all those faces. Extend through the actual induced retractions and finite constructions, and totalize with the same uniform coefficient bound. This proves the Betti assertion on the genuine envelopes, with their specified objects and actions. ∎ #### C.27.3. The actual ordered convolution diagrams **Theorem C.27.3.1.** The induced spherical envelope is closed under actual derived convolution. The comparisons of C.27.2.2 preserve its associative convolution, its unit, and every ordered finite convolution correspondence with all its coherence maps. The full coordinate structures are retained. This establishes the ordered monoidal comparison; the moving fusion diagrams which determine its symmetric constraint are an additional assertion. **Proof.** Fix a coherent genuinely spherical second factor \(Q\) in the family envelope. On a bounded convolution source the final finite frame supplies its actual associated kernel \(K_Q\). Coherent étale descent constructs it as in C.27.1.1. The resulting exact functor on the whole underlying first-factor category is \[ \mathcal C_Q(A)=Rm_*(p^*A\otimes_E^LK_Q). \tag{C.27.7} \] Both frame bounds and supports are finite. Arbitrary first-factor maps, finite cones and coherent splittings are therefore transported by this original enhanced functor. Genuine actions in the first factor give the transported left action on its output; the second factor supplies the frame descent. Restrict \(Q\) to the Iwahori root-word envelope using C.27.1.1. For a root-word first factor \(B_{\omega,s_1,\ldots,s_r}\), the actual proper pullback and projection calculations on the finite frame give \[ \mathcal C_Q(B_{\omega,s_1,\ldots,s_r}) =(n_\omega\cdot)_*\Theta_{s_1}\cdots\Theta_{s_r}(Q)[r]. \tag{C.27.8} \] This is the same contracted-product identity as The Mirković–Vilonen theorem: identifying the dual group, equation C.3.2.2, now for the specified coherent derived \(Q\); its proof only uses its kernel and the proper operations, so does not require \(Q\) to be perverse. Each unshifted wall operation preserves the root-word envelope. On a root-word generator it concatenates one wall to the source, with the compensating shift \([-1]\) relative to the newly shifted generator. On arbitrary envelope objects its actual exact functor preserves the finite cones and coherent splittings. Length-zero translation also preserves this envelope, conjugating the wall roots by its normalizing action. Hence (C.27.8) lies in it. The underlying first factor \(P\) of a spherical-envelope pair also restricts to that envelope. Apply (C.27.7) through its finite construction. It follows that \(\operatorname{Res}(P*Q)\) lies in the root-word envelope. The output is genuinely spherical, with full coordinates when given, by the same coherent left and frame actions. C.27.1.1 gives its unit retraction from \(\operatorname{Ind}\operatorname{Res}(P*Q)\). Induction of that root-word construction belongs to the induced-generator envelope. This proves convolution closure without requiring a separate paving of an induced pair correspondence. Geometric inverse image preserves the associated kernel: pull its entire étale frame nerve, including its action identifications and diagonal contraction, to the fibre. It becomes exactly the fibre kernel. Inverse image preserves tensor, and the endpoint proper base-change comparison is its actual adjunction mate. They therefore give \[ b^*(P*Q)\xrightarrow{\sim}b^*P*b^*Q. \tag{C.27.9} \] For a longer ordered word, use one common sufficiently deep frame. Its successive contracted products and their endpoint maps are the same before and after pullback. Every change of parentheses is the original associativity of tensor and proper-image composition on that full frame, followed by coherent descent. Actual base-change naturality preserves those entire diagrams, in every higher degree. The length-zero unit source and its unit maps are preserved as well. This proves the associative enhanced monoidal assertion. For the complex Betti comparison, the corresponding statement may be checked on root-word first factors and then on their finite constructions. Equation (C.27.8) expresses each such operation through the original integral wall correspondences. On a root-word second generator a wall is the proper image of the concatenated smooth source. Its actual étale–Betti comparison is C.25's generator comparison; the two shifted source conventions give the same \([-1]\) above. Proper pullback and projection naturality identify the operation's actual comparison map with that generator map. Extend through the finite mapping and object triangles and coherent retracts of the second factor, then the first. This proves the kernel comparison on the envelopes without assuming a general proper Betti theorem for arbitrary coefficients. The ordered full-frame argument gives all its associativity coherences and unit comparisons as above. Coordinates act on those same frames by Lemma C.27.2.1, so their complete action diagrams are retained. There is as yet no point-interchange argument in this proof. In particular the unit projection in C.27.2 is a splitting of an actual coefficient functor, not a tensor-compatible averaging rule. The symmetric comparison requires the actual ordered moving families and their fusion interchange maps. ∎ ![The actual flag unit projection yields spherical retracts, every finite coordinate coefficient and nerve level is retained, and ordered convolution uses its original second-factor kernel](assets/spherical-induction.png) Theorem C.27.1.1 proves the unit and induced-word splittings. Lemma C.27.2.1 and Theorem C.27.2.2 retain full coordinates and genuine spherical mapping objects. Theorem C.27.3.1 compares the original ordered convolution diagrams and their associative coherences. Editable SVG source. #### C.27.4. Two tests of the induction projection **Exercise C.27.4.1.** For \(G=SL_2\), compute \(\operatorname{Ind}\operatorname{Res}P\) from the actual flag Chern basis. Identify the unit retraction and the shift of the concatenated generator in C.27.1.1. **Solution.** The finite flag is \(\mathbf P^1\), and its basis is \(1,c_1\mathcal O(1)\), of polynomial degrees zero and one. Hence the actual decomposition is \(P\oplus P(-1)[-2]\). The unit is the first inclusion; the retraction is projection onto that same summand after the inverse of the Chern comparison. The longest finite word has length one. Convolving its unshifted resolution image with a root-word generator shifted by \([r]\) gives the concatenated generator, whose defining shift is \([r+1]\), followed by \([-1]\). No flag dimension shift is hidden in the definition of induction. **Exercise C.27.4.2.** Normalize \(H_T^*(\mathrm{pt},E)=E[u]\) and \(H_{SL_2}^*(\mathrm{pt},E)=E[q]\) by \(q=u^2\). The flag basis is \(1,u\). Is the projection \(F(q)+uG(q)\mapsto F(q)\) multiplicative? What does this test exclude from the proof of C.27.3.1? **Solution.** It sends \(u\) to zero and \(u^2=q\) to \(q\ne0\), so it is not multiplicative. It is the coefficient-linear projection of the flag basis summand containing the unit. Thus the existence of this coherent induction retraction alone proves neither a tensor averaging functor nor a symmetric convolution comparison. C.27.3.1 instead compares ordered convolution by its actual kernels and frame maps. The symmetric constraint must still be compared through the moving fusion diagram. ### C.28. Configuration complements and moving generator diagrams Keep the trait \(S\) of C.26 and the pinned integral group. For \(n\geq1\), let \[ B_n=\mathbf A^n_S,\qquad U_n=\{x_i-x_j\text{ is invertible for all }i\ne j\}\subset B_n. \tag{C.28.1} \] An ordered moving generator is the proper endpoint image of a chain whose \(i\)-th step is an induced root-word resolution at the section \(u=x_i\). Each step retains its finite flag reduction and every wall in its word. The moving-endpoint and finite-frame constructions are those of Fusion and the commutativity constraint, Sections 1–2, using its full monic completion and the all-ring gluing theorem of Beauville–Laszlo gluing and the moduli interpretation, Sections 1–3 and Theorem 8.1. They work over \(S\): their equations, splitting, monic operator resolution and detecting-line conditions are integral. We prove the geometric and coefficient comparisons required for these actual moving generators; the full symmetric fusion diagram is [C.29](#c29-actual-complex-fusion-and-its-coherent-symmetric-enhancement). #### C.28.1. Compact constants for diagonal arrangements **Theorem C.28.1.1.** The compact coefficient object of \(U_n/S\) is a perfect constant object, with actual geometric base-change comparisons, finite coefficient reductions and their complete models. The same holds for a scheme with a finite relative closed filtration whose differences are finite disjoint unions of \[ U_k\times_S\mathbf A^a_S\times_S\mathbf G_{m,S}^b \tag{C.28.2} \] up to closed nilpotent immersions. All conclusions of C.26.1 concerning its compact object, maps and coefficient diagram remain valid for this larger filtration class. **Proof.** Let \(D\subset B_n\) be the closed union of the diagonals \(D_{ij}:x_i=x_j\). Every nonempty finite intersection \(D_J\) of those diagonals is exactly \(\mathbf A^{c(J)}_S\), where \(c(J)\) is the number of connected components of the graph of its equalities. A spanning forest eliminates one variable at each edge by a coefficient \(1\) or \(-1\); all remaining equalities become zero. This is a scheme isomorphism over every parameter ring, including in characteristic two. It does not infer the intersection from its geometric points. The augmented finite closed-cover complex \[ E_D\longrightarrow \bigoplus_{|J|=1}i_{J*}E_{D_J}\longrightarrow \bigoplus_{|J|=2}i_{J*}E_{D_J}\longrightarrow\cdots \tag{C.28.3} \] has alternating restriction maps. It is an equivalence from the left term to the total complex on the right. Indeed, at a geometric point of \(D\), retain exactly the diagonals containing it. That set is nonempty; the augmented cochain complex of its full simplex is contractible by inserting any selected vertex. Every other stalk summand is zero. Stalks are exact and conservative, so this verifies the original sheaf complex. The same construction applies with \(O_n\), and is compatible with coefficient maps. The closed immersions \(i_J\) are proper. Take relative compact images in (C.28.3). Every intersection's compact object is \(O_{n,S}(-c(J))[-2c(J)]\), by the affine-line calculation of C.26.1. Finite totalization gives a perfect constant object for \(D\). Its actual maps and all their coherences lift through the fully faithful perfect constant functor (C.26.2). The compact localization triangle for \(U_n\subset B_n\) then gives a perfect constant object for \(U_n\) as well. Its actual comparison with any geometric fibre is the pullback of the constant counit (C.26.4), followed by compact base change. Thus the construction identifies the map, rather than only the ranks of its cohomology. For (C.28.2), compact projection and Künneth multiply this finite perfect model by the toric compact model of C.26.1. Products of the projective-line compactifications supply the actual Künneth map and its boundary diagram, just as there. Finite closed localization then proves the filtered assertion. Nilpotent closed immersions preserve the étale coefficient site and all these maps. The complete coefficient models follow by doing this same finite complex and finite cones with the compatible \(O_n\)-diagram: C.26.1's perfect constant mapping calculation lifts each connecting map and gives a finite projective \(O\)-model with its derived reductions. Derived inverse limit and exact localization give the stated complete and rational assertions. This proof makes no toric-cell paving assumption about the configuration space itself. ∎ #### C.28.2. Flag and wall chains on a collision block **Theorem C.28.2.1.** A finite same-point chain of induced root-word steps has smooth projective source, with dimension the sum of its flag and wall dimensions. Its reduced endpoint fibres admit relative affine pavings, in the precise nilpotent-filtration sense of C.26. For any two such chains, their pair correspondence over the Grassmannian admits the same type of relative affine paving. The assertions include nonreduced words, length-zero steps and different orders of the same steps. **Proof.** Keep the current endpoint in the affine flag whenever a flag has been chosen. There are three elementary source operations. Inserting a flag at a Grassmannian endpoint pulls back the proper smooth map \(\operatorname{Fl}_G\to\operatorname{Gr}_G\), whose fibre is \(G/B\). Appending a wall is its contracted \(\mathbf P^1\) bundle, with the determinant-one charts of Loop groups and the affine Grassmannian, Section 7.4. A length-zero step is right multiplication by its normalizing representative on the affine flag; it is an isomorphism there. The normalizing condition makes it well defined on the quotient by \(I\). Starting at the trivial endpoint, these operations give the chain, followed by forgetting its final flag. Thus its source is smooth projective over \(S\); each flag adds \(d_F=\dim(G/B)\), each wall adds one, and a length-zero step adds zero. Their finite pole bounds and detecting-line conditions give a finite separated endpoint target. The graph of the endpoint map is closed, so properness of the source makes that map proper. We prove the fibre assertion during this construction, retaining fibres over normalizer sections in both endpoint spaces. At a flag-insertion step, the fibre over a fixed affine-flag section is exactly the previous fibre over its underlying Grassmannian section, by the defining pullback square. No assertion about triviality of a general flag bundle is required. At a length-zero step the fibre is the previous fibre at the inverse-translated normalizer section. At a wall step, forgetting the final factor identifies the fibre over \(yI\) with the previous source's inverse image of \(yP_s/I\). For the shorter \(v\) of \(y,ys\), this line has the closed section \(vI\) and open root chart \(z\mapsto x_{v(a_s)}(z)vsI\). The root is positive and belongs to \(I\). Its left action on the previous source gives the actual product trivialization of the open inverse image. The two pieces are therefore \[ F_{\mathrm{prev}}(v),\qquad \mathbf A^1_S\times_SF_{\mathrm{prev}}(vs). \tag{C.28.4} \] This is C.26.4's integral fibre recurrence, and is valid whether or not the total word is reduced. Inductively insert their closed filtrations, adjoining the whole closed piece to each closed stage in the open piece. When forgetting a flag, the inverse image of a normalizer Grassmannian endpoint \(t^\nu\) is the pullback over its finite flag fibre \(t^\nu G/B\). Choose the finite Borel by the signs of \(\nu+\epsilon\gamma\), as in The Mirković–Vilonen theorem: identifying the dual group, Lemma C.2.1.1. Its root sections \(v(z)\) have conjugates \(t^\nu v(z)t^{-\nu}\in I\). They fix the Grassmannian endpoint, since conjugating back gives an integral loop. Hence their left action identifies the source above each of these relative flag cells with that affine cell times the source's fibre at the corresponding affine-flag normalizer endpoint. All these fibres were handled by the preceding recurrence. The finite flag closed ordering and its root sections are Automorphisms, forms and parabolic subgroups, Theorem 7.1. Inserting its finite closed stages proves the claimed Grassmannian fibre paving. Every intermediate source and map has a genuine left loop action, at sufficiently deep common finite bounds: flag insertion, right wall multiplication and right normalizer multiplication commute with that action. In particular it has the Iwahori action used in the trivializations. Over an actual Grassmannian Iwahori cell \(A_S^e\), its endpoint root section identifies a pair of chain fibres simultaneously, giving \[ (Y\times_XZ)|_{A_S^e} =A_S^e\times_SF_Y(x)\times_SF_Z(x). \tag{C.28.5} \] The inverse is action by the inverse section. Products of the two fibre pavings and the finite closed endpoint-cell ordering give the pair paving. We retain the scheme qualification on those closed stages. Endpoints are in finite proper word images. The allowed labels and their downward order are the integral wall and flag root-coordinate calculation: each reachable normalizer section is obtained by the closed or open wall endpoint and a finite flag normalizer, and its whole root cell factors through the proper image by the source action. On each geometric fibre these are exactly its labels. The corresponding locally closed cell immersion into an open difference of stages consequently has the full underlying image. Its defining ideal is in the nilradical and is finitely generated, so it is nilpotent, exactly as proved in C.26.4.1. Pulling these immersions to the pair space preserves nilpotence. Thus (C.28.5) yields the precise permitted filtration, without a flatness assumption on every closed image or a coproduct identity on all test-ring points. This proves all assertions. ∎ #### C.28.3. The complete moving divisor-frame comparison **Theorem C.28.3.1.** Fix \(n\) and a finite collection of ordered moving induced generators, including any selected permutations of their factors. Geometric restriction compares their actual enhanced finite envelopes over \(B_n\). This includes the full moving divisor-frame action nerve, restriction to any collision partition and to the all-distinct locus, and all original proper-operation diagrams between the selected generators. The analogous complex étale–Betti comparison holds. No symmetric fusion constraint is asserted by this theorem alone. **Proof.** Put \[ f(u)=\prod_{i=1}^n(u-x_i),\qquad Q_N(A)=G\bigl(A[u]/f(u)^N\bigr) \tag{C.28.6} \] for a \(B_n\)-algebra \(A\). The algebra in this formula is finite free of rank \(nN\), by monic division. Expanding a finite affine presentation of \(G\) in its basis \(1,u,\ldots,u^{nN-1}\) represents \(Q_N\) by an affine finitely presented \(B_n\)-scheme, with its actual group laws. This group is smooth. A square-zero ideal \(J\subset A\) gives a square-zero quotient of the displayed finite free algebra, so formal smoothness of the affine group algebra lifts a \(G\)-point through it. Here the requisite global affine formal smoothness follows from *Formally smooth, unramified and étale ring maps*, Theorem 3.1: local standard smooth presentations give the injective conormal sequence; its differentials are finite projective on the affine smooth scheme, so that sequence splits globally. The theorem's explicit derivation correction then gives the global square-zero lift. Consequently the represented algebra of \(Q_N\) is formally smooth and finitely presented, hence smooth in the local standard form proved in that lesson's Theorem 5.1. Its tangent module is \(\operatorname{Lie}(G)\otimes A[u]/f^N\), of rank \[ D=nN\dim G \tag{C.28.7} \] over \(A\). Thus its relative dimension over \(B_n\) is this same \(D\), including on collisions. We verify that its finite action is the actual one on the moving chain. Let a common \(f\)-pole bound \(M\) contain every required faithful frame and inverse frame, at every step of the finite diagram. The monic lattice construction of Lesson 8, Section 1, places them in a finite \(f\)-window on all parameter rings; multiplication of successive bounds gives one such \(M\). If a formal loop is \(1\pmod {f^N}\), conjugating it by such a frame gives \[ h^{-1}gh=1\pmod {f^{N-2M}}. \tag{C.28.8} \] This follows entry by entry from the two pole bounds, without dividing by a difference \(x_i-x_j\). Choose \(N-2M\) at least every right congruence depth in the diagram. Since \(f\) is divisible by each \(u-x_i\), (C.28.8) kills the kernel on every flag, wall and endpoint frame quotient. Formal lifts of a jet exist successively by the just-proved square-zero lifting property; different lifts differ by this kernel. The action therefore factors through exactly \(Q_N\), with all parameter-ring identities and higher frame maps. The finite quotient action is algebraic: its coefficient operations in the finite monic window are polynomial with the same permitted unit inverses as the original frame charts. Each generator source is constructed by associating its induced smooth projective word source to the current formal frame at \(x_i\), then proceeding to the next point. The proper moving construction of Lesson 8, Section 2, gives the actual finite target and endpoint map. The associated source is smooth proper over the preceding source, since its frame torsor is smooth and its fibre is the specified smooth projective induced word. Descending its equivariant projective embedding gives properness, as in that construction. Over \(B_n\) it has relative dimension \(\sum_i(d_F+r_i)\), where \(r_i\) are its wall lengths. Its dimension over \(S\) is therefore \[ R_Y=n+nd_F+\sum_i r_i. \tag{C.28.9} \] The original unshifted moving-generator coefficient has shift \(\alpha_Y=\sum_i r_i\). Thus every selected \(n\)-factor generator has the same deficit \(R_Y-\alpha_Y=n(1+d_F)\). Adding the common moving base shift \([n]\), when desired for IC normalization, changes none of their mapping objects. Stratify \(B_n\) by exact collision partitions. For a partition with \(k\) blocks, the stratum is an actual copy of \(U_k\): coordinates in each block are equal, and all differences between blocks are invertible. Order partitions by number of blocks. Unions of the closed partial diagonals with at most \(k\) blocks are closed, and their differences are these finitely many disjoint strata. The equations are the integral equalities already treated in C.28.1. On a partition stratum, the Chinese remainder theorem identifies (C.28.6) with \[ \prod_{\text{blocks }C}J_{N|C|}G. \tag{C.28.10} \] Indeed \(f^N\) becomes the product of the relatively prime powers \((u-x_C)^{N|C|}\). This identity includes every parameter algebra on that stratum. The same completion splitting and the all-ring gluing equivalence separate the moving source into one same-point flag/wall chain for each block, in the original order within that block. Modifications in different blocks commute by that actual gluing equivalence. Translation supplies the fixed parameter \(u-x_C\), so these are products over the configuration stratum, not merely fibre descriptions. C.28.2 therefore gives the relative affine pair filtrations of any two selected moving sources on that stratum. The group in (C.28.10) has the toric-cell filtration of C.27.2.1. At nerve level \(q\) their product pair is \(Q_N^q\times_{B_n}(Y\times_XZ)\). Its partition differences consequently have finite filtrations by \(U_k\) times affine and multiplicative cells. Insert their finite closed stages into the partition ordering. A closed stage in an open difference becomes closed in the whole space after adjoining its preceding closed boundary, because its complement is open. Possible nilpotent stage immersions are those of C.28.2 and remain nilpotent under these pullbacks. This verifies the exact larger-filtration hypothesis of C.28.1 for every level pair correspondence. The smooth level sources have dimensions \(qD+R_Y,qD+R_Z\) over \(S\), but shifts \(\alpha_Y,\alpha_Z\). Since their shift deficit is common, the relative trace adjunction of C.26.2 gives \[ \operatorname{RHom}_{X_q}(B_{Y,q},B_{Z,q}) \simeq D_EK_{\,Q_N^q\times_{B_n}(Y\times_XZ)} [-2qD-R_Y-R_Z](-qD-R_Z). \tag{C.28.11} \] Equivalently its unshifted adjunction has shift \(-2(qD+R_Z)+\alpha_Z-\alpha_Y\), which is the same integer. C.28.1 supplies the actual compact comparison and perfect complete coefficient models on the right. Evaluation and relative trace identify the actual forward restriction with the dual of the inverse compact comparison, exactly (C.26.9). Hence actual mapping comparison holds at every nerve level. All those maps are the original inverse-image, proper, multiplication and identity comparisons; thus their entire cosimplicial diagrams compare. For coefficient totalization, choose a common fibre-dimension bound \(d_C\) of \(Y\times_XZ\) over \(S\), from the finitely many partition-cell dimensions. The level pair has bound \(qD+d_C\). Formula (C.28.11) has no cohomology below \(R_Y+R_Z-2d_C\), independently of \(q\). The finite-window, Mittag–Leffler completion and exact localization proof of C.26.3 applies. Derived totalization proves the genuine divisor-frame mapping comparison. Finite cones in both variables and coherent retracts extend it to the stated enhanced envelopes. Every fibre generator is the actual restriction of its family generator; full faithfulness lifts the finite construction and gives essential surjectivity onto that fibre generator envelope. The same proof on a selected partition, or the all-distinct open part, uses its configuration-times-toric compact model and the same relative adjunction. On a \(k\)-block partition replace \(R_Y\) by \(R_Y-n+k\); its shift deficit becomes \(k+nd_F\), still common to all selected \(n\)-step sources. Its group dimension \(D\) remains (C.28.7). Restriction of the whole family diagram commutes with these scheme maps by actual base-change naturality, so it retains all the stated restriction and proper-operation diagrams. On the complex fibre, the compact comparison for an arrangement complement uses the same finite closed-cover complex (C.28.3), projective-line affine compact models and localization. Their maps are the actual comparisons of C.25.1, so it proves the Betti version of C.28.1 as well. The moving-source proper comparison is an equivalence on every geometric stalk: on a collision partition its endpoint fibre is the product of the same-point chain fibres of C.28.2, with their actual affine pavings. Proper base change and C.25.1 compare those fibres. This identifies the Betti generator with the original ordinary moving endpoint image. We also check trace when a smooth moving source itself has not been given a toric paving. Over a geometric field the smooth exceptional formula of C.26 and its compact adjunction identify the dual of its top compact group with its degree-zero ordinary constant-coefficient group, one coefficient line per connected component. With \(O_n\), the self-injective coefficient dual separates elements: a nonzero element of a cyclic submodule has a nonzero functional to the corresponding ideal of \(O_n\), and injectivity extends it to the whole module. If the dual is one finite coefficient line, the module therefore injects into its finite double dual and is finite. Biduality on finite cyclic summands of the principal Artin ring then gives that same top coefficient line. The classical smooth orientation and compact adjunction proved in C.25.2.2 give these same top lines on the complex manifold. A supported point class is their trace-one generator: the actual positive Kummer boundary and local complex orientation comparison is C.25.2.1, and the chart transitivity and trace counit preserve its normalization. Naturality sends that class to the analytic supported point class. A finite coefficient derived trace to \(O_n[-2d]\) is determined by this top group, using the self-injective dual; thus the original trace diagrams compare on every connected component. Taking compatible complete models and then localizing gives the same rational trace comparison. This does not assume a paving or general proper Betti theorem for that smooth source. The level pair filtrations, whose compact objects are perfect, now give C.25's trace-normalized inverse-dual Hom comparison, with the shifts in (C.28.11). Full divisor-frame totalization and finite coherent constructions finish the Betti assertion. This proves a comparison of the actual moving generator diagrams. Turning its permutation maps into the symmetric fusion comparison still requires identifying the actual moving IC coefficients and their full fusion diagram. ∎ ![Configuration compact support uses the actual diagonal complex; collision flag/wall fibres give pair filtrations; the moving divisor-frame nerve retains longer jets at collisions and the correct Hom shifts](assets/moving-generators.png) Theorem C.28.1.1 proves the configuration compact comparison without a toric paving assumption. Theorem C.28.2.1 proves the actual flag/wall chain fibre and pair filtrations. Theorem C.28.3.1 compares the complete moving frame and partition-restriction diagrams with their relative dimensions. Editable SVG source. #### C.28.4. Two configuration and frame calculations **Exercise C.28.4.1.** Compute the compact cohomology of \(U_3\). Explain why a toric-cell paving cannot replace the diagonal-arrangement argument in C.28.1. **Solution.** The coordinates \[ (x_1,\delta=x_2-x_1,\tau=(x_3-x_1)/\delta) \tag{C.28.12} \] give the all-ring isomorphism \(U_3=\mathbf A^1\times\mathbf G_m\times(\mathbf A^1\setminus\{0,1\})\). The punctured line has compact groups \(E^2\) in degree one and \(E(-1)\) in degree two: in the projective-line localization complex, quotient the three point classes by their diagonal unit. Multiplying by the affine and multiplicative compact complexes gives \[ H_c^4(U_3,E)=E(-1)^2,\quad H_c^5(U_3,E)=E(-2)^3,\quad H_c^6(U_3,E)=E(-3), \tag{C.28.13} \] and zero otherwise. In fact over an algebraic closure of a finite field, \(U_3\) has no finite paving by affine-times-multiplicative cells. If it did, its finitely many closed stages and cell isomorphisms would descend to one finite field. Counting points over all its finite extensions would give the polynomial identity \[ q(q-1)(q-2)=\sum_{\text{cells}}q^a(q-1)^b. \] There are infinitely many such \(q\), so this is an identity of integer polynomials. At a real \(q\) strictly between one and two the left side is negative and every right summand is positive. This is impossible. The actual point count on the left merely chooses three distinct field elements; no cohomological trace theorem was used in this obstruction. The finite closed-cover calculation proves the needed compact comparison nonetheless. **Exercise C.28.4.2.** For \(G=\mathbf G_m\), \(n=2\), \(N=1\), compute the all-ring frame group \(Q_1\) and its distinct-point and collision restrictions. **Solution.** Write \(f(u)=(u-a)(u-b)\). An element of \(A[u]/f\) is \(c_0+c_1u\). Its multiplication matrix has determinant \[ (c_0+c_1a)(c_0+c_1b). \tag{C.28.14} \] It is a unit exactly when this determinant is a unit: an invertible multiplication matrix sends its inverse image of \(1\) to a multiplicative inverse, and the converse is immediate. Thus \(Q_1\) is the determinant-open part of the two-dimensional coefficient affine space over \((a,b)\), and is smooth of relative dimension two. When \(a-b\) is invertible, evaluation at \(a,b\) gives \(\mathbf G_m^2\), with its actual Chinese remainder inverse. When \(a=b\), put \(t=u-a\). Units of \(A[t]/t^2\) are uniquely \(r(1+vt)\), \(r\in A^\times\), \(v\in A\). Their multiplication is \((r,v)(r',v')=(rr',v+v')\), so the collision group is \(\mathbf G_m\times\mathbf G_a\). Both have dimension two. Replacing the whole moving group by a constant two-dimensional torus would therefore discard the actual collision group and its complete nerve. ### C.29. Actual complex fusion and its coherent symmetric enhancement We use finite \(E/\mathbf Q_\ell\) first. The geometric group in this section is the loop group \(G\), with its complete finite divisor-frame nerves. The moving curve is the affine line with its specified coordinate. Its existence gives an actual moving functor on the \(G\)-equivariant derived category: the chain coefficient descends through the \(G\)-frame torsors of Convolution and rigidity, A.4, and Fusion and the commutativity constraint, C.1. Coordinate equivariance of a perverse input is available for the intrinsic curve description in Lesson 8. It is not used to claim that every derived \(G\)-equivariant morphism lifts to the category with a rotation action. Write \(\mathcal C_a,\mathcal C_s,\mathcal C_c\) for the enhanced finite spherical envelopes on the arithmetic geometric fibre, over the strict arithmetic trait, and on the complex geometric fibre. The middle notation refers to the generator envelope, rather than to a perverse heart over the trait. The actual restriction and comparison functors give \[ \mathcal C_a\ \xleftarrow{\,r_a\,}\ \mathcal C_s\ \xrightarrow{\,r_c\,}\ \mathcal C_c. \tag{C.29.1} \] C.27 proves that both arrows are enhanced equivalences with their original ordered convolution, associators and units. The complex right arrow includes the actual étale–Betti comparison. We retain these particular functors throughout. #### C.29.1. The moving IC objects, including their strict extension tests **Theorem C.29.1.1.** Let \(P_1,\ldots,P_n\) be spherical IC objects, or finite perverse convolution words in them, and take their corresponding objects through (C.29.1). Their actual moving endpoint coefficients on the affine line correspond through C.28.3, with the common shift \([n]\). On each geometric fibre they are the intermediate extensions of their all-distinct external products. Comparison retains their ordinary and exceptional stratum restrictions and the strict boundary inequalities. **Proof.** For fixed finite supports, take the moving chain with its specified away frame. The coefficient on it is constructed on the finite frame charts as the external product of the \(n\) input coefficients; the transition maps are their genuine \(G\)-actions. The original action cocycles and frame refinements provide the entire descent diagram. This gives a multilinear exact enhanced functor, followed by the proper endpoint image. Exactness means that in each input it retains finite cofibres, shifts and coherent retracts. It follows directly from exact external tensor over the field \(E\), exact derived inverse image, descent and exact derived proper image. Neither the definition nor this verification involves lifting an arbitrary morphism through a coordinate-forgetting functor. C.27 expresses each input as a coherent finite construction from induced wall-word generators. Apply the moving functor to these constructions one variable at a time. Its resulting objects and maps lie in the enhanced envelope of precisely the moving induced generators in C.28.3. That theorem compares the original coefficient descent, proper endpoint image and the whole divisor-frame nerve. Thus the moving object of corresponding inputs corresponds as an enhanced object, rather than merely having isomorphic stalk groups. It also corresponds on the all-distinct locus and on every selected collision partition. Adding the same base shift \([n]\) gives the normalized objects \(F_n\). We check the perverse assertions without assuming that every finite construction on a configuration space is constant. Choose the finite stratification by collision partitions and endpoint root cells on each block. A stratum is a configuration space \(U_k\) times the finite product of these cells. The Chinese remainder and root-section identifications of C.28.3 and C.28.2 identify the ordinary restriction of a moving generator with the actual constant compact coefficient of its paved endpoint fibre, with its original shift. Proper base change identifies the map defining this restriction. C.26.1 and C.28.3 compare this coefficient and its geometric and Betti restrictions. This proves comparison of the restriction functors on generators. Exactness, and coherent retracts in the source and target, extend that comparison to the finite envelope. It does not replace the resulting complexes on \(U_k\) by constant complexes: for example a cone using a positive-degree class of \(U_k\) can retain a nontrivial local system. The exceptional restriction comparison follows from actual duality. On a smooth moving source \(Y\) of dimension \(R_Y\) over the trait, its shifted constant coefficient \(E_Y[\alpha_Y]\) has relative dual \[ D_Y(E_Y[\alpha_Y]) =E_Y[2R_Y-\alpha_Y](R_Y). \tag{C.29.2} \] Proper dual exchange therefore identifies the dual of its endpoint generator with that generator shifted by \(2(R_Y-\alpha_Y)\) and twisted by \(R_Y\). Evaluation, rather than an unspecified isomorphism, defines this identification. The relative trace comparison in C.28.3 preserves that evaluation on the geometric fibres and on the complex manifold. Hence duality compares on generators and, contravariantly, on every finite cofibred construction and coherent retract. For a stratum immersion \(i\), the identity \(i^!D=D i^*\) follows from the compact adjunction and projection formula as in C.26.4. Its evaluation diagram is retained. On the smooth stratum, the duality of the constant restricted generators is its original smooth orientation; C.28.3 compares this orientation and trace. The ordinary restriction comparison applied to dual generators consequently proves the exceptional restriction comparison as well. These arguments are statements about functors and their evaluations, so remain valid for the nonconstant finite constructions just mentioned. On the arithmetic geometric fibre, Fusion and the commutativity constraint, Theorem C.2.1.1, proves that \(F_n\) is perverse and is the strict intermediate extension of its all-distinct product. Its proof supplies both inequalities: on a partition with \(k\) blocks, of codimension \(c=n-k\), ordinary perverse restriction is in degrees at most \(-c\), and exceptional perverse restriction in degrees at least \(c\). The inputs on the complex fibre are their actual IC objects by C.27's two IC restriction tests. Comparison of ordinary and exceptional stratum functors preserves the vanishing groups defining these inequalities. Thus they hold for the actual complex moving object as well. On the open part its coefficient is the actual independent external product, by disc gluing. The closed-open intermediate-extension criterion proved in Fusion and the commutativity constraint, A.1.2.1, now identifies the complex object as that product's intermediate extension. Finite perverse convolution words are allowed because Lesson 7 proves their perversity and the same moving proof applies to those perverse inputs. This proves every assertion. ∎ #### C.29.2. Comparison of the geometric permutation components **Theorem C.29.2.1.** On the IC and perverse convolution-word objects, the components of the arithmetic fusion permutations compare with the actual complex geometric fusion permutations. This includes all partial-diagonal regroupings, their further collisions, their supported maps, and the component parity correction. This assertion identifies the components of the constraints; their full enhanced naturality is supplied separately in C.29.3. **Proof.** Fix finitely many inputs, permutations and partition maps occurring in a desired diagram. Include their permuted moving chains among the finite collection of generators. The objects constructed in C.29.1 lie in their envelope. A permutation also pulls back the base \(B_n\) by the actual coordinate permutation; it carries \(f(u)=\prod_i(u-x_i)\) and its complete frame group to that of the permuted chain. C.28.3 applies to this pullback diagram. Full faithfulness of its actual arithmetic restriction equivalence lifts each arithmetic moving fusion component, with all compositions in this selected diagram, to the trait envelope. More explicitly, an equivalence induces an equivalence on the mapping spaces of the finite diagram; the fibre over its specified map, homotopy or iterated homotopy is contractible. This allows us to lift the diagram itself, retaining its given identifications. It is not an assertion that an enhanced space of natural transformations between all functors is contractible. Restrict this lift to \(U_n\). The proper and open restriction squares of C.28.3 commute as their original functor squares. On \(U_n\), independent-disc gluing defines the actual external permutation of coefficients, with the ordinary derived Koszul sign. Its arithmetic restriction is the external permutation of the arithmetic inputs. Full faithfulness of the *open* comparison therefore identifies the restricted lift with that actual external permutation over the trait. Its complex restriction is consequently the actual complex external permutation, with the same sign. C.29.1 identifies source and target of the complex moving map as intermediate extensions. The intermediate-extension functor on the perverse heart is fully faithful on morphisms of its open objects: a kernel or cokernel supported on the closed boundary would be a forbidden subobject or quotient, by the same closed-open criterion; existence and the image construction are proved in Fusion and the commutativity constraint, A.1.2.1. Hence a map extending the specified external permutation is unique. The compared component is exactly the complex geometric fusion component. This uses uniqueness of a degree-zero perverse map to identify that component, without drawing a conclusion about all enhanced natural transformations. For a partition \(\pi\) with \(k\) blocks, put \(c=n-k\), order blocks by their least labels, and preserve order within each block. The proper and supported comparisons of C.28.3 retain the actual diagonal maps in Fusion and the commutativity constraint, Theorem C.3.1.1: \[ i_\pi^*F_n[-c]=F_k\bigl(*_{i\in\pi_1}P_i,\ldots, *_{i\in\pi_k}P_i\bigr),\qquad i_\pi^!F_n[c]=F_k\bigl(*_{i\in\pi_1}P_i,\ldots, *_{i\in\pi_k}P_i\bigr)(-c). \tag{C.29.3} \] That theorem is proved on the entire partial diagonal. Its supported class consists of the \(c\) independent differences of graph coordinates, and its Kummer orientation and trace compare by C.28.3. Thus the exceptional map, including its twist, compares as well. For noncontiguous blocks, the all-ring independent-disc regrouping is the original chain isomorphism, and its coefficient shuffle is the original derived external shuffle. It compares before intermediate extension. On the all-distinct block locus the lifted map is therefore the specified regrouping; the strict extension proof in Fusion and the commutativity constraint, C.3, identifies it at every further collision. It retains the supported map and its evaluation, not only the two objects in (C.29.3). Finally the correction of Fusion and the commutativity constraint, C.5–C.6, multiplies the geometric permutation on connected input components by the parity sign determined by their component labels. Those labels are the same integral coweight classes on every base: their normalizer representatives and central quotient were retained in C.26–C.28. The correction is therefore the same scalar \(1\) or \(-1\) on the lifted and compared component. With that correction the fibre functor has its ordinary symmetric constraint, as proved by the explicit cohomology shuffle computation in Fusion and the commutativity constraint, C.4–C.6. Its actual cohomology comparison is C.17 and C.22, so the normalization is retained too. This proves the stated component comparisons. ∎ #### C.29.3. The full enhanced symmetric equivalence **Theorem C.29.3.1.** The actual ordered comparison (C.29.1) carries an enhanced symmetric monoidal structure on the complex finite spherical envelope. On IC objects and their perverse convolution words its symmetry is the actual corrected geometric fusion symmetry. With this enhancement it yields the symmetric monoidal equivalence \[ D^b_G(\operatorname{Gr}_{G,\mathbf C},E) \ \simeq\ \operatorname{Perf}_H \bigl(\operatorname{Sym}_E(\mathfrak h[-2])\bigr), \tag{C.29.4} \] where the left category has finite support and the full loop-action descent of C.27, and \(H\) is the full split dual group. The equivalence sends \(\mathrm{IC}_\lambda\) to \(\operatorname{Sym}_E(\mathfrak h[-2])\otimes V_\lambda\). Its ordered convolution, associators and units are the original geometric ones. **Proof.** We first describe exactly which higher coherence is being used. C.23.4 constructs the entire arithmetic enhanced symmetric diagram on normalized convolution words: its Weil coefficient replacement, all moving proper maps, restrictions, permutation maps, block maps and their homotopies. C.23.3 applies simultaneous formality to that full diagram, and C.23.5 extends it through finite cones and coherent retracts. Thus \(\mathcal C_a\) already carries an enhanced symmetric monoidal structure on its original ordered convolution, and an enhanced symmetric equivalence to the right side of (C.29.4). An enhanced equivalence with its ordered tensor comparison has an ordered monoidal inverse. To see what this statement requires, choose inverse objects and the unit and counit equivalences. Lift its binary tensor comparison and unit through the equivalence on mapping spaces. Their associativity diagrams, and the homotopies between those diagrams, lift by that same equivalence. At each stage the space of lifts of the *specified* datum is contractible. Continue over the skeleta of the diagram of finite ordered strings; the coherent unit and counit identify the composite with the original ordered functor. This constructs the inverse with all ordered coherence, rather than choosing unrelated binary isomorphisms. This is the mapping-space construction used for the finite coherent envelopes in C.23.5. Choose such an inverse \(r_a^{-1}\) and form the ordered equivalence \(T=r_c r_a^{-1}:\mathcal C_a\to\mathcal C_c\). Transport the arithmetic symmetric structure through this ordered equivalence. Concretely, for each finite ordered string apply \(T\) to the original arithmetic operation and use its given ordered tensor comparisons to express the result in the existing complex ordered tensor product. Do this for every permutation, regrouping, unit insertion and their higher compositions. A relation between operations is carried to its specified homotopy; a relation between those homotopies is carried to its specified next homotopy. The coherent inverse identifies the composite diagram back with the original one. Hence this construction is an enhanced symmetric structure on the *existing* complex ordered tensor operation. It does not assert uniqueness of arbitrary enhanced natural-transformation spaces, or reconstruct a higher homotopy from a vanishing negative Hom group of perverse objects. For IC and perverse-word inputs the transported symmetry has exactly the compared arithmetic components. C.29.2 identifies these with the actual corrected complex fusion components, including their diagonal identifications and normalization. Thus the constructed enhanced structure extends the geometric Satake symmetry on its heart and perverse convolution words. Its enhanced naturality on all finite derived constructions comes from transport of the complete arithmetic diagram. Its ordered part remains the actual C.27 comparison, whose associator and unit come from the original finite-frame chain maps. This proves the claimed relation to the geometric tensor operation. Compose the inverse of \(T\) with C.23.5's arithmetic enhanced symmetric model. C.27 proves that the finite spherical envelope is the whole finite-support bounded spherical derived category: IC coherent retracts lie in the generator envelope, every perverse object is a finite sum of them, and its finite perverse truncation tower puts every bounded object there. This gives (C.29.4). C.23.5 identifies every IC image as the indicated free equivariant \(A\)-module, for \(A=\operatorname{Sym}_E(\mathfrak h[-2])\). The group and algebra are retained with their actual normalizations. The root lattice, central characters and full \(H\) were proved in C.18–C.22, including the central directions of \(\mathfrak h\). The graded generator map is the actual cohomology and supported cup construction in C.21–C.22. The trace-normalized comparisons in C.26–C.28 commute with these original proper, external and supported maps; C.29.2 retains their fusion normalization. Consequently composition uses this particular graded generator identification, with its actual cup classes and Kostant coordinates, before the coherent formality step. It does not replace \(H\) by its adjoint quotient or select an unrelated isomorphism of Ext ranks. This theorem concerns the loop-equivariant category on the complex Grassmannian. A derived rotation-equivariant category has an additional action nerve and is a separate object; the full coordinate comparison in C.27 remains valid on its own stated envelopes. No fully faithful coordinate-forgetting functor, nor an enhanced intrinsic construction on every uncoordinated curve, is deduced here. ∎ #### C.29.4. Flat coefficient extensions and complex coefficients **Theorem C.29.4.1.** For a field extension \(E\to L\), the coefficient extension of the left side of (C.29.4), including idempotent completion, is the full finite-support \(L\)-coefficient spherical category. Its actual ordered tensor product and moving fusion components compare, so (C.29.4) extends to \[ D^b_G(\operatorname{Gr}_{G,\mathbf C},L) \ \simeq\ \operatorname{Perf}_{H_L} \bigl(\operatorname{Sym}_L(\mathfrak h_L[-2])\bigr) \tag{C.29.5} \] with the transported enhanced symmetric structure. In particular this gives the complex-coefficient statement after choosing an embedding \(E\hookrightarrow\mathbf C\). Descent to characteristic-zero fields admitting no such \(E\)-embedding is [Corollary C.80.2.1](#c802-arbitrary-characteristic-zero-coefficients). **Proof.** On two induced generators, every finite group-nerve level mapping object is the shifted coefficient dual of the compact pair model of C.26–C.27. That model is a finite perfect \(E\)-complex. Its coefficient extension is its actual compact \(L\)-model, by the finite affine/projective-line and localization construction. Finite perfect dual commutes with \(E\to L\), and evaluation and trace give the same mapping map. Thus generator mapping objects compare at each level. One must check the entire action nerve before commuting coefficient extension with it. The lower bound \(L_0\) in C.26.3 is independent of the nerve degree. In a fixed total cohomological degree \(d\), levels greater than \(d+1-L_0\) contribute neither the terms nor their incoming differential relevant to that degree. The totalization truncation in that finite window computes it, by the finite-window proof there. Field extension is exact and commutes with these finite complexes. It therefore compares the full mapping objects degree by degree. This argument uses the uniform lower bound; it does not commute tensor with an arbitrary infinite product. Finite cofibres in both variables and coherent retracts then prove full faithfulness of coefficient extension on the enhanced finite envelope. Its essential image is all of the \(L\)-category. Ordinary and exceptional IC tests are the finite restriction complexes of C.27, so flat coefficient extension preserves their vanishing, their rank-one open coefficient and the strict boundary tests. It sends \(\mathrm{IC}_{\lambda,E}\) to the actual \(\mathrm{IC}_{\lambda,L}\). The IC projector used to produce it was a coherent projector over \(E\), so its extension is already in the completed image. Over \(L\), the classical semisimplicity proof of The Mirković–Vilonen theorem: identifying the dual group, Theorem 8.3, identifies every spherical perverse object as a finite sum of these IC objects tensored by finite-dimensional \(L\)-spaces. That theorem is proved over the original arbitrary characteristic-zero coefficient field, without assuming a scalar-extension equivalence of hearts. Finite perverse truncation then gives every bounded finite-support object. The same coefficient check on the ordered first-factor wall kernels of C.27 compares their actual proper and tensor operations. On each finite moving collection, C.28's configuration-times-toric pair models are again finite perfect, and its full divisor-frame nerve again has the uniform bound. The preceding finite-window argument therefore compares actual moving objects and maps after extension as well. C.29.1–C.29.2 identify the \(L\)-perverse geometric fusion components by the same strict extension tests. Extend the entire \(E\)-enhanced symmetric diagram through these exact coefficient functors, or equivalently through their ordered enhanced equivalence. Its heart components are consequently the actual \(L\)-fusion components. The free-generator model on the right extends to the displayed \(L\)-model: its finite free modules, equivariant maps, cofibres and coherent retracts are the same constructions after extension. This proves (C.29.5). For completeness, existence of an abstract \(E\)-embedding into \(\mathbf C\) does not require a continuous embedding. Both fields have cardinality the continuum. For \(\mathbf Q_\ell\), digit sequences supply this cardinality, and a finite extension has the same cardinality; the elementary expansions also give the upper bound. The real and imaginary expansions give it for \(\mathbf C\). A transcendence basis over \(\mathbf Q\) exists by maximal algebraic independence. If its cardinality is \(\kappa\), the rational-function field and its algebraic closure have cardinality \(\max(\aleph_0,\kappa)\): expressions use finitely many basis elements, and each nonzero polynomial has only finitely many roots. Since the two fields are uncountable of continuum cardinality, each such basis has continuum cardinality. Match their bases to embed the rational-function subfield of \(E\) into \(\mathbf C\). Here is also the needed algebraic closedness of \(\mathbf C\). For a nonconstant complex polynomial \(p\), its leading term gives \(|p(z)|\to\infty\). It attains a minimum on a sufficiently large closed disc. If at its minimizer \(z_0\) the value \(c\) were nonzero, write \(p(z_0+w)/c=1+aw^m+O(|w|^{m+1})\) with \(a\ne0\) and \(m\ge1\). Choose the argument of \(w\), of small modulus \(r\), so \(aw^m=-|a|r^m\). The remainder is bounded by \(Cr^{m+1}\), giving \(|p(z_0+w)/c|\le1-|a|r^m+Cr^{m+1}<1\). This contradicts minimality. Thus every nonconstant polynomial has a root; division and induction split it. The compact-disc minimum used here follows directly from sequential compactness in \(\mathbf R^2\): a minimizing sequence has a convergent subsequence and continuity gives its limit value. Finally order the extensions of the rational-function embedding to intermediate algebraic subfields of \(E\). Unions of chains remain embeddings. A maximal one has domain all of \(E\), since otherwise an omitted algebraic element has a minimal polynomial whose transported polynomial has a root in \(\mathbf C\); quotienting by that irreducible polynomial extends the embedding. This proves the asserted abstract embedding and the complex-coefficient consequence. It supplies no descent to, for example, \(\overline{\mathbf Q}\), which cannot contain \(\mathbf Q_\ell\) since it is countable. The rational tensor descent and its extension to arbitrary characteristic-zero fields are [Theorem C.80.1.1 and Corollary C.80.2.1](#c801-the-rational-represented-tensor-comparison). ∎ ![The original ordered comparison retains geometric convolution; strict moving extension identifies the geometric permutation components, while the entire arithmetic diagram supplies enhanced coherence; diagonal normalizations retain their supported maps](assets/coherent-complex-fusion.png) Theorem C.29.1.1 compares the actual moving IC objects and strict extension tests. Theorem C.29.2.1 identifies geometric permutation and supported partial-diagonal components. Theorem C.29.3.1 retains the entire higher coherence through the original ordered equivalence; Theorem C.29.4.1 proves flat coefficient extensions. Exercise C.29.5.1 computes the negative-degree diagonal compatibility datum. Editable SVG source. #### C.29.5. Two coherence and diagonal calculations **Exercise C.29.5.1.** Let \(A=E[u]\), with \(|u|=2\) and zero differential. Compute the derived diagonal endomorphism complex \(\operatorname{RHom}_{A\otimes A}(A,A)\). Explain why discrete negative-degree maps between perverse word objects cannot by themselves justify discarding all higher coherence. **Solution.** Write \(A^e=E[u_l,u_r]\), and use its free graded resolution \[ K=A^e\otimes\Lambda(\epsilon),\qquad |\epsilon|=1,\qquad d\epsilon=u_l-u_r. \tag{C.29.6} \] The differential has degree one. Multiplication by \(u_l-u_r\) is injective in the polynomial ring, so the augmentation \(K\to A\) is a quasi-isomorphism: its surviving quotient is exactly \(u_l=u_r=u\). The two-stage free filtration makes \(K\) an appropriate derived free resolution. Applying \(\operatorname{Hom}_{A^e}(-,A)\) gives \[ \operatorname{RHom}_{A^e}(A,A) =A\oplus A\theta,\qquad |\theta|=-1,\qquad d=0. \tag{C.29.7} \] The differential becomes zero because \(u_l-u_r\) acts by zero on the diagonal. In particular its cohomology in degree \(-1\) is \(E\theta\), nonzero, although \(H^{<0}(A)=0\). The diagonal adds compatibility with the algebra action; it is not the plain underlying generator endomorphism complex. An enhanced compatibility problem can therefore retain a higher datum even when the objectwise mapping spaces used to identify perverse components are discrete. C.29.3 retains the already constructed arithmetic coherent diagram and transports it through an enhanced equivalence; C.29.2 uses perverse uniqueness only for the component identification. **Exercise C.29.5.2.** For three inputs, compare first identifying labels \(1,3\) and then colliding that block with label \(2\), with imposing the full diagonal at once. Compute the total ordinary shift, exceptional shift and twist. Compute the open coefficient shuffle needed to put the noncontiguous block first. **Solution.** The first partition has two blocks, so \(c_1=1\); the next has one block, so \(c_2=1\). Ordinary restriction in (C.29.3) has total normalization \([-2]\). Exceptional restriction has total normalization \([2]\) and twist \((-2)\), the two trace-one normal orientations. The full diagonal has codimension two and gives exactly these same integers. Its independent equations may be chosen \(x_3-x_1,x_2-x_1\). Their successive supported cup and trace are the composite supported class, as in Fusion and the commutativity constraint, C.3, including its actual transitivity map. For homogeneous open coefficients of degrees \(a,b,c\), regrouping the ordered list \((1,2,3)\) into \((1,3,2)\) crosses only the second and third factors and gives sign \((-1)^{bc}\). Independent-disc gluing supplies this actual shuffle before any collision. The corrected symmetry further supplies the same component-parity factor as Fusion and the commutativity constraint, C.5, when the indicated factors are in specified components. Both paths retain that shuffle and the supported composition. Once the intermediate extensions and their actual supported maps are identified, (C.29.3) agrees on further collisions. Matching the integers alone would not have checked the shuffle or the supported map. ### C.30. Relative degree-zero normalization for tensor descent The remaining coefficient problem must retain the whole tensor diagram. We first remove a specific obstruction to a finite bar argument: convolution words have matrix spaces in degree zero, whereas C.24.3 assumed scalar identities on distinct simple generators. The following relative normalization is a proved algebraic input. It does not yet encode or descend the full geometric tensor coherence. #### C.30.1. Splittings over a split semisimple coefficient algebra **Lemma C.30.1.1.** Let \(k\) have characteristic zero and let \(S=\prod_{\lambda=1}^m M_{d_\lambda}(k)\). Every surjection of \(S\)-bimodules has an \(S\)-bimodule section. If a cochain complex \(C\) of \(S\)-bimodules contains \(S\) in its degree-zero cycles and that inclusion identifies \(S\) with \(H^0(C)\), then it has an \(S\)-bimodule contraction onto its cohomology whose degree-zero cohomology inclusion is that specified inclusion of \(S\). The contraction, its side conditions and all its chosen matrices remain a contraction under any field extension. **Proof.** Write \(e^\lambda_{ij}\) for the matrix units of a factor, viewed as elements of \(S\). The finite sum \[ e=\sum_{\lambda=1}^m\frac1{d_\lambda} \sum_{i,j=1}^{d_\lambda} e^\lambda_{ij}\otimes e^\lambda_{ji} =\sum_\alpha a_\alpha\otimes b_\alpha \tag{C.30.1} \] satisfies \(\sum_\alpha a_\alpha b_\alpha=1\) and \(\sum_\alpha s a_\alpha\otimes b_\alpha =\sum_\alpha a_\alpha\otimes b_\alpha s\). Check the second identity on a matrix unit \(s=e^\lambda_{pq}\): both sides are \(d_\lambda^{-1}\sum_j e^\lambda_{pj}\otimes e^\lambda_{jq}\). Different factors have product zero. For the first, each fixed \(i\) has \(d_\lambda\) summands \(e^\lambda_{ii}\); summing over \(i,\lambda\) gives the unit. Let \(q:M\to N\) be a surjection of bimodules. Choose a \(k\)-linear section \(t\), by lifting a basis of \(N\). First average its left action: \[ t_l(n)=\sum_\alpha a_\alpha t(b_\alpha n). \tag{C.30.2} \] The second identity for \(e\) gives \(t_l(sn)=s t_l(n)\); applying \(q\) gives \(q t_l(n)=n\). Next average the right action: \[ t_{lr}(n)=\sum_\alpha t_l(n a_\alpha)b_\alpha . \tag{C.30.3} \] The same tensor identity gives \(t_{lr}(ns)=t_{lr}(n)s\). It remains left linear because \(t_l\) was left linear and left and right actions commute. Applying \(q\) again gives \(q t_{lr}(n)=n\). Thus it is the desired bimodule section. In particular every submodule of a bimodule has a bimodule complement, by splitting its quotient. For the complex write \(B^r=\operatorname{im}(d:C^{r-1}\to C^r)\) and \(Z^r=\ker(d:C^r\to C^{r+1})\). The first assertion splits both \(B^r\subset Z^r\) and \(Z^r\subset C^r\). Choose \[ C^r=B^r\oplus H^r\oplus L^r,\qquad Z^r=B^r\oplus H^r,\qquad d:L^r\xrightarrow{\sim}B^{r+1}. \tag{C.30.4} \] At degree zero choose \(H^0\) to be exactly the given \(S\): it meets \(B^0\) in zero and its image fills \(Z^0/B^0\), by the hypothesis, so it is already a bimodule complement. Let \(i:H\to C\) be inclusion, \(p:C\to H\) projection, and let \(h:C^r\to C^{r-1}\) be the inverse of the last isomorphism on \(B^r\), zero on \(H^r\oplus L^r\). On the three summands in (C.30.4) direct calculation gives \[ dh+hd=1-ip,\qquad pi=1,\qquad h^2=hi=ph=0. \tag{C.30.5} \] These are bimodule maps and retain the specified degree-zero unit and matrix inclusions. Tensoring the direct sums, the displayed isomorphisms and the identities with any extension field preserves each assertion. In particular the very same contraction is available before and after coefficient extension. ∎ #### C.30.2. Finite positive-degree relative bar stages **Lemma C.30.2.1.** Let \(M\) be a graded \(S\)-bimodule, with \(S\) as in C.30.1, concentrated in degrees at least two and finite-dimensional in every degree. Give \(sM\) the suspension lowering degrees by one. The relative tensor coalgebra \[ B_S(M)=S\oplus\bigoplus_{r\ge1}(sM)^{\otimes_S r} \tag{C.30.6} \] has finite-dimensional subcoalgebras of total suspended degree at most \(D\). Their coalgebra automorphisms preserving degree and the \(S\)-bimodule structure, fixing \(S\) and inducing the identity on their one-letter quotient, form reduced closed subgroups of finite unitriangular groups. This remains true when a finite collection of additional polynomial compatibility equations, satisfied by the identity and stable under composition and inverse, is imposed. **Proof.** Each letter has suspended degree at least one, so a word in the selected stage has length at most \(D\); all of its letters have degrees at most \(D\). There are only finitely many compositions of an integer at most \(D\) into positive integers. Each summand for a composition is a quotient of a finite tensor product of finite-dimensional \(k\)-spaces, hence finite-dimensional. Adding \(S\), which is finite-dimensional, proves stage finiteness. Deconcatenation splits a word into two subwords of no greater degree and gives the coalgebra structure over \(S\); thus it restricts to this stage. The one-letter projection of a coalgebra map determines its higher components: for a word, each output word is obtained by partitioning its consecutive letters into nonempty blocks and applying the chosen one-letter component to each block. This formula is verified by deconcatenation, or inductively by taking the first cut. Fixing the component on one-letter words to be the identity leaves the leading length unchanged, and every other component lowers length. Order a basis by length. Such a map is therefore unitriangular, with each nonidentity entry strictly lowering length. Conversely its inverse is the finite sum for a length-lowering endomorphism, and the inverse of a bijective coalgebra map is a coalgebra map by applying its inverse to the coalgebra equation. Commuting with the coproduct, with the left and right \(S\)-actions, and with the degree projections imposes finitely many polynomial equations on this finite matrix. Fixing \(S\) and the leading quotient imposes further linear equations. Their solution scheme is a closed subgroup of the indicated unitriangular group; take its reduced closed subgroup, as in C.24.1. Additional polynomial equations with the stated composition and inverse stability cut out a reduced closed subgroup there as well. This proves the assertion. Positive degree, not omission of only the overall identity, is essential. For \(S=M_2(k)\), the matrix unit \(e_{12}\) is a degree-zero nonidentity. If it were left as a letter in an ordinary suspended bar construction, its suspension would have degree \(-1\), and an arbitrary string of copies would defeat this positive-degree length bound. The relative construction places all matrix operations in \(S\), rather than pretending they are positive letters. It has exactly the finiteness used above. ∎ #### C.30.3. Descent of a verified filtered multi-operation presentation **Theorem C.30.3.1.** Suppose a countable structured comparison over \(k\) has the following presentation. At stage \(D\) there is a finite-dimensional graded space \(V_D\), a reduced closed unitriangular coordinate group \(U_D\), and finitely many pairs of prescribed multilinear operations \(\mu_{D,i},\nu_{D,i}\). An operation can have several inputs and several outputs. Its input and output subspaces in the tensor powers of \(V_D\) are invariant under \(U_D\). The coordinate changes satisfy \[ g^{\otimes s_i}\mu_{D,i} =\nu_{D,i}g^{\otimes r_i} \tag{C.30.7} \] on those input subspaces. Allow \(r_i=0\) for a specified unit. Assume that the stage restrictions are homomorphisms \(U_{D+1}\to U_D\), send solutions of every displayed equation to solutions of the preceding equations, and exhaust the coordinate entries and operation equations of the original comparison. If an extension field gives a compatible solution with the prescribed leading coordinate components, there is a compatible solution over \(k\). For this statement, the presentation includes every operation and higher compatibility datum required by the comparison. Supplying just its composition operations proves only a composition comparison. **Proof.** The tensors in (C.30.7) are finite matrices, and every entry of a tensor power of \(g\) is a polynomial in its entries. These equations cut out a closed solution scheme in \(U_D\). Let \(T_D\) be its reduction. Its geometric nonemptiness follows from the extension solution. Let \(G_D\) be the reduced closed subgroup of \(U_D\) defined by \[ a^{\otimes s_i}\nu_{D,i} =\nu_{D,i}a^{\otimes r_i} \tag{C.30.8} \] for all operations at that stage. The equations are closed polynomial equations. The identity satisfies them. Composing two solutions of (C.30.8) gives a solution, and multiplying the equation on both sides by the tensor powers of \(a^{-1}\) proves it for the inverse. Invariance of the designated input and output subspaces makes these restrictions legitimate. The left action of \(G_D\) preserves \(T_D\). If \(g,h\) are two geometric solutions of (C.30.7), then \[ (hg^{-1})^{\otimes s_i}\nu_{D,i} =h^{\otimes s_i}\mu_{D,i}(g^{-1})^{\otimes r_i} =\nu_{D,i}(hg^{-1})^{\otimes r_i}. \tag{C.30.9} \] Thus \(hg^{-1}\in G_D\), and it is the unique group element carrying \(g\) to \(h\). This verifies the torsor scheme assertion as well as transitivity on points: after choosing a geometric \(g\), the polynomial isomorphism \(a\mapsto ag\), with inverse \(h\mapsto hg^{-1}\), changes the defining equations (C.30.8) into exactly (C.30.7). Taking reductions preserves this isomorphism. Hence \(T_D\) is a geometrically nonempty finite-type torsor under \(G_D\). The restrictions in the hypothesis induce \(G_{D+1}\to G_D\) and \(T_{D+1}\to T_D\) and are equivariant. Theorem C.24.2.1 now supplies compatible \(k\)-points. Every fixed original coordinate appears at a stage and is retained by further restriction. Define that coordinate to be its value in the compatible system. Every original operation equation is finite at its designated stage, so it holds for these limiting coordinates. If the coordinate groups arise from C.30.2, their inverses are defined on each finite word by the finite length-lowering recursion; the compatible limiting map therefore has its compatible inverse. In any other verified presentation the same conclusion is exactly its stipulated exhaustion of coordinate maps and inverses. The resulting map is the required structured comparison over \(k\). This theorem permits tensor, permutation, unit and multi-output compatibility equations in one tower. It does not assert that a given geometric tensor category has such a presentation. That construction must prove stage finiteness, invariant equation domains, genuine coordinate-group restrictions, and exhaustiveness of the full higher diagram. C.30.1–C.30.2 provide its relative algebra ingredients; C.29 provides an extension-field solution. The missing geometric presentation is still a separate proof obligation. ∎ ### C.31. Finite ordered-tree indices and coherence degree bounds An ordered tensor construction has two sources of higher data: its input mapping complexes and its associativity diagrams. A positive input degree bounds the former. A finite tree index bounds the latter. We prove those bounds with the relative resolutions that retain the actual diagrams. They will be used to build the full tensor presentation; its coordinate groups and comparison equations must still be verified. #### C.31.1. Trees, contractions and the terminal corolla **Lemma C.31.1.1.** Let \(n\ge2\). The category \(T_n\) of planar rooted trees with \(n\) leaves in their fixed order and with every internal vertex of arity at least two is a finite poset under internal-edge contraction. It has terminal object the \(n\)-corolla. A strict chain of contractions has length at most \(n-2\). Its augmented normalized nerve chains are contractible by a homotopy of that same finite complex, over \(\mathbf Z\). **Proof.** Every subtree occupies an interval of the ordered leaves. Associate to each internal edge its proper leaf interval of length at least two. These intervals form a nested or disjoint collection. Conversely a collection of proper intervals with this property determines the tree: include the full interval as root and the singleton leaves; attach each interval to its smallest strictly containing interval. Order its children by their least leaves. A proper child cannot be its parent's only child, since it omits at least one leaf of that parent. Thus every internal vertex has at least two children. The constructions are inverse. There are only finitely many possible intervals and hence finitely many trees. Contracting an internal edge deletes its proper interval. Keeping a selected subset of the original intervals produces the corresponding contraction tree, with its ordered leaves. Thus two different contractions with the same result have deleted the same intervals. There is at most one morphism between two objects, and contraction corresponds to reverse inclusion of their interval collections. This makes \(T_n\) the claimed poset. The empty collection is its terminal corolla. If the internal vertices have arities \(r_1,\ldots,r_v\), then counting edges gives \[ \sum_{j=1}^v(r_j-1)=n-1. \tag{C.31.1} \] Indeed the number of edges is \(n+v-1\), also \(\sum_jr_j\). Since each \(r_j-1\ge1\), \(v\le n-1\). There are \(v-1\) internal edges, so at most \(n-2\) strict contraction steps. The same bound follows by counting proper intervals. Write \(c_n\) for the terminal corolla. A normalized degree-\(q\) simplex is a strict chain \([t_0\to\cdots\to t_q]\). If its last object is not \(c_n\), append the unique map to \(c_n\) and multiply by \((-1)^{q+1}\); if it already ends there, send it to zero. In augmented degree \(-1\), send the generator to \([c_n]\). This defines a homotopy \(h\). In the boundary of the appended chain, deleting its last vertex gives the original chain with coefficient one. All other deletions cancel the corresponding terms in \(h\partial\). For a chain already ending at \(c_n\), only the deletion of that last vertex contributes to \(h\partial\), again giving the original chain. In degree zero the augmentation and the chosen \([c_n]\) give the same calculation. Consequently \(\partial h+h\partial=1\). Appending the corolla adds at most one contraction; any chain already of maximal length ends at the corolla, so the homotopy stays within the finite normalized complex. This proves all assertions integrally. ∎ #### C.31.2. A bounded relative resolution for a directed index **Theorem C.31.2.1.** Let \(I\) be a finite skeletal category, with finite morphism sets. Suppose each endomorphism is invertible and every nonisomorphism strictly increases an integer height whose range has length \(H\). Let \(k\) have characteristic zero. A \(k\)-linear diagram on \(I\) has a projective relative bar resolution of length at most \(H\). The same resolution computes mapping complexes between strict complex-valued diagrams, retaining all their diagram compatibilities. For \(I=T_n\), one can take \(H=n-2\). **Proof.** Let \(A=k[I]\), with multiplication given by composition when endpoints match and zero otherwise. Its unit is the sum of the identity arrows. Let \(S\) be the subalgebra spanned by the endomorphisms and let \(R\) be the span of all nonisomorphisms. Because the category is skeletal, its only isomorphisms are endomorphisms. Therefore \[ S=\prod_{x\in I} k[\operatorname{Aut}_I(x)], \qquad A=S\oplus R. \tag{C.31.2} \] Left or right multiplication of a nonisomorphism by an automorphism preserves its endpoints and its strictly increasing height. A composite containing a nonisomorphism is still a nonisomorphism, by height. Thus \(R\) is an ideal and an \(S\)-bimodule, and projection \(A\to S\) is an algebra map. Every \(S\)-module is projective. For each finite automorphism group \(\Gamma\), average a vector-space section \(t\) of a surjection by \[ t_\Gamma(v)=|\Gamma|^{-1}\sum_{g\in\Gamma}g\,t(g^{-1}v). \tag{C.31.3} \] It remains a section and is equivariant: translating the sum just reindexes it. Decompose a module into the finitely many object-idempotent components and apply this argument there. The same argument for the two commuting groups proves the bimodule version. It works for infinite-dimensional modules as well because the sums are finite. A covariant diagram is a left \(A\)-module \(M\): its object component is \(1_xM\) and each arrow acts by the prescribed map. Conversely these components and actions recover the diagram and its composition identities. Set \[ P_q(M)=A\otimes_S R^{\otimes_S q}\otimes_S M \quad(q\ge0). \tag{C.31.4} \] The augmentation \(P_0(M)\to M\) is action. The differential multiplies adjacent factors, with the alternating signs, including first the action of the first \(R\)-factor on the left \(A\)-factor and finally the action of the last \(R\)-factor on \(M\). Associativity cancels the two possible orders of each multiplication, so its square is zero. Each term is induced from an \(S\)-module and is projective over \(A\): a section of a surjection to its coefficient module lifts by projectivity over \(S\), and the induced \(A\)-map is then defined by the left action. Equivalently \(\operatorname{Hom}_A(A\otimes_SN,-)=\operatorname{Hom}_S(N,-)\), which is exact. We verify exactness without assuming it from a bar citation. Write \(a=\epsilon(a)+\bar a\) using (C.31.2). In augmented degree \(-1\) send \(m\) to \(1\otimes m\). At degree \(q\ge0\) send \[ a\otimes r_1\otimes\cdots\otimes r_q\otimes m \longmapsto 1\otimes\bar a\otimes r_1\otimes\cdots\otimes r_q\otimes m . \tag{C.31.5} \] These are well-defined \(S\)-linear maps on the balanced tensors: projection to \(R\) is an \(S\)-bimodule map. Multiplying the first inserted factor gives \(\bar a\) back; all successive multiplication terms cancel in \(\partial h+h\partial\). The remaining projection term is \(1\otimes\epsilon(a)\) balanced into the next factor, which equals that part of the original tensor. For \(q=0\), the calculation is explicitly \[ \bar a\otimes m-1\otimes\bar a m+1\otimes a m =a\otimes m. \tag{C.31.6} \] For \(q>0\) the same cancellation uses that \(\epsilon(r_1)=0\). Thus (C.31.5) contracts the augmented complex as \(S\)-modules and proves it exact. Tensoring over the object idempotents forces adjacent arrow endpoints to match. A tensor of \(q\) nonisomorphisms therefore lies on a path with \(q\) strict height increases. There is none for \(q>H\). Quotienting further by the endomorphism-group relations does not create such a path. Consequently \(R^{\otimes_Sq}=0\) for \(q>H\), giving the bound. For a strict diagram \(F\) of cochain complexes, apply (C.31.4) in every internal degree and totalize the finite resolution with its standard internal differential signs. Formula (C.31.5) remains a contracting homotopy of the augmented total complex, so it is equivalent to \(F\). Its terms are induced complexes from \(S\). They are suitable for derived Hom: an acyclic complex of \(S\)-modules is contractible by the splitting proof above, so applying \(\operatorname{Hom}_S(N,-)\) for any coefficient complex \(N\) preserves acyclicity by that contraction. The finite resolution therefore computes the derived \(A\)-mapping complex. Its ordinary differential records internal maps; its relative bar differential records all diagram compatibility and its successive homotopies. No deletion of those higher compatibility terms is made. Finally for \(T_n\), use height \(n-2\) minus the number of its proper intervals. Each strict contraction increases it, by C.31.1, and its range is from zero to \(n-2\). This gives the asserted tree bound. ∎ #### C.31.3. Positive mapping degrees combined with a finite diagram **Lemma C.31.3.1.** Consider finitely many object labels, with reduced mapping spaces \(M(X,Y)\) concentrated in degrees at least two and finite-dimensional in each degree. Let an additional directed diagram index have maximum nonidentity chain length \(H\). Suppose a component is a homogeneous map from a relative suspended input bar to an output mapping complex suspended by \(\epsilon\), where \(\epsilon=0\) or \(1\) and suspension lowers degree. If its total mapping degree is \(h\), its input word has total suspended degree \(w\), and its diagram bar term has degree \(q\), the output's ordinary degree is \[ d=w+h-q+\epsilon,\qquad 0\le q\le H. \tag{C.31.7} \] If outputs are retained only in ordinary degrees at most \(D\), then \(w\le D+H-h-\epsilon\). There are finitely many coordinate slots for these components, provided each of the finitely many output mapping spaces is degreewise finite-dimensional. The same assertion holds for several input relative bars, with \(w\) their summed suspended degree. **Proof.** In the normalized mapping bar, a letter of ordinary degree \(a\) has suspended degree \(a-1\). Its word has degree \(w\). A homogeneous map of internal degree \(p\) sends that word to an element of degree \(w+p\) in the suspended output complex, so that element's ordinary degree is \(w+p+\epsilon\). The additional diagram bar term contributes cochain degree \(q\) in the derived mapping totalization of C.31.2. Thus a component of total degree \(h\) has \(p=h-q\), proving (C.31.7). In unsuspended Taylor notation with \(r\) input letters, this component has degree \(h+\epsilon-r-q\). The case \(\epsilon=1\) retains the extra degree for a suspended functor component; it must not be confused with the unsuspended output convention for a natural transformation. The inequality \(d\le D\) and \(q\le H\) give \(w\le D+H-h-\epsilon\). Each letter has suspended degree at least one, so word length is at most that integer, and each individual letter's suspended degree is bounded by it too. With finitely many labels and degreewise finite mapping spaces, there are only finitely many labelled degree profiles, each with finite-dimensional coefficients. The index has finitely many objects and arrows and its relative resolution has only the finitely many terms \(q\le H\). There are therefore finitely many component-coordinate slots. With several bars each letter still adds at least one to their sum, so the same reasoning bounds their total number of letters and all their profiles. For ordered \(n\)-input tensor comparisons the additional tree index contributes \(H=n-2\), by C.31.2. Thus, for a fixed finite list of input labels, fixed \(n\) and fixed \(D,h\), there is an actual finite coefficient range. This observation bounds the profile spaces; it does not by itself construct a compatible finite coordinate-group quotient of the whole infinite tensor presentation. Such a quotient must also retain the domains of composition and inverse and all required equations, as C.30.3 specifies. ∎ ![The relative degree-zero matrix algebra is separated from positive bar letters; the actual three-leaf trees contract to their common corolla; the finite diagram bound and output suspension determine the retained mapping profiles](assets/tensor-coherence-bounds.png) Lemma C.30.1.1 proves the matrix averaging and fixed-unit contraction. Lemma C.30.2.1 bounds positive relative bar stages, and Theorem C.30.3.1 descends every operation of a verified filtered presentation. Lemma C.31.1.1 gives the displayed actual tree contractions; Theorem C.31.2.1 proves the finite projective diagram resolution. Lemma C.31.3.1 proves the profile bound with both output suspension conventions. Editable SVG source. #### C.31.4. Two finite-index tests **Exercise C.31.4.1.** Describe \(T_3\) and the relative resolution of its constant \(k\)-diagram. For \(n=3\), \(D=2\), \(h=0\), compute the possible suspended input-degree range for unsuspended and suspended output conventions in C.31.3. **Solution.** The three trees are \[ a=((12)3),\qquad b=(1(23)),\qquad c=(123), \tag{C.31.8} \] with only the nonidentity arrows \(a\to c\) and \(b\to c\). Hence \(H=1\). In the constant diagram each arrow acts by the identity. The \(P_0\)-basis in C.31.2 consists of its three identity arrows and two nonidentity arrows, tensored by the source's constant generator, so \(\dim P_0=5\). The \(P_1\)-basis consists of those two nonidentity arrows with the outer factor \(1_c\), so \(\dim P_1=2\). Higher terms vanish. Its differential sends each nonidentity-arrow basis vector to that arrow's \(P_0\)-basis vector minus \(1_c\)'s vector. Augmentation sends any arrow to the constant generator at its target. These two columns are independent and span the two-dimensional augmentation kernel, proving the exact projective resolution \[ 0\longrightarrow k^2\longrightarrow k^5 \longrightarrow k^3\longrightarrow0 \tag{C.31.9} \] with its specified diagram actions. For \(\epsilon=0\), the profile bound is \(w\le3\). More precisely at \(q=0\) it is \(w\le2\), and at \(q=1\) it is \(w\le3\). For \(\epsilon=1\) the ranges are \(w\le1\) and \(w\le2\), respectively. For example a degree-two ordinary input letter has \(w=1\); at \(q=0\) its unsuspended degree-zero component lands in ordinary output degree one, whereas its suspended-output component lands in degree two. If that degree-one output mapping space vanishes, the former coordinate is absent. Counting both conventions identically would insert the wrong coordinate slots into the formality equations. **Exercise C.31.4.2.** Why is characteristic zero essential to the bounded projective-resolution assertion in C.31.2, even if the index has no nonisomorphisms? **Solution.** Let \(k\) have characteristic \(p>0\), and let the index have one object with automorphism group \(C_p\). It has no nonisomorphisms and would have \(H=0\). Its category algebra is \[ A=k[C_p]=k[t]/t^p,\qquad t=g-1. \tag{C.31.10} \] The equality follows from \((g-1)^p=g^p-1=0\), and the \(p\) basis elements give the inverse presentation. Its trivial module \(k=A/(t)\) has the free resolution with alternating maps \[ \cdots\xrightarrow{\,t\,}A\xrightarrow{\,t^{p-1}\,}A \xrightarrow{\,t\,}A\longrightarrow k\longrightarrow0. \tag{C.31.11} \] In the basis \(1,t,\ldots,t^{p-1}\), multiplication by \(t\) has kernel \(kt^{p-1}\) and image \((t)\); multiplication by \(t^{p-1}\) has kernel \((t)\) and image \(kt^{p-1}\). Thus the complex is exact. Applying \(\operatorname{Hom}_A(-,k)\) makes every differential zero, and gives \(\operatorname{Ext}_A^r(k,k)=k\) for every \(r\ge0\). In particular the trivial module is not projective and no length-zero projective resolution computes its mapping complex. The missing step is exactly averaging (C.31.3): \(p\) is not invertible. Its degree-zero relative resolution over \(S=A\) is exact but not projective, so cannot replace the displayed derived resolution. Finiteness of the tree index alone never licenses dropping automorphism-group cohomology without the stated coefficient hypothesis. ### C.32. Directed relative bars with degree-zero arrows A bound on total suspended degree alone is not enough when a diagram has degree-zero arrows. Their suspensions have degree \(-1\), and deconcatenation can increase the degree of a subword. We instead count the positive letters' suspended degree and retain the directed arrow bound. This gives subcoalgebras genuinely preserved by the coordinate groups. #### C.32.1. Positive-weight windows in a directed coefficient diagram **Lemma C.32.1.1.** Let \(S=\prod_{j=0}^H S_j\) be a finite-dimensional split semisimple \(k\)-algebra. Let \(M\) be a graded \(S\)-bimodule, degreewise finite-dimensional, with the following conditions. Its degrees below zero and its degree one vanish. Its degree-zero part has components only from height \(j\) to strictly higher height \(i\). All its positive-degree components go from height \(j\) to height \(i\ge j\). Form the relative tensor coalgebra \[ B=S\oplus\bigoplus_{r\ge1}(sM)^{\otimes_Sr}. \tag{C.32.1} \] Give a positive-degree letter its suspended degree as weight, and give a degree-zero letter weight zero. A window \(B_{\langle D\rangle}\) of summed positive weight at most \(D\) is a finite-dimensional subcoalgebra. Any nonempty word in it has at most \(D+H\) letters. A degree-preserving \(S\)-relative coalgebra automorphism whose one-letter component is the identity preserves each window, as does its inverse. **Proof.** Call a suspended degree-zero letter a zero-arrow letter; it has cohomological degree \(-1\). A positive letter has ordinary degree at least two, hence suspended degree at least one. If a word has \(q\) zero-arrow letters and positive weight \(w\), its cohomological degree is \[ |{\rm word}|=w-q. \tag{C.32.2} \] Balanced tensoring forces successive height endpoints to match. All letters are nondecreasing in height and every zero-arrow letter strictly increases it. Thus \(q\le H\). Positive letters number at most \(w\). In weight at most \(D\), total length is at most \(D+H\); each positive letter has ordinary degree at most \(D+1\). The labels, permitted degrees and coefficient spaces are finite, so the window is finite-dimensional. Deconcatenation splits the weight into two nonnegative weights, each at most the original, proving the subcoalgebra assertion. Consider a one-letter component of a degree-preserving coalgebra map applied to a block. If its output is positive, it has weight \(w-q\), since its degree equals (C.32.2). This is no greater than \(w\). If its output is a zero-arrow letter, preservation of degree gives \(w-q=-1\), so \(q=w+1\ge1\); its output weight is zero, again no greater than \(w\). The coefficient component to the coaugmentation is zero, since the map is relative to \(S\) and preserves the counit. The cofree coalgebra formula partitions a word into consecutive blocks, applies these one-letter components and tensors their outputs. Summing the inequalities proves that it cannot increase positive weight. Its leading component is the identity, and all other components lower length. On a fixed finite window its inverse is therefore the finite length-lowering recursion of C.30.2. It preserves degree and the \(S\)-relative structure, so the same block argument proves weight preservation for the inverse. Both assertions hold on each word of the full coalgebra, by its finite length. ∎ #### C.32.2. The coderivation output bound **Lemma C.32.2.1.** In C.32.1, every degree-one \(S\)-relative coderivation whose outputs are in \(sM\) increases positive weight by at most one: \[ b(B_{\langle D-1\rangle}) \subset B_{\langle D\rangle}. \tag{C.32.3} \] Its object heights are retained. In particular this applies to the reduced relative bar coderivation of a strictly \(S\)-unital minimal \(A_\infty\) category with these graded mapping conditions, and to the coderivation recording its ordinary graded composition. **Proof.** A one-letter coderivation component applied to a block of positive weight \(w\) and zero-arrow count \(q\) has cohomological output degree \(w-q+1\). If the output is positive, its weight is \(w-q+1\le w+1\). If it is a zero-arrow letter, then \(w-q+1=-1\), so \(q=w+2\); its weight zero is again at most \(w+1\). Its \(S\)-bimodule condition retains the block endpoints and hence the height ordering. A coderivation applies such a component to one consecutive block at a time, retaining all other letters. Each term consequently has total weight at most the input weight plus one. This proves (C.32.3). We also justify why the relative category coderivation has no extra coaugmentation output. An output in \(S\) would have equal initial and final heights. Nondecreasing heights then force every input letter to stay at that height, so no degree-zero letter of \(M\) occurs. Every ordinary input degree is at least two. An \(r\)-ary category operation has degree \(2-r\); its ordinary output degree would be at least \(2r+2-r=r+2>0\), incompatible with an output in \(S\). Minimality makes the unary differential zero, and strict \(S\)-unitality already places the prescribed degree-zero coefficients in \(S\). Thus the reduced coderivation outputs only in \(sM\), with the asserted degree one. The ordinary composition coderivation has the same property and the same bound. ∎ #### C.32.3. Compatible directed relative composition descent **Theorem C.32.3.1.** Let a strictly \(S\)-unital minimal \(A_\infty\) category have the finite directed graded mapping description of C.32.1. Suppose an extension field gives a relative \(A_\infty\) comparison with its ordinary graded category, with first component the identity. Then that comparison exists over \(k\), with the same prescribed first component. The result also holds for a countable exhaustion by full finite labelled directed relative subcategories, provided their bar-word embeddings, coefficient corners, heights and operations restrict compatibly. Additional marked operations can be included when their window equations and coordinate-group restrictions have been verified as in C.30.3. **Proof.** Let \(b\) be its reduced bar coderivation and \(b_2\) the ordinary composition coderivation. At finite stage \(D\), use the actual subcoalgebra \(B_{\langle D\rangle}\) of C.32.1. Its coordinate changes preserve degree, height labels and the \(S\)-bimodule structure, fix the coaugmentation and have identity one-letter component. The coalgebra equations on its finite matrices are polynomial. With a basis ordered by length, every such coordinate change is unitriangular, and its inverse is the finite length recursion. They therefore form a reduced closed subgroup \(U_D\) of a finite unitriangular group. Impose \[ g b=b_2g \quad\hbox{on }B_{\langle D-1\rangle}. \tag{C.32.4} \] This is a finite polynomial system with outputs in \(B_{\langle D\rangle}\), by C.32.2 and the weight preservation in C.32.1. We do not apply the coderivation to the top window and claim it stays there. The extension-field comparison gives a solution at every stage. If \(g,h\) are solutions, \(hg^{-1}\) commutes with \(b_2\) on that same input range, by the intertwining calculation in C.30.3. The inverse preserves the input window, so every occurrence of that calculation is legitimate. Conversely any coordinate change commuting with \(b_2\) sends a solution to a solution. The reduced solution variety is therefore a geometrically nonempty finite-type torsor under its reduced closed stabilizer in \(U_D\). Restriction \(B_{\langle D+1\rangle}\to B_{\langle D\rangle}\) here means restricting its coordinate maps to the subcoalgebra. It gives group homomorphisms, because every map and inverse preserves that subcoalgebra. It sends solutions and stabilizers to the preceding ones: the previous equations have input weight at most \(D-1\) and output weight at most \(D\), both retained. The compatible-point theorem C.24.2.1 supplies solutions over \(k\) at all stages. Each word has finite positive weight, so the compatible coordinates define a coalgebra automorphism on all of \(B\). Its inverse is the wordwise finite length recursion. Each coderivation equation is checked at a sufficiently large window. This proves the comparison. For the countable assertion enumerate the object labels and at stage \(D\) retain the first \(D\) labels and positive weight at most \(D\). Use the maximum height range on this finite set to bound its zero-arrow count; no uniform height bound on the entire countable family is required. A bar word using the selected labels remains among them under deconcatenation. A coalgebra component retains its endpoints, and the cofree formula uses only block endpoints of its original word, so its outputs also use those labels. Fullness and the stipulated compatible coefficient corners consequently give the restriction homomorphisms as the finite label set grows. All operation equations of input weight at most \(D-1\) on those labels have their outputs in the retained window, as before. The same torsor tower proves the assertion. Every original finite word and coefficient occurs at some stage. Finally, for additional marked multilinear operations, C.30.3 applies to this particular coordinate tower once their input subspaces, output windows and restriction equations are verified. Their reduced stabilizers are closed subgroups of these same unitriangular groups. A comparison over the extension field then descends the entire marked presentation. This last statement includes only the operations actually supplied and verified. To obtain the full tensor comparison we still need to construct the geometric tensor collages and their marked copies, external maps and associativity diagrams; composition descent of the directed category alone does not assert that construction. ∎ ### C.33. Rational Betti coefficients and actual convolution graph categories The next constructions use classical sheaves on the complex Grassmannian with rational coefficients. They concern the actual mapping complexes and convolution kernels. Formalizing a graph category compares a functor along some comparison of its input category. To prove a strong monoidal comparison, one must additionally identify that input comparison with the tensor powers of one common output comparison. We keep that condition explicit. #### C.33.1. The rational coefficient generator category **Theorem C.33.1.1.** Let \(\mathcal G_{\mathbf Q}\) be the enhanced category of the complex spherical IC generators with rational Betti coefficients, and let \(k\) be any characteristic-zero field. Actual coefficient extension induces an enhanced equivalence \[ \operatorname{Perf}(\mathcal G_{\mathbf Q}\otimes_{\mathbf Q}k) \ \simeq\ D^b_G(\operatorname{Gr}_{G,\mathbf C},k). \tag{C.33.1} \] It retains the original ordered convolution and its complete finite-frame operation diagrams on the corresponding generator envelopes. On rational generator mapping cohomology there are no negative or odd degrees; the degree-zero maps between distinct simple generators vanish, each simple has endomorphism field \(\mathbf Q\), and every degree is finite-dimensional. **Proof.** Start with the actual induced flag/wall generators of C.27. Their classical mapping complexes are the trace-normalized shifted duals of their compact pair complexes, with the entire finite action nerve retained. These compact complexes have a rational perfect model. Indeed an affine cell's compact model is the rational orientation line in its top degree. A multiplicative cell is the two-term projective-line localization complex with the actual unit and positive first Chern class. Products, finite closed localization and the actual pair filtrations in C.25–C.27 give finite complexes of finite-dimensional rational spaces. Their connecting morphisms are the morphisms of the original localization triangles. With rational coefficients their finite cofibres are still perfect, and extending coefficients preserves each of these triangles and its morphisms. This verifies coefficient comparison on the compact pair complexes themselves, with their evaluation and trace, rather than by equality of dimensions. Dualizing a finite perfect rational complex commutes with \(k\)-extension. The shifted dual mapping formula therefore gives the actual coefficient comparison on every finite action-nerve level. The level lower bound in C.26.3 and C.27 is uniform. In any fixed total degree its full totalization is consequently computed by a finite initial nerve window, including its incoming differential. Exact coefficient extension commutes with that finite window. Thus the full equivariant generator mapping complexes compare. This argument does not tensor an unrestricted infinite product with \(k\). Finite cofibres in the two variables and coherent projectors extend full faithfulness to the induced-generator envelope. Rational IC generators belong to that envelope. The classical IC parity and coherent root-word projector construction of C.25 and C.27 uses the rational Kummer and orientation classes and the finite split flag Chern basis; its rational projector has both strict IC restriction tests. Extending it gives the actual \(k\)-IC: flatness preserves those restriction vanishing groups, its constant rank-one open coefficient and its intermediate-extension criterion. The same construction over \(k\) lies in the extended envelope. Every spherical perverse \(k\)-object is a finite sum of these IC objects with finite-dimensional multiplicity spaces, by the arbitrary-characteristic-zero classical semisimplicity proof in *The Mirković–Vilonen theorem: identifying the dual group*, Theorem 8.3. Its bounded perverse truncation tower gives every bounded finite-support object. This proves essential surjectivity and (C.33.1). For operations, the original finite-frame coefficient extension is exact, commutes with tensor and carries the wall kernel and its endpoint proper maps to their \(k\)-coefficient versions. The proper maps between the rootwise generators compare by their same compact fibre models. The first-factor wall-kernel identity (C.27.8) then extends this comparison through the whole induced envelope in each variable. On longer ordered strings, use the same common finite frame before and after extension. Its tensor associativity, projection maps, proper composition and descent comparisons remain the original maps with extended coefficients. Their higher diagrams remain the original diagrams as well, since coefficient extension acts on the enhanced resolution diagram. This proves the stated ordered operation assertion. Finally take \(k=E\), a finite extension of \(\mathbf Q_\ell\). C.29's enhanced comparison and its graded generator identification give the mapping cohomology of \(\mathcal G_E\): \[ H^\bullet\operatorname{Hom}(\mathrm{IC}_\lambda,\mathrm{IC}_\mu) =\bigl(\operatorname{Sym}_E(\mathfrak h[-2]) \otimes V_\lambda^\vee\otimes V_\mu\bigr)^H. \tag{C.33.2} \] Its negative and odd degrees vanish and its degreewise spaces are finite. In degree zero, the split simple-module endomorphism calculation gives \(E\) for \(\lambda=\mu\) and zero otherwise. The coefficient comparison just proved, together with faithfulness of extension of vector spaces from \(\mathbf Q\) to \(E\), gives all these assertions over \(\mathbf Q\). In particular a rational group is zero if its extension is zero, and its dimension equals the finite extended dimension. This proves the final assertions without claiming tensor formality over \(\mathbf Q\). ∎ #### C.33.2. The graph construction retains the enhanced functor **Lemma C.33.2.1.** Let \(F:\mathcal A\to\mathcal C\) be an enhanced functor between categories given by their unital \(A_\infty\) models over a field. There is a two-level graph category \(\operatorname{Gr}(F)\) with the original \(\mathcal A,\mathcal C\) as full subcategories, no maps from the target level to the source level, and cross mapping complex \[ \operatorname{Hom}_{\operatorname{Gr}(F)}(x,y) =\operatorname{Hom}_{\mathcal C}(F(x),y) \quad(x\in\mathcal A,\ y\in\mathcal C). \tag{C.33.3} \] Its higher operations retain the graph's complete left and right actions. If all \(F(x)\) lie in the finite target envelope, those actions recover \(F\) on that envelope. Compatible functor and natural-transformation diagrams induce compatible graph diagrams. **Proof.** The right action is induced by the entire functor, not only by its first Taylor component. On a consecutive string of source arrows, partition it into nonempty consecutive blocks and apply the respective components of \(F\) to the blocks. Insert the resulting target arrows before the cross arrow, followed by any target arrows in the original composable string, and apply the corresponding target operation. Sum over the partitions, using the tensor Koszul signs after suspension. This defines the cross operations. A composable word can cross from the source to the target at most once, so no additional type of mixed operation is needed. On words lying in one level use its original operations. Check the square-zero bar identity by separating the terms for such a crossed word. Terms with an operation in the source portion cancel the differential of the partitioned functor expression by the functor's bar identity. Terms with a target operation before or after the cross arrow cancel by the target bar identity. A partition cut produces the same two consecutive applications as composing two right actions, with the opposite sign in the differential expansion. These are exactly the remaining terms. Thus the mixed operations satisfy the \(A_\infty\) identities. On words without a cross arrow this is the original source or target identity. Normalized identities give the strict units; inserting an identity in a higher component is zero. This constructs the category and its full two levels. For fixed \(x\), its cross module in the target variable is the represented module \(\operatorname{Hom}_{\mathcal C}(F(x),-)\). Its right source action is the whole enhanced action just constructed. Evaluation at the representing object recovers the identity of \(F(x)\), and precomposition by an enhanced source map recovers the corresponding functor map and its higher components. More formally, a transformation between two represented cross modules is determined by its evaluation on that identity: composing it with any target map gives its other values. For higher components the same statement applies to each source bar word, and the module identity is precisely the functor's compatibility equation on that word. The bar version of this evaluation gives an equivalence of the represented-module mapping complex with the target mapping complex. Its extra identity-degenerate terms contract by insertion of the identity, as in the normalized bar contraction of C.31.2. Hence it recovers the functor, including higher actions, up to enhanced equivalence. If the representing object is a finite cofibred or retract construction, the same evaluation extends through its mapping cones and commuting projectors in each variable, exactly the enhanced Yoneda argument of A.4. This proves recovery on the finite envelope. Finally a coherent functor diagram applies to the same consecutive-block formula; its natural-transformation components supply the cross-module transformations and their homotopies. The bar identities of that diagram give their compatibility. This proves the diagram assertion without imposing a strict dg lift of every functor or replacing a graph module by its cohomology alone. ∎ #### C.33.3. Actual ordered-tree convolution collages **Theorem C.33.3.1.** Fix a positive arity \(n\) and a finite set of spherical simple IC labels. The actual ordered convolution functor on their input generator category, with its ordered-tree associativity diagram, has a directed graph-category presentation over \(\mathbf Q\). Its graded mapping spaces satisfy C.32's hypotheses after its input tuples and target objects are chosen as simple labels and finite sums are retained in the envelope. There is a countable compatible system of these presentations as the finite label sets, finite output sets and arities grow. On each fixed arity the diagram retains the actual finite-frame associativity maps. **Proof.** Let \(\mathcal A_n\) be the generator tensor-product category. Its objects are ordered tuples of simple labels and its mapping complex is \[ \operatorname{Hom}_{\mathcal A_n} ((P_1,\ldots,P_n),(Q_1,\ldots,Q_n)) =\bigotimes_{i=1}^n \operatorname{Hom}_{\mathcal G_{\mathbf Q}}(P_i,Q_i). \tag{C.33.4} \] Its interchange signs are the ordinary dg tensor signs. We keep this actual input category, before replacing its mapping complexes by cohomology. For a tree \(t\in T_n\), evaluate the iterated ordered convolution indicated by \(t\). The enhanced finite-frame construction of Convolution and rigidity, Appendices A.4-A.6 and C.27 gives these functors with the original coherent edge-contraction comparisons. To describe them on one model, take the full \(n\)-step common frame chain. Flattening a tree to that chain is the actual tensor and proper-composition comparison, with inverse given by its original parenthesized evaluation. Every edge contraction compares two evaluations through this same flattened chain. Thus every finite contraction diagram is the original coherent diagram on the common frame. Passing to a resolution model gives its \(A_\infty\) diagram; its components, rather than arbitrary chosen homotopies, are retained in the graph construction. The input diagram category has objects \((t,\mathbf P)\). For a contraction \(t\to t'\) its mapping complex is the mapping complex (C.33.4); if no contraction exists it is zero. Composition is the input composition together with the actual composition of contractions. The common-chain comparisons just described give the enhanced functor from this category to the spherical target envelope. Apply C.33.2 to it, using the target's simple IC generator category and retaining the convolution output as its represented cross module. Its cross mapping complex is the original \[ \operatorname{Hom}\bigl(F_t(P_1,\ldots,P_n),Q\bigr), \tag{C.33.5} \] not an invented matrix complex. All target summands needed by these finitely many convolution words are finite: those words are perverse with finite support, and the proved semisimple heart expresses each as a finite sum of simple ICs. Enlarge the finite target label set by those summands. By C.33.1, the input complexes have nonnegative even cohomology, finite in each degree. Their degree-zero maps vanish between different tuples and are the scalar identity on an equal tuple. This follows from finite tensor Künneth over a field, obtained by splitting each complex into cohomology plus a contractible summand. The same assertion holds for the target simple generators. Formula (C.33.5), with the finite semisimple decomposition of its source, has nonnegative even finite-dimensional cohomology as well. Give an input tree the height \(n-2\) minus its number of proper intervals, as in C.31.2, and give the target level height \(n-1\). Every proper contraction strictly increases height; every cross map strictly increases it. Maps within a fixed tree or the target stay at its height. Their only degree-zero endomorphism classes are scalar identities of the simple tuple or target labels. Distinct trees of the same height have no contraction maps. It follows that every degree-zero nonidentity mapping class strictly increases height, and all positive mapping classes are nondecreasing. The finite relative coefficient algebra is therefore the product of these scalar identity lines. Its reduced mapping cohomology satisfies exactly C.32.1, including vanishing in degree one. Choose minimal models on all the Hom complexes of this graph. The transfer proof A.3.1.1 applies to the graph and retains its source/target labels. Choose contractions once on the countable original label set and restrict them to each finite full corner. Scalar identities have their fixed representatives, and the homotopies vanish on them. The resulting normalized transfer is strictly relative to those identity lines. Its consecutive-word formulas never introduce a label absent from the original word, so full corners and their bar embeddings are compatible. Enumerate all simple labels, all finite ordered tuples and the finite target summands of their products. At a stage take finitely many explicitly selected input tuples, their selected output labels including all summands of those finitely many products, and all trees of the finitely many selected arities. Take full corners on these selected objects. Do not impose closure under arbitrary tensor products of the resulting label set: that would generally make the set infinite. The objects for different arities are kept in separate input components with a shared output category; no map is added between two different input arities. Use the globally compatible height \(-1-\#\{\text{proper intervals of }t\}\) on every input tree and height zero on the shared target. Every contraction and cross arrow strictly increases this height. It has a finite range on each finite stage. Full corner inclusions and the already chosen contractions give the stipulated compatible bar embeddings. Every finite input tuple, every chosen target label and its original convolution graph occurs at a stage. This proves the countable compatible presentation. The theorem presents the actual ordered functors and their fixed-arity associativity diagrams. Maps identifying different arities under further tensor substitution are additional marked maps of these presentations. Likewise (C.33.4) must be tied to tensor powers of the same output comparison in any eventual monoidal comparison. Neither condition is erased by forming the graph category. ∎ #### C.33.4. What whole-graph descent actually supplies **Theorem C.33.4.1.** The countable rational convolution graph system of C.33.3 admits a compatible relative formality comparison over \(\mathbf Q\), with identity leading component, preserving its full source/target corners and its fixed-arity tree graph actions. In this comparison its input categories are compared by the input corner of the graph equivalence. This assertion alone does not identify that corner with the tensor powers of one common output functor. **Proof.** Extend coefficients to the finite \(E/\mathbf Q_\ell\) of C.29. By C.33.1 this gives the actual complex \(E\)-convolution graph diagram. C.29's ordered enhanced symmetric equivalence compares that diagram with the diagram of free \(\operatorname{Sym}_E(\mathfrak h[-2])\)-modules. On the free generator category its differential is zero, its ordered tensor is the genuine tensor over that algebra, and its tree identifications are its usual associativity maps. Its input tensor-category complexes and its represented cross modules therefore have zero differential, with their original graded compositions and graph actions. This is the ordinary graded graph identified by C.33.2 and C.33.3. The enhanced ordered comparison induces an equivalence of the whole graph system, not merely separate equivalences of its corner categories. Pass to the fixed rational minimal graph model and extend its chosen contractions to \(E\). The equivalence just obtained gives a relative \(A_\infty\) equivalence to its graded graph. Its leading component is an invertible graded graph functor, so composing with that component's graded inverse makes the leading component the identity. The inverse retains the graph's object and height labels. Normalization at scalar identities gives a strictly relative equivalence. C.33.3 verified every directed mapping and compatible-corner hypothesis of C.32.3. The positive-weight windows of that theorem are genuine finite subcoalgebras, preserved by coordinate changes and inverses; the coderivation raises weight by at most one. Thus their whole-graph intertwining equations are the finite torsor equations (C.32.4), with the extension solution just constructed. The compatible-point theorem descends that entire countable graph system over \(\mathbf Q\). Its restrictions retain every full source and target corner. Since its cross operations are the enhanced graph actions, those are retained as well. The graph recovery in C.33.2 identifies the resulting represented functor comparison, including its fixed-arity tree diagram. Let \(g_1\) be the output-corner equivalence and \(g_n\) the input-corner equivalence on (C.33.4). Whole-graph descent has produced a comparison of the actual convolution graph along \(g_n\) and \(g_1\). A strong ordered monoidal equivalence requires, in addition, the coherent identifications \[ g_n\simeq g_1^{\otimes n} \tag{C.33.6} \] and the substitution identifications between the arities, together with their units and permutations. These are marked external-product and graph-substitution equations. An arbitrary formality coordinate change on the graph can have extra input higher components even when its first component is the identity, so (C.33.6) does not follow from the first-component condition. The proof above has not imposed these additional equations. They remain the exact condition needed before this graph comparison can be used to finish the full arbitrary-field tensor theorem. ∎ ![Directed degree-zero arrows have bounded length and positive-weight windows preserve the actual coalgebra maps; the binary convolution graph retains the input polynomial algebra, output algebra and its representable cross action; a separate tensor-power condition is required](assets/directed-convolution-graphs.png) Lemmas C.32.1.1-C.32.2.1 prove genuine finite windows and the coderivation output bound. Theorem C.32.3.1 proves compatible directed relative descent. Theorems C.33.1.1-C.33.4.1 compare rational coefficients, construct the actual graph and ordered-tree diagram, and descend that whole graph. Exercises C.33.5.1-C.33.5.2 compute the actual torus graph and explain why graph descent alone does not impose the tensor-power condition. Editable SVG source. #### C.33.5. Two actual graph calculations **Exercise C.33.5.1.** For the unit weight of \(G=\mathbf G_m\), put \(A=k[u]\), \(|u|=2\), and let the binary input algebra be \(B=k[u_1,u_2]\). Compute the binary convolution graph algebra, its degree-zero idempotents, its graded centre and its degree-two dimension. **Solution.** The unit's actual mapping algebra is \(A\), and ordered convolution is tensor over \(A\), by the torus proof in Section 5. On the input tuple the tensor mapping algebra is \(B\). Its action on the convolved unit is \(f:B\to A\), \(f(u_1)=f(u_2)=u\). The graph cross module is \(M=A\), with its regular output action and that input action. The graph algebra is the lower triangular algebra with underlying triples \((a,b,m)\in A\oplus B\oplus A\) and multiplication \[ (a,b,m)(c,d,n)=(ac,bd,an+m f(d)). \tag{C.33.7} \] There are no maps in the reverse cross direction. Its unit is \((1,1,0)\). The output and input idempotents are \(e_o=(1,0,0)\) and \(e_i=(0,1,0)\). The degree-zero cross arrow is \(z=(0,0,1)\); it satisfies \[ e_o z=z=z e_i,\qquad e_i z=z e_o=0,\qquad z^2=0. \tag{C.33.8} \] It strictly increases the graph height, so its suspended degree \(-1\) is permitted by C.32's directed bound. All homogeneous degrees here are even. A central element must commute with \(e_i,e_o\), forcing \(m=0\). Commuting with \(z\) then forces \(a=f(b)\). Conversely those conditions commute with every triple, since \(A,B\) are commutative and the two cross actions agree. Thus the centre is \[ Z(\operatorname{Gr}(f)) =\{(f(b),b,0):b\in B\}\simeq B. \tag{C.33.9} \] The class \(u_1-u_2\) is a nonzero central graph class acting by zero on the cross module. This is an actual input direction, not an additional output parameter. Degree two has dimensions \(1\) in the output corner, \(2\) in the input corner and \(1\) in the cross module, hence total dimension four; degree zero has dimension three. **Exercise C.33.5.2.** In the preceding graph let \(\sigma:B\to B\) interchange \(u_1,u_2\). Does the graph automorphism \((a,b,m)\mapsto(a,\sigma(b),m)\) come from tensor powers of its output-corner map? What does this show about using a graph comparison as a monoidal comparison? **Solution.** Since \(f\sigma=f\), substitution into (C.33.7) verifies that this is a graph algebra automorphism. It fixes the output corner, both idempotents, the cross arrow and every degree-zero graph map. Its input corner interchanges two independent degree-two classes. The tensor square of its output-corner identity fixes both \(u_1,u_2\), so is different. Consequently a whole-graph comparison fixing the output and the degree-zero heart need not be the tensor comparison determined by that output. This example's input linear component on positive cohomology is nonidentity. C.33.4 imposes identity leading components and therefore excludes this particular automorphism, but that normalization still does not impose the higher tensor-power equations (C.33.6). Those are enhanced marked equations, not just a statement about the degree-zero heart. To conclude a monoidal comparison one must impose them and their cross-arity compatibilities on the actual bar coordinates. The graph retains the data where they can be checked; it does not supply an automatic inference of them. ### C.34. Strict positive models and tensor bar shuffle maps We now construct the bar maps which express the common-input condition of (C.33.6). They require a strict positive model before taking tensor products; a shuffle formula on separately chosen \(A_\infty\) tensor models is not presumed to be strict. #### C.34.1. A strict positive replacement of the rational IC category **Theorem C.34.1.1.** Let \(\mathcal M\) be the strictly unital minimal rational IC category obtained in C.33.1 and A.3.1.1. Its reduced Hom spaces have degree at least two, and its degree-zero coefficient category \(S\) consists of the scalar identity lines. It has a strictly augmented dg replacement \(\mathcal U=\Omega_S B_S\mathcal M\), with no reduced mapping terms in degrees below two. There is an \(A_\infty\) quasi-equivalence \(\mathcal M\to\mathcal U\) retaining its objects, units and all its higher compositions. The construction is defined over \(\mathbf Q\), commutes with field extension and with full finite-label restrictions before applying the cobar functor. **Proof.** Let \(B_S\mathcal M=T_S^c(s\overline{\mathcal M})\) have its reduced bar coderivation \(b\). A cohomology operation on \(r\) reduced letters has ordinary output degree at least \(2r+2-r=r+2>0\), so cannot have an output in \(S\). Thus it defines a genuinely augmented bar coalgebra. Let \(\overline B\) omit its coaugmentation. Define \[ \mathcal U=T_S(s^{-1}\overline B). \tag{C.34.1} \] An element is a string of blocks of suspended \(\mathcal M\)-letters; each block is desuspended once. Its multiplication concatenates blocks. Its differential has two parts: the desuspended bar coderivation applied inside a block, and splitting a block at each nonempty deconcatenation. Signs are fixed by moving the degree-one differential past the preceding desuspended factors. Extend it as a derivation. Its square is zero. Two internal coderivations cancel by \(b^2=0\); two successive cuts of the same block appear in opposite orders with opposite signs, by coassociativity and the desuspension. An internal operation and a cut cancel by the coderivation identity \(\Delta b=(b\otimes1+1\otimes b)\Delta\). On different blocks they cancel by the graded derivation rule. These exhaust the terms, proving the dg assertion. Projection onto \(S\) kills all blocks and is a strict augmentation. If the underlying ordinary letter degrees are \(a_1,\ldots,a_r\) and there are \(v\) blocks, the element's degree is \[ a_1+\cdots+a_r-r+v\ge r+v. \tag{C.34.2} \] A nonempty term has \(r,v\ge1\), so its reduced degree is at least two. The universal degree-one map from \(B_S\mathcal M\) to \(\mathcal U\) sends a whole bar word to its one desuspended block. Its differential and its products along cuts are exactly the two terms just defined. Equivalently it satisfies the twisting identity. The consecutive-partition formula therefore defines a coalgebra map \[ B_S\mathcal M\longrightarrow B_S\mathcal U: \quad [a_1|\cdots|a_r]\longmapsto \sum_{\text{consecutive partitions}} [\,\text{block}_1|\cdots|\text{block}_v\,], \tag{C.34.3} \] with the induced suspension signs. Cutting a partition verifies the coalgebra identity. The twisting identity verifies its differential identity. It is the stated \(A_\infty\) functor; its first component sends a letter to its one-letter one-block term. We prove that first component is a quasi-isomorphism. Filter \(\mathcal U\) by total underlying letter length. The splitting differential preserves that length; any nonzero reduced \(m_r\), \(r\ge2\), lowers it. For fixed underlying letters of length \(r\), the splitting complex has one basis term for each subset of the \(r-1\) possible cuts. After rescaling the basis signs, it is the exterior complex on these cuts with differential wedge by their sum. If \(r\ge2\), contraction by the functional which is one on the first cut and zero on the others satisfies \[ \iota\,(\theta\wedge-)+(\theta\wedge-)\,\iota=1, \qquad \theta=\sum_{j=1}^{r-1}\text{cut}_j. \tag{C.34.4} \] The identity follows by the exterior contraction rule; it also follows by cancelling all insertions except that first cut. Hence these pieces are acyclic. Length one has no cuts and is exactly the original \(\mathcal M\)-letter, and length zero is \(S\). The filtration calculation converges in every degree: (C.34.2) bounds underlying length by that degree, and by one less for a nonempty block string. The same bound applies to the degree before and after its differential. One may therefore eliminate the acyclic higher-length pieces in a finite filtered complex in each required degree window. Only the one-letter cohomology and the units survive. The first component in (C.34.3) identifies them with the original \(\mathcal M\)-mapping cohomology. This proves the quasi-equivalence. All formulas use finite consecutive partitions, rational coefficients and tensor products over the specified identity algebra. They commute with field extension. A fixed finite object word uses only its listed endpoints and intermediate labels; restricting the minimal category to a full finite label set restricts the same bar and cobar formulas. No compatibility of unrelated choices of strict tensor models is being assumed. The dg replacement retains the actual higher structure because its internal coderivation is the entire transferred \(b\), not just \(b_2\). ∎ #### C.34.2. Shuffle maps for the actual strict tensor input **Theorem C.34.2.1.** Let \(\mathcal U_1,\ldots,\mathcal U_n\) be strictly augmented dg categories over a field, with reduced mapping terms of degree at least two and scalar degree-zero identity algebras. There is a natural dg coalgebra map \[ \mathrm{sh}_n: B\mathcal U_1\otimes\cdots\otimes B\mathcal U_n \longrightarrow B(\mathcal U_1\otimes\cdots\otimes\mathcal U_n). \tag{C.34.5} \] It interleaves the factor bar words, preserving order within each word, with the Koszul signs of their suspended letters; a factor letter changes only that coordinate of the object tuple. It preserves summed positive suspended weight. Its maps are associative under regrouping factors and equivariant for their graded permutations, with the empty words supplying the units. **Proof.** Embed each factor arrow in the strict tensor category by inserting the identity arrows of the other object coordinates. For words in the factors, interleave those embedded letters in every way preserving each factor's order. Assign the sign obtained by exchanging their suspended letters from their original factor-block ordering to that interleaving. This gives the displayed linear map. The object endpoints after a partial interleaving are its coordinate endpoints, so it is well-defined for categories and not only for one-object algebras. Every sum is finite. For its coalgebra identity, cut an interleaved word. In each factor its letters preceding the cut are a prefix and those following it a suffix. Conversely choose a cut of every factor word and an interleaving of its prefixes and of its suffixes; these determine exactly one interleaving with that total cut. The Koszul sign is the product of the prefix and suffix signs and the sign exchanging each later prefix past earlier suffixes. The last is exactly the tensor-coalgebra coproduct sign. Thus the two coproduct formulas agree term by term, including the empty cuts. The internal differential commutes with interleaving by the tensor differential rule. For the multiplication part of the bar differential, first consider adjacent letters from the same factor. Replacing them by their product corresponds to precisely the same replacement in that factor word before interleaving; its bar and Koszul signs agree. Adjacent letters from different factors occur in two interleavings, one in each order. Their products agree in the tensor category with the ordinary tensor interchange sign. Their suspended exchange sign and bar multiplication sign differ by a minus sign after that interchange, so the two resulting terms cancel. These are all adjacent pairs. This proves the differential identity. Every factor letter keeps its ordinary degree after embedding and hence its suspended degree. Interleaving changes neither the list nor the sum of those degrees, proving the weight assertion. To check regrouping, observe that a nested interleaving is uniquely the final interleaving of all letters, together with its order inside the selected factor blocks. The composite Koszul sign is the sign of that final permutation. Thus any parenthesization yields the same sum and sign. Permuting factor blocks first yields the same calculation with its graded block exchange sign; this proves equivariance. An empty factor contributes only its identity coordinate and has no interleaving choices or extra sign, proving the unit assertion. These identities are the actual coalgebra and chain identities, not identities only after cohomology. Applying them to the strict replacement of C.34.1 therefore gives rational maps on its actual tensor input categories. Transferring the complete diagrams through fixed minimal-model maps gives the marked maps which must be retained with the convolution graphs. The theorem constructs those maps and their regrouping identities; it does not yet assert that an arbitrary graph-formality coordinate change commutes with them. ∎ #### C.34.3. The transferred common-input maps and their window equations **Theorem C.34.3.1.** Use the strict positive dg replacement \(\mathcal U\) of C.34.1. Choose normalized mapping contractions for \(\mathcal U\) and its strict tensor powers, and use those same choices in the corresponding source corners of the convolution graph model. Write \(\mathcal M_1,\mathcal M_n\) for their minimal models and \(I_1,P_1,I_n,P_n\) for the bar inclusion and projection maps of A.3.1.1. Then the actual transferred common-input map is \[ \sigma_n=P_n\,\mathrm{sh}_n\,I_1^{\otimes n}: (B\mathcal M_1)^{\otimes n}\longrightarrow B\mathcal M_n. \tag{C.34.6} \] It is a chain coalgebra map, with its actual induced one-letter external-product maps, and it preserves summed positive suspended weight. Thus the strict marked equation \[ g_n\sigma_n=\sigma_n^{\,0}\,g_1^{\otimes n} \tag{C.34.7} \] for a prescribed target common-input map \(\sigma_n^{\,0}\) of the same type is a finite polynomial equation on every positive-weight window. Such equations, when an extension-field solution to all of them exists, can be included in the compatible directed coordinate torsors of C.32.3. This last claim requires a strict solution to the complete marked system; an equivalence of its maps up to an unspecified homotopy is not substituted for that hypothesis. **Proof.** A.3.1.1 supplies the bar maps with \(P_jI_j=1\), their coalgebra identities and their coderivation identities. They are obtained from the mapping contractions by the finite wordwise perturbation recursion. Choose scalar identities as their degree-zero cohomology representatives and make the contractions zero on them. Normalization gives maps relative to the scalar identity algebras. They therefore apply to the reduced bars used here. The entire construction stays in positive reduced degrees. The dg replacement has no degree-one reduced terms by C.34.1; its minimal model has reduced degree at least two by C.33.1. For a reduced word in either bar, every letter is consequently positive and its positive suspended weight equals its cohomological degree. The inclusion and projection are coalgebra maps of degree zero, so every output term has exactly the input's weight. A component into the coaugmentation is zero by counit preservation. This proves weight preservation for \(I_j,P_j\). The strict tensor-power category is positive for the same reason: a reduced tensor arrow has at least one positive factor, all other nonzero factors have degree zero or at least two, so its reduced degree is at least two. C.34.2 gives the chain coalgebra shuffle between these strict tensor powers, with the same weight property. Composing these three actual maps proves every asserted property of \(\sigma_n\). Its leading maps are the cohomology maps of the original embedded factor arrows and their external tensor operations, because \(I_j,P_j\) have the fixed inclusion and cohomology projection as first components. Nothing has replaced those operations by an arbitrary vector-space map. For a finite selected label system, a positive-weight window is finite by C.30.2. For the n-fold domain use summed weight; each factor's word length and every letter degree are bounded by that sum, so there are finitely many terms. Coordinate maps on a source corner of the graph preserve this corner and its weight: they retain all object labels, and pure source words have only those labels. Their restriction is therefore its actual degree-zero reduced coalgebra coordinate map, with identity leading component. The preceding degree argument shows exact weight preservation there. The induced tensor coordinate map \(g_1^{\otimes n}\) has entries that are products of the finite entries of \(g_1\). Both sides of (C.34.7) are consequently finite matrices with polynomial entries on the same window. The map \(\sigma_n\) sends that window into the retained source window of \(g_n\), so its equation domain is invariant under all coordinate maps and inverses. Restricting coordinates to a smaller label-and-weight window retains the corresponding equation. When selecting labels, include the finitely many tuples and factor labels actually occurring in these equations; increasing the finite stages then exhausts all of them. Finally take two solutions \(g,h\) of the whole graph equations and (C.34.7), for all selected arities. Their difference \(a=hg^{-1}\) satisfies \[ a_n\sigma_n^{\,0} =\sigma_n^{\,0}a_1^{\otimes n}. \tag{C.34.8} \] To check it, substitute \(\sigma_n^{\,0}=g_n\sigma_n(g_1^{-1})^{\otimes n}\) on the left, then use the equation for \(h\). The inverse and all substitutions remain in the invariant window just proved. Conversely a target coordinate stabilizer satisfying (C.34.8) sends every solution to a solution. The stabilizer equations are polynomial, are satisfied by the identity, and are stable under composition and inverse by the same calculation. They give reduced closed subgroups of the finite unitriangular groups in C.32.3. Thus the nonempty solution varieties are torsors under those stabilizers, just as in C.30.3. The compatible-point theorem descends this marked system whenever its complete strict extension-field solution is supplied. There is a specific remaining distinction. The contraction identities give \(P_jI_j=1\), whereas \(I_jP_j\) is generally only homotopic to the identity. Transporting a dg tensor functor through these maps therefore gives coherent comparisons of (C.34.6), but does not automatically give a strict equality (C.34.7). Their actual homotopies, cross-arity substitutions and higher relations must be included or strictified with proof in the full marked presentation. The theorem verifies the common-input maps and the finite strict equation windows; it does not silently replace that remaining coherent system by these binary equalities. ∎ ### C.35. Actual transfer homotopies and their contraction cubes The strict marked equation in C.34.3 must be distinguished from its transferred homotopy. We now give that homotopy and the finite cube identities connecting different orders of contraction. They are equations of the actual bar-complex maps. Their coalgebra and multi-input compatibility must still be included in the full tensor presentation. #### C.35.1. The perturbed contraction retains its homotopy **Lemma C.35.1.1.** In the normalized tensor-bar contraction of A.3.1.1, retain its homotopy \(H\), and let \(\delta\) be the original composition coderivation. The actual perturbed homotopy is \[ H_\infty=H(1+\delta H)^{-1} =(1+H\delta)^{-1}H. \tag{C.35.1} \] With the bar maps \(I_\infty,P_\infty\) and the transferred coderivation \(b\) there, it satisfies \[ (d_0+\delta)H_\infty+H_\infty(d_0+\delta) =1-I_\infty P_\infty,\qquad H_\infty^2=H_\infty I_\infty=P_\infty H_\infty=0. \tag{C.35.2} \] On the strict positive input models of C.34, all the sums are finite wordwise, and \(H_\infty\) lowers positive suspended weight by one. **Proof.** Put \(D=d_0+\delta\), \(E=IP\), and \(A=(1+H\delta)^{-1}\), \(B=(1+\delta H)^{-1}\). The original identities are \(d_0H+Hd_0=1-E\), \(PI=1\), \(H^2=HI=PH=0\). The two inverses are finite geometric sums on any word because \(\delta\) lowers word length and \(H\) preserves it. Expanding those sums gives \(AH=HB\), so the displayed definitions agree. The already verified perturbation recursion in A.3.1.1 gives \[ I_\infty=AI,\quad P_\infty=PB,\quad DI_\infty=I_\infty b,\quad P_\infty D=bP_\infty, \quad P_\infty I_\infty=1. \tag{C.35.3} \] For completeness put \(L=1+H\delta\), \(R=1+\delta H\). Multiply the proposed homotopy identity on the left by \(L\) and on the right by \(R\); since \(LH_\infty=H=H_\infty R\), its left side becomes \[ LDH+HDR =d_0H+Hd_0+\delta H+H\delta +H(\delta d_0+d_0\delta)H+2H\delta^2H. \tag{C.35.14} \] The square-zero identity gives \(\delta d_0+d_0\delta=-\delta^2\). The displayed expression is consequently \(1-E+\delta H+H\delta+H\delta^2H=LR-E\). The right side after the same multiplication is also \(LR-E\), because \(LI_\infty P_\infty R=IP=E\). Inverting the finite \(L,R\) proves the exact homotopy identity. Each summand of \(H_\infty\) begins and ends with \(H\), so adjacent copies of \(H\) make \(H_\infty^2=0\). Expanding \(H_\infty I_\infty\) uses either \(HI=0\) or an adjacent \(H^2\); expanding \(P_\infty H_\infty\) uses either \(PH=0\) or \(H^2\). This proves its side conditions. All these are finite identities on each word, compatible with word enlargement. On a strict positive input bar every reduced letter has positive suspended degree, so a word's cohomological degree equals its positive weight. Every term of \(H_\infty\) has degree \(-1\). The normalized homotopy preserves the reduced part and has no component in the coaugmentation; hence its nonzero outputs have exactly weight one less. This proves the bound, with zero output whenever no such reduced degree exists. The identity retains \(I_\infty P_\infty\) as a retract up to its specified homotopy, rather than asserting that it is the identity map. ∎ #### C.35.2. The original homotopy of the transferred shuffle square **Theorem C.35.2.1.** Let \(f:\mathcal U\to\mathcal V\) be a strict dg functor between the positive models of C.34, with their selected normalized contractions and tensor-power contractions. Write \(F=Bf\), \(F_n=B(f^{\otimes n})\), and use tildes for the target contraction maps. Put \[ g_1=\widetilde P_1 F I_1,\qquad g_n=\widetilde P_n F_n I_n, \tag{C.35.4} \] and use the transferred shuffle maps \(\sigma_n,\widetilde\sigma_n\) of (C.34.6). There is a specified degree-\(-1\) bar-complex homotopy \[ dK_n+K_nd =g_n\sigma_n-\widetilde\sigma_n g_1^{\otimes n}. \tag{C.35.5} \] It is given by the actual contraction maps and lowers summed positive weight by one. It is an identity of chain maps and their specified homotopy; no strict equality of the two coalgebra maps is inferred. **Proof.** Write \(H_n,\widetilde H_1\) for the actual perturbed homotopies of C.35.1, and let \(\widetilde E_1=\widetilde I_1\widetilde P_1\). The ordered target tensor homotopy is \[ \widetilde H_{1,n} =\sum_{j=1}^n \widetilde E_1^{\otimes(j-1)} \otimes\widetilde H_1\otimes1^{\otimes(n-j)}. \tag{C.35.6} \] Tensor products of maps carry their ordinary graded evaluation signs. Since \(d\widetilde H_1+\widetilde H_1d=1-\widetilde E_1\), its differential telescopes to \[ d\widetilde H_{1,n}+\widetilde H_{1,n}d =1-\widetilde E_1^{\otimes n}. \tag{C.35.7} \] Indeed the \(j\)-term gives \(\widetilde E_1^{\otimes(j-1)}\otimes(1-\widetilde E_1) \otimes1^{\otimes(n-j)}\), and successive terms cancel. Define \[ K_n=-\widetilde P_n F_nH_n\,\mathrm{sh}_n I_1^{\otimes n} +\widetilde P_n\widetilde{\mathrm{sh}}_n \widetilde H_{1,n}(F I_1)^{\otimes n}. \tag{C.35.8} \] All the maps except the homotopies have degree zero and commute with their differentials. Using (C.35.2), the differential of the first summand is \[ g_n\sigma_n-\widetilde P_nF_n\mathrm{sh}_n I_1^{\otimes n}. \tag{C.35.9} \] Naturality of the actual strict shuffle gives \(F_n\mathrm{sh}_n=\widetilde{\mathrm{sh}}_nF^{\otimes n}\). By (C.35.7), the differential of the second summand is therefore \[ \widetilde P_nF_n\mathrm{sh}_n I_1^{\otimes n} -\widetilde P_n\widetilde{\mathrm{sh}}_n \widetilde E_1^{\otimes n}(F I_1)^{\otimes n}. \tag{C.35.10} \] The last term is precisely \(\widetilde\sigma_n g_1^{\otimes n}\), by their definitions. Adding the two equations proves (C.35.5), including the signs of its two homotopy summands. The inclusion, projection, strict functor bar maps and shuffles preserve positive weight, by C.34.2–C.34.3. Each nonzero homotopy term lowers it by one, by C.35.1 and (C.35.6). Thus (C.35.8) has the asserted invariant finite-window bound. These formulas give the exact chain homotopy of the transferred common-input square. The coalgebra compatibility of a homotopy, its compatibility with graph-module substitution, and the relations among such homotopies remain part of the required full marked diagram. A proof of (C.35.5) alone is not a proof that the strict marked torsor in C.34.3 has an extension-field point. ∎ #### C.35.3. Contraction cubes and all their face identities **Theorem C.35.3.1.** Given finitely many chain contractions \((E_j,h_j)\), with \(E_j\) a degree-zero chain projection and \(dh_j+h_jd=1-E_j\), the tensor products of their \(h_j\)'s give a coherent system of \(q\)-dimensional cube maps of degree \(-q\), with every face the difference between inserting \(1\) and inserting \(E_j\) at that position. These identities include the homotopy between two orders of tensor contraction and all higher faces. For the positive bars of C.35.1 they lower positive weight by the number of homotopy factors. **Proof.** In the Hom complex of maps, the differential of a homogeneous map \(v\) is \(\partial v=dv-(-1)^{|v|}vd\). Thus \(\partial h_j=1-E_j\) and \(\partial E_j=0\). For a cube choose a subset \(J\) of positions carrying \(h_j\), and at every other position choose a fixed endpoint \(v_j=1\) or \(E_j\). Put \(v_j=h_j\) at the positions in \(J\) and define its cell map to be the graded tensor \(\bigotimes_jv_j\). The tensor differential rule gives \[ \partial\!\left(\bigotimes_jv_j\right) =\sum_{j\in J} (-1)^{\,\#\{i\in J:i0\) it is positive. If \(k=0\), it is \(|J'|-|J|\), positive unless the arrow is the identity. Thus there are no nonidentity endomorphisms. Since \(|J|\le V\), \(h\le2L-V\le2L-1\); the lower bound is immediate. Strictly increasing integer height gives the chain bound. To reach the completed corolla one must contract all \(V(t)-1\) internal edges; it is the unique edge set with a one-vertex quotient. Its completed target ideal contains every image of \(J\), so it gives the unique arrow. This proves terminality. For sharpness put an \(n\)-corolla at the bottom of a chain of \(L-1\) unary vertices. Execute its \(L\) vertices from bottom to top, one at a time, and then contract the \(L-1\) edges one at a time in completed states. All these arrows have height increase one, giving \(2L-1\) arrows. At \(n=0\) the bottom vertex is genuinely nullary, so the same chain proves that case too. The normalized nerve contraction obtained by appending the unique terminal arrow is exactly C.44.1's full contraction, retaining parallel arrows and their compositions. Thus the finite projective resolution proof of C.31.2 applies to this category with its new height bound. ∎ #### C.47.2. The actual typed enhanced diagram The frontier of \((t,J)\) is the ordered list obtained by stopping at each executed operation output and otherwise descending to external inputs; unexecuted nullary vertices contribute no input. C.44.3 proves its arity \[ m_J=n+\sum_{v\in J}(1-r(v)),\qquad 0\le m_J\le n+L. \tag{C.47.5} \] Its generator category is the finite additive envelope of the actual rational IC input category \(\mathcal G_{\mathbf Q}^{\otimes m_J}\), and its perfect enhancement when needed. The zero-frontier category is the coefficient category. Executing a nullary vertex selects the actual \(IC_0\) object in the arity-one output; its full endomorphisms are retained. **Theorem C.47.2.1.** For the joint arrow \(f=(K,J'):(t,J)\to(t/K,J')\), put \(I_f=\pi_K^{-1}(J')\). The actual partial forest \(F^t_{I_f,J}\), followed by its canonical frontier relabelling, gives a functor from the source frontier category to the target frontier category. The rational finite-frame tensor, proper and unit maps give its coherent enhanced identity and composition comparisons, including all higher cells. They retain the selected original contraction labels. The complete comparison of C.29 carries this finite native diagram after extension to \(E\). **Proof.** First the frontier at a cluster-saturated ideal \(I\) has a canonical ordered bijection with the frontier at \(\pi_K(I)\) in the quotient. A completed maximal cluster output is represented by the original top vertex of that cluster. All vertices below a completed cluster are also completed, by the ideal condition. A cluster not completed has no vertices in \(I\), so one descends to exactly the same surviving boundary inputs and their same order. External input labels are unchanged by contraction. This constructs the bijection directly, including nullary clusters. Numerically a cluster has boundary arity \(r_{\rm cl}\) satisfying \[ 1-r_{\rm cl}=\sum_{v\in\mathrm{cluster}}(1-r(v)), \tag{C.47.6} \] since its internal edges number one less than its vertices. Summing over completed clusters gives \(m_I=m_{\pi_K(I)}\). Thus the relabelling is between categories with the same number of factors; it deletes no arbitrary factor. The arrow condition gives \(J\subseteq I_f\), so the partial forest of C.44.3 is defined on the source's actual arbitrary frontier factors. Those already completed at \(J\) are its variables. Every newly executed vertex acts by its ordered convolution, by the identity in arity one, or by the actual unit functor in arity zero. The canonical frontier bijection identifies its output with the target's frontier, proving the type assertion. For composable arrows \(f=(K,J')\) and \(g=(K',J'')\), let \(I=\pi_K^{-1}(J')\) and \(P=(\pi_{K'}\pi_K)^{-1}(J'')\). These are nested ideals of the original tree with \(J\subseteq I\subseteq P\); both \(I,P\) are saturated under \(K\). The second arrow's quotient partial forest has a canonical native comparison with \(F^t_{P,I}\). Indeed each \(K\)-cluster is either already a completed input variable at \(I\), or has no executed vertex there. If newly completed at \(P\), its whole original operation tree is executed. Comparing that complete operation tree with its collapsed corolla is exactly C.44.3's native unit/associativity contraction on this fixed input profile. The completed-variable case is the identity. The comparison therefore works on arbitrary frontier inputs, not only on inputs that came from an earlier convolution. Literal substitution of the two original partial forests gives the same typed forest, with the original enhanced comparison \(F^t_{P,I}F^t_{I,J}\simeq F^t_{P,J}\), as proved by the typed forest calculation in C.44.3. The latter is the functor of the composite contraction and final ideal. This proves the composition comparison. Empty contraction and unchanged state have the identity forest, so give the identity comparison. For any longer string of joint arrows, pull all its intermediate ideals back to the first tree. They are nested and each required contraction is saturated at the profiles where it is used. Both ways of grouping its comparisons execute the same partial forests and contract the same selected clusters. The expanded finite chain proof of C.44.3 retains the original proper composition, tensor interchange and identity-unit higher diagrams on these forests. Its actual higher cells thus supply the entire longer-string comparison, with the same ordered edge and state labels. This invokes that proved full native diagram; it does not use an objectwise negative-Hom vanishing assertion or identify parallel arrows. Coefficient extension commutes with those same enhanced finite-frame tensor, proper and unit maps by C.33.1. The full strong comparison of C.29.3 carries their ordered composites and all their higher unit/associativity cells. It therefore carries the entire finite joint native diagram. This is the actual enhanced diagram comparison, not yet a simultaneous point in a chosen common-core polynomial coordinate group. Finally a finite selected generator pool can be closed under the actual forward images at this stage. Process its finitely many objects by increasing height, adding the finite IC summands of each outgoing forest image to its target pool. Every edge goes strictly forward, so this finite process terminates. These functors are perverse exact and send each finite selected IC tuple to a finite additive IC sum by the earlier Satake heart results. Their represented cross mapping cohomology is therefore nonnegative, even and degreewise finite by C.33.1. This supplies the bounded finite-corner data for the native diagram without demanding an infinite tensor-closed pool. ∎ #### C.47.3. The exact compatible grafting domain Take an outer tree \(t_0\) with \(m\) external leaves and cap \(L_0\), and inner trees \(t_i\) of arity \(n_i\) and cap \(L_i\), one at each outer leaf. Keep all their operation vertices and graft seams as in C.44.2. For ideals \(J_0,J_i\), form the disjoint union of their embedded operation sets in the grafted tree. Call the tuple compatible when this union is an ideal. Equivalently, every inner tree attached below an executed outer vertex has been completed in its entirety; other inner ideals can be arbitrary. **Theorem C.47.3.1.** The full subcategory of compatible objects in the product of the joint categories admits the actual grafting functor into \(\mathcal E_{\sum_i n_i,\sum_{i=0}^mL_i}\). It retains every contraction-set label and is noncollapsing on arrows. Its height is the exact sum of the factor heights. Nested grafts associate on their common compatibility domain. If the outer ideals are empty at both ends of an arrow, the native functor is exactly the external product of the inner partial-forest functors. General compatible graft comparisons retain the original enhanced unit, tensor and proper maps. All statements include \(m=0,n_i=0\). **Proof.** The union is an ideal precisely under the stated condition: an executed outer operation has each attached inner operation among its operation descendants, and the original ideals already include its outer descendants. An executed inner operation has no new operation descendant outside its own tree. This proves both directions, with no condition to add at an empty outer leaf set. For a factor arrow embed its chosen original contracted edges in the grafted tree. None is a new seam. The old factor edge sets are disjoint, and their union has quotient exactly the graft of the factor quotients, by C.44.2. Its vertex projection restricts to the original factor vertex projections. Each executed source vertex consequently maps into the embedded executed target union, by the factor arrow condition. The compatible target union is an ideal. This proves the joint-arrow condition. Composition is the same disjoint embedded lifted-edge union in both orders. Identities are retained, so this is a functor on that full subcategory. All operation vertices remain distinct in the object graft. Hence \(V_{\rm graft}=\sum_{i=0}^mV_i\) and \(|J_{\rm graft}|=\sum_{i=0}^m|J_i|\). Using the stated sum of caps, \[ h_{\rm graft} =2\left(\sum_iL_i-\sum_iV_i\right)+\sum_i|J_i| =\sum_i h_i. \tag{C.47.7} \] If a factor contracts an edge, its embedded union is nonempty and strictly reduces grafted vertex count. If no factor contracts an edge but a state increases properly, the disjoint executed union increases properly. Thus no nonidentity factor arrow becomes an identity under grafting. The same disjoint embeddings retain parallel contraction labels. Nested substitution keeps exactly the same operation vertices, leaf order and executed union in either grouping. Its compatibility condition is exactly that this final union be an ideal of that one grafted tree. Thus the common domain and result agree, proving associativity with no suppressed seams. For empty outer ideals, the source and target frontiers are the concatenations of their inner frontiers. In the saturation preimage for a grafted arrow no outer vertex is executed: the factor contraction clusters do not cross a seam. Its partial forest therefore consists exactly of the disjoint inner partial forests, in their original order. This proves the external-product assertion on its actual variable factor categories. An inner nullary functor can create the actual unit in its own ordered position; the empty product is the coefficient category. Neither case discards a unit endomorphism complex. For general compatible ideals the partial forest is literal substitution of these inner forests and the newly executed outer forests. C.44.3's expanded chain maps compare their two groupings by the actual tensor, proper and unit maps. The identical operation vertices and selected contractions retain every higher diagram, just as in C.47.2. This proves the native graft comparison on the stated exact domain. At \(m=0\) there are no inner factors and grafting is the outer identity. Zero inner external arities keep all their nullary vertices and satisfy the same ideal criterion. ∎ #### C.47.4. Solved exercises **Exercise C.47.4.1 (intermediate).** For \(L=3,n=0\), put a nullary unit under two unary vertices. Give a chain of five nonidentity joint arrows and compute every height and frontier arity. Explain why no longer chain exists, and why nullary inputs cannot be omitted from this example. **Solution.** Let the operation vertices from bottom to top be \(u,i_1,i_2\). Start with no executed vertex, execute them in that order, and then contract the two unary edges one at a time, keeping every remaining vertex executed. The vertex counts, executed counts, heights and frontier arities are \[ \begin{array}{c|rrrrrr} \text{stage}&0&1&2&3&4&5\\ V&3&3&3&3&2&1\\ |J|&0&1&2&3&2&1\\ h=2(3-V)+|J|&0&1&2&3&4&5\\ m&0&1&1&1&1&1. \end{array} \tag{C.47.EX1} \] Every arrow raises height by one. Theorem C.47.1.1 bounds it by \(2L-1=5\), proving maximality. The first operation is the actual unit functor from the coefficient category to the arity-one IC category. Its output has the full mapping complex of \(IC_0\). Treating the bottom nullary operation as an external input would change the initial frontier from zero to one and would describe a different diagram. **Exercise C.47.4.2 (advanced).** With cap \(L=3\), take \(t=\mu(u_\ell,u_r)\) and the state in which just \(u_\ell\) is executed. Compare the smallest allowed target states after contracting the left unit edge and after contracting the right unit edge. Compute the height jumps and the actual functors on the old one-factor frontier. Also decide whether an executed binary outer vertex may be grafted to two inner unit trees when only the left one is completed. **Solution.** The old state has \(V=3,|J|=1,h=1,m=1\); its factor is an arbitrary variable \(X\), not a fixed unit input. Contracting the left edge puts the already executed left unit in the root cluster. The quotient root must consequently be executed. Its descendant right unit must then be executed too. Thus the smallest target is the complete two-vertex quotient, with \(V'=2,|J'|=2,h'=4,m'=1\). The saturation \(I\) is all three original vertices. Its native functor is \(F_2(X,IC_0)\), with its original right-unit comparison to \(X\). Here \(k=1,k_{\rm ex}=1,|I|-|J|=2\), so the exact height jump is \(1+0+2=3\). Contracting the right edge instead leaves the executed left unit as the quotient's bottom vertex. The smallest target ideal consists of just that vertex; its root cluster is unexecuted. Thus \(V'=2,|J'|=1,h'=3,m'=1\), with jump two. Its saturation is the original \(J\); its partial forest is the identity on \(X\). Here \(k=1,k_{\rm ex}=0,|I|-|J|=0\), giving \(1+1+0=2\). The two contractions have the same quotient shape but different forced states and functors. For the graft, the executed outer root has both inner unit vertices as operation descendants. Leaving the right one unexecuted violates the ideal condition, so that tuple is outside the compatibility domain of Theorem C.47.3.1. Completing both inner units makes it compatible. Keeping the outer ideal empty allows arbitrary inner ideals and gives the exact ordered external product sector proved there. ![Joint execution and selected-edge contraction have sharp increasing height; the two unit-edge choices force different target ideals and functors; compatible grafting adds the heights and retains exact product sectors](assets/joint-execution-contraction-index.png) Theorem C.47.1.1 proves the joint category, terminal corolla and sharp height bound. Theorem C.47.2.1 gives its actual enhanced partial-forest diagram on arbitrary frontier inputs. Theorem C.47.3.1 proves grafting on its exact compatible ideal domain. Exercises C.47.4.1–C.47.4.2 compute the nullary sharp chain and the two different unit contractions shown here. The native maps and input profiles were proved in C.44.1.1–C.44.3.1. Editable SVG source. ### C.48. Calibrated native cells in fixed state models #### C.48.1. Fix the rational cohomological target Let \(\mathcal G_{\mathbf Q}\) be the actual rational IC generator category, with its finite additive envelope. Define \(H_{\mathbf Q}\) to have those generator labels, mapping spaces \[ H_{\mathbf Q}(P,Q) =\bigoplus_a H^a\operatorname{RHom}_{\mathcal G_{\mathbf Q}}(P,Q) \tag{C.48.1} \] with their original cohomological grading, zero differential and the actual Yoneda composition. Units are the original scalar identities. In positive degree this is a genuinely rational graded space; no basis of an extension-field vector space has been descended. For every selected finite convolution tuple choose its actual finite semisimple IC decomposition over \(\mathbf Q\), including the degree-zero isomorphism and its inverse. Such a decomposition exists by the proved rational Satake heart. Carry those choices into every larger selected corner. The actual convolution maps on mapping cohomology then define graded dg functors \[ F_r^H:H_{\mathbf Q}^{\otimes r}\longrightarrow \operatorname{Add}(H_{\mathbf Q}),\qquad r\ge0. \tag{C.48.2} \] They use the fixed decompositions on objects and conjugate the original mapping-cohomology action by those exact object isomorphisms. At arity zero the object is \(IC_0\), with its full mapping spaces. The zero-input source is the coefficient field. At arity one the chosen model can be the identity. For arbitrary outputs retain finite additive matrices rather than treating an IC sum as a new simple object. **Theorem C.48.1.1.** The rational graded category \(H_{\mathbf Q}\), with these ordered tensor operations, their actual cohomological associators and unit maps, is a graded monoidal dg category. The full enhanced comparison of C.29 has a representative with target \(\operatorname{Perf}(H_{\mathbf Q}\otimes E)\), whose first map on every generator mapping cohomology is the identity. It carries the entire finite native enhanced diagram of C.47, rather than just its edge functors. Its induced minimal core comparison \[ g:B\mathcal M_E\longrightarrow B(H_{\mathbf Q}\otimes E) \tag{C.48.3} \] is an actual chain coalgebra isomorphism with identity primitive component. Thus the induced pure state maps can be taken to be the literal powers \((\Omega g)^{\otimes m}\), including \(m=0\), and the canonical common input squares of C.43 and C.46 strictly commute. **Proof.** Yoneda composition in mapping cohomology is associative and unital because the original enhanced composition is associative with its coherent homotopies: applying cohomology to the given homotopies gives the two required equalities. Tensor Künneth over the field identifies the cohomology of a tensor of generator mapping complexes with the tensor of their cohomologies; the split contraction proof is C.33.1. Therefore the cohomology action of each actual multilinear convolution is the displayed graded dg functor, with its graded tensor interchange signs. The actual native associators and unit comparisons give degree-zero closed components between the chosen output objects. Their enhanced naturality gives strict graded naturality on the mapping cohomology. The pentagon and triangle are equalities on this graded category, because their original specified enhanced homotopies become equalities after passing to cohomology. Conjugating by the fixed decomposition isomorphisms retains these equalities: every intermediate isomorphism is followed by its inverse. This proves the graded monoidal dg assertion. We are constructing this graded target; we have not asserted that the original rational enhanced category is formal, or that its arbitrary natural transformation spaces have no higher homotopy. Let \(\Phi\) be the actual full comparison of C.29.3, restricted first to IC generators and their finite additive sums. Its spectral target generator category \(S_E\) has mappings \(\operatorname{Hom}_{A,H}(A\otimes V_\lambda,A\otimes V_\mu)\), with zero differential, for \(A=\operatorname{Sym}_E(\mathfrak h[-2])\). These are the actual graded mapping spaces identified in C.21–C.22. The induced graded functor \[ L:H_{\mathbf Q}\otimes E\longrightarrow S_E \tag{C.48.4} \] is an isomorphism on mappings and identifies the chosen object labels with their corresponding spectral generators and sums. Its tensor and unit comparisons are the cohomological components of the full \(\Phi\). By the first part of the proof they are graded natural isomorphisms satisfying the strict graded pentagon and triangle. They thus make \(L\) a genuine strong monoidal dg equivalence between these zero-differential generator categories. Relabel \(S_E\)'s selected objects by these generator labels and sums. Define \(J\) on each mapping by the inverse of \(L\)'s mapping isomorphism. It preserves composition and identities exactly because \(L\) does. For its tensor comparison take the unique preimage under the same mapping isomorphism of the inverse tensor comparison of \(L\), with the appropriate relabelled source and target. Applying the faithful \(L\) to each naturality, associativity and unit equation for \(J\) gives the already proved inverse equation for \(L\); it therefore holds before applying \(L\). This constructs the entire strong monoidal dg inverse. Both \(JL\) and \(LJ\) have their specified coherent monoidal identity comparisons. Extending through finite cones and coherent retracts gives their perfect-envelope inverses by the complete enhanced-envelope argument of A.4. Now \(J\Phi\) is a full enhanced strong monoidal comparison into the specified rational-cohomology target extended to \(E\). Its first mapping cohomology map is \(L^{-1}L=1\). It carries every selected native operation, unit, regrouping and all the given higher cells, because it is a composite of the actual full enhanced comparison and a genuine monoidal dg functor. This is how the entire C.47 diagram is retained. No separate choice of the comparison for each edge is made. Although \(J\) is defined over \(E\), the target category and its displayed tensor operations were fixed over \(\mathbf Q\) at the start; we do not claim that \(J\)'s individual coefficients descend. Use the fixed minimal model \(\mathcal M_E\) of the rational generator category and the zero-differential target \(H_{\mathbf Q}\otimes E\). Transport \(J\Phi\)'s whole generator functor through the actual normalized mapping contractions, exactly as in A.3.1.1 and C.38.1. Its bar map is normalized and has primitive component the identity, by the preceding calibration. It is augmented: a Taylor component on \(r\) reduced ordinary inputs, each of degree at least two, has output degree at least \(2r+1-r=r+1>0\). It cannot have a scalar output. Its consecutive-partition bar formula is a chain coalgebra map. At a fixed word its nonleading terms strictly lower primitive length, so the finite triangular inverse exists. The inverse preserves coproduct and commutes with the differential by the original identities. This is the asserted actual isomorphism \(g\). Applying \(\Omega\) gives the strict dg isomorphism \(q=\Omega g\) of positive core replacements. Transport the given full comparison along the canonical bar-cobar equivalences, retaining their units, rather than making an unrelated choice of its power at each state. On a tensor-product state its pure map is exactly \(q^{\otimes m}\). C.43.1 proves that the actual canonical unit and shuffle maps commute strictly with this power and the input \(g^{\otimes m}\); C.46.3 proves the corresponding rectified state equations. The zero-fold map is the coefficient identity. Thus this step calibrates the full enhanced extension-field datum and its canonical pure cores. It does not yet identify that datum with a chain coalgebra point in the complete native marked raw presentation. ∎ #### C.48.2. Simultaneous face-compatible strict cells Put \(U_m=U^{\otimes m}\), \(Q_m=\Omega B U_m\), and retain the canonical counit \(\epsilon_m:Q_m\to U_m\) of C.46.3. For a finite linearly ordered parameter category \([p]\), put \[ R_{n,[p]}=\Omega B(U_n\otimes k[p]). \tag{C.48.5} \] The relative coefficient category keeps the distinct parameter vertices, even when a native comparison maps two of them to the same output object. An increasing injection of parameter vertices induces the actual full-face embedding on these free cobar categories. Parallel native cells are assigned different cell labels, even when their parameter shape is the same. Here is the precise parameter data used. For each fixed source and target frontier arity \(n,m\), take the actual finite native family of functors and enhanced comparisons in C.47 with that input and output type. Record a chosen resolved \(p\)-simplex of this family by its normalized full unital components on \(U_n\otimes k[p]\), with values in \(\operatorname{Add}(U_m)\). A mixed input is a string of actual source arrows with a nondecreasing parameter-vertex string. The component assigned to that mixed input is the corresponding component of the **given** native enhanced cell, transported through the one fixed input and output mapping models as in C.33.2 and C.43.2. It is not an arbitrarily selected map on that string. To spell out why this is a unital functor, expand the native family on each such mixed string. Its boundary terms apply a source differential, compose two adjacent source arrows, compose the adjacent parameter arrows, or apply one target composition to two consecutive output blocks. These are exactly the terms in the normalized unital functor equation C.43.6. Every term keeps the same ordered source labels and parameter vertices. A parameter face simply restricts the allowed vertex labels, so applying the fixed model transport commutes with that restriction. An ordinary source identity is handled by normalization. A nonidentity parameter arrow is retained as a reduced input even when its output is a scalar identity. Thus C.43.2 gives an actual strict dg functor \[ F_\alpha^\sharp:R_{n,[p]}\longrightarrow \operatorname{Add}(U_m) \tag{C.48.6} \] for the entire supplied cell, strictly compatible with its face functors. This recording uses the complete enhanced cells provided by C.47.2 and C.29.3. It does not infer any new higher cell from individual edge data. **Theorem C.48.2.1.** The native \(F_\alpha^\sharp\)'s admit dg lifts \[ \widetilde F_\alpha:R_{n,[p]}\longrightarrow \operatorname{Add}(Q_m),\qquad \epsilon_m\widetilde F_\alpha=F_\alpha^\sharp, \tag{C.48.7} \] with all face restrictions retained exactly. Given compatible previously chosen lifts on a collection of faces, one can retain those same lifts. The construction uses one \(Q_m\) at each state arity, includes zero inputs and scalar outputs, and needs no iterated strictification of its own target. It is compatible along a chosen countable increasing family of finite corners, retaining all earlier assigned generator images. **Proof.** The counit \(\epsilon_m\) is surjective and a quasi-isomorphism on mappings by C.46.3. Its mapping kernels have split contractions, as in C.45.1. The source \(R_{n,[p]}\) is a free cobar. Filter its free generator complex by the number of original bar letters. The length-preserving part is the original unary differential. Binary source composition reduces that length, and a proper coproduct splits it into products of generators at smaller individual lengths. These are precisely the nonminimal hypotheses of C.46.2. The parameter arrows have degree zero before suspension; this does not invalidate the original-length induction. The finite parameter height bounds their number on a composable input. At parameter dimension zero apply C.46.2 to choose the lift. Suppose all proper faces of a cell have already been lifted. Each face is the free subcategory generated by the bar words whose parameter vertices lie in that face. Its generator subcomplex is differential-stable: the unary differential retains the vertices; merging adjacent input arrows can only remove vertices; each proper coproduct piece has vertices from the original string. Thus the union of these face generator subcomplexes is differential-stable too. Their given lifts agree on overlaps by the preceding dimension induction. They therefore prescribe one well-defined lift on the free subcategory generated by their union. There can be products mixing generators from different faces; those products are assigned multiplicatively in this free subcategory, so no extra compatibility is assumed for them. Apply the relative form of C.46.2 on each remaining generator complex. Its defect vanishes on the prescribed subcomplex and is a closed map out of the quotient generator complex into the contracted mapping kernel. The postcomposition kernel homotopy corrects it while staying zero on that subcomplex. This gives the lift of the whole cell with all prescribed faces unchanged. Induction on cell dimension proves the assertion for the entire finite family. On a countable increasing family, choose the same source generator lifts once as they first occur, include their finitely many target images in later target corners, and extend relatively. Every finite input receives its value at a finite stage. At \(n=0\) the source still retains its parameter category; at \(m=0\) the counit is the identity of the coefficient category, so there is nothing to lift. A functor creating a unit has \(m=1\), and its target has the full \(IC_0\) mappings. Source scalar identities and any explicitly prescribed constant-cell generator images are kept in the relative subcomplex. There is no assumption that a nonidentity index arrow maps to a reduced target arrow. In particular the scalar outputs discussed in C.43.2 are retained in this free unital dg lift. This proves all the asserted cases. ∎ #### C.48.3. Retain both endpoint systems in the full prisms Over \(E\), identify the target positive cores using C.48.1's actual strict isomorphism \(q_m=(\Omega g)^{\otimes m}\). Its canonical state isomorphism is \(\widehat q_m=\Omega B(q_m)\). The counit square \(\epsilon'_m\widehat q_m=q_m\epsilon_m\) is strict by C.46.3. The corresponding parameter-input square is strict under \(\Omega B(q_n\otimes1_{k[p]})\), by the canonical bar construction. The calibrated enhanced comparison supplies a full comparison prism for every native cell and its faces, with the chosen \(q_n,q_m\) on the pure input and output models. To record a prism one may use the ordered product \([p]\times[1]\). Its chains are its finite standard simplex subdivision. On each such simplex use the actual enhanced comparison components, with their original face restrictions; this is the mixed-input construction of C.48.2 applied to the supplied full comparison. Its two ends are the original two native cell functors, after the displayed strict core identifications. The comparison prism includes its higher components; it is not formed from unrelated one-cross maps. **Theorem C.48.3.1.** Given the two rational systems of native cell lifts chosen in C.48.2, their extension-field comparison prisms have lifts to the canonical \(Q_m\) state targets which retain both endpoint systems and every common prism face exactly. Each lifted prism projects under \(\epsilon_m\) to its supplied actual comparison. This realizes the calibrated comparison as one face-compatible family in fixed canonical state models. **Proof.** Identify the target model at one prism end with the other by \(\widehat q_m\), and the source by the displayed canonical input map. The strict counit square shows that the prescribed endpoint lifts project to the correct endpoint functors. At dimension zero a prism has its two endpoint subcategories already specified. The mixed-input graph uses the full target mapping space, so its connecting parameter arrow can have the actual unit or other scalar output. Apply the relative lifting proof of C.48.2 to extend the endpoint lifts across that prism. For higher-dimensional native cells, prescribe the two endpoint cells and the already lifted lower-dimensional comparison faces. They agree on intersections: endpoint intersections were fixed by C.48.2, and the remaining intersections are lower-dimensional prism faces already assigned in this induction. In the standard chain subdivision of \([p]\times[1]\), add its top simplices one at a time. Their overlaps and the already assigned boundary are unions of full faces, hence are differential-stable free generator subcomplexes by the explicit argument in C.48.2. The same relative generator-complex lift extends over each new simplex. Its kernel homotopy is zero on the prescribed overlap. This proves compatibility without changing either endpoint system. Induction over parameter dimension and the finite simplex subdivision constructs the whole prism family. For countably many selected cells, choose the same nested finite-stage construction as before. No averaging over independently selected cells is used, and every lift still projects exactly to its supplied entire prism. The canonical input squares then pull this family to the literal common cores \(C^{\otimes n}\) and their actual index factors, with the single core coordinate \(g\). This proves the statement. It does not by itself prove that the family is a point satisfying all tensor-product and grafting equations in a chosen finite coordinate group; those marked operations and their full domains are still to be given. ∎ #### C.48.4. Solved exercises **Exercise C.48.4.1 (intermediate).** Take the graded rational algebra \(H=\mathbf Q[x]/(x^3)\), with \(|x|=2\), and \(E=\mathbf Q(\sqrt2)\). A graded comparison to an \(E\)-algebra model acts by \(x\mapsto\sqrt2x\). Calibrate it against the original rational target, check the tensor multiplication map, and identify exactly which data have been defined over \(\mathbf Q\). **Solution.** On the ordered basis \(1,x,x^2\), its matrix and inverse are \[ L=\operatorname{diag}(1,\sqrt2,2),\qquad J=\operatorname{diag}(1,1/\sqrt2,1/2). \tag{C.48.EX1} \] If \(\mu:H\otimes H\to H\) is multiplication, then \(L\mu=\mu(L\otimes L)\): on \(x^a\otimes x^b\) with \(a+b<3\), both sides give \((\sqrt2)^{a+b}x^{a+b}\), and for \(a+b\ge3\) both are zero. Thus \(J\mu(L\otimes L)=\mu\) and \(JL=1\). The category of rational graded mappings, its product and unit were fixed over \(\mathbf Q\) before this calibration. The matrices \(J,L\) here have coefficients in \(E\), and their individual entries have not been descended to \(\mathbf Q\). The calibrated first cohomology map is exactly the identity on the already rational target. This is the distinction used in Theorem C.48.1.1. **Exercise C.48.4.2 (advanced).** Let a source generator complex have elements \(w_0,u\) in degree two and \(w_1,v\) in degree three, with \(dw_0=w_1\), \(du=w_1+v\), and all other differentials zero. The subcomplex spanned by \(w_0,w_1\) is an already fixed face. The target surjection has these quotient generators and a square-zero kernel \(x,y\) in degrees two and three, with \(dx=y\). Let a provisional lift agree with the quotient generator names except that \(u\) is sent to \(u+x\). Correct it while keeping the fixed face, and explain how this applies to two adjacent simplices in a comparison prism. **Solution.** The provisional lift has defect zero on \(w_0,w_1,v\), while on \(u\) it is \[ o(u)=d(u+x)-(w_1+v)=y. \tag{C.48.EX2} \] This defect vanishes on the face, belongs to the kernel and is closed in the Hom complex with the whole source differential. With \(h(y)=x\), subtracting \(h o\) sends \(u\) back to \(u\) and retains the other images. The corrected lift is a chain map and has the same quotient projection. It leaves the whole face unchanged even though the incoming unary differential \(du\) meets that face through \(w_1\). In \([1]\times[1]\), the two standard maximal simplices are \[ (0,0)<(1,0)<(1,1),\qquad (0,0)<(0,1)<(1,1). \tag{C.48.EX3} \] Their intersection is the full diagonal face \((0,0)<(1,1)\). On either simplex the bar generators belonging to that face form a differential-stable subcomplex, since differentials and adjacent merges cannot add a new parameter vertex. Lift the first simplex with its prescribed boundary. Lift the second relative to its boundary and that same diagonal face. The relative correction is zero on this overlap, just as in the generator calculation, so the two functors agree there exactly. Arbitrary independent lifts on both triangles would not establish that agreement. Neither construction forces additional tensor/grafting equations unless those actual marked operations are also supplied. ![The actual rational graded target is fixed before its extension-field comparison is calibrated; native cell defects are corrected relative to prescribed faces; comparison prism triangles retain the same diagonal](assets/calibrated-native-cell-lifts.png) Theorem C.48.1.1 proves the calibrated identity leading map and canonical pure state powers. Theorem C.48.2.1 lifts the full supplied native cells, keeping their faces. Theorem C.48.3.1 lifts their entire comparison prisms while retaining both endpoint systems. Exercises C.48.4.1–C.48.4.2 compute the nonrational calibration and the unary correction on a shared face. The input and kernel constructions were proved in C.43.2.1, C.45.2.1 and C.46.2.1–C.46.3.1. Editable SVG source. ### C.49. Native indexed prisms and invariant marked domains #### C.49.1. Nonnegative mapping-degree windows Take a finite selected directed path system. Its unindexed pure cores are literal tensor powers of \(C=B\mathcal M\), whose reduced letters have strictly positive suspended degree. Parameter cores are relative bars \(D_I=B(kI)\) of finite directed index categories. A nonidentity parameter letter has suspended degree minus one. Only finitely many strictly directed index steps can occur on a path in this selected system. Distinct parallel index arrows retain their separate labels. Each cross central complex is nonnegative and degreewise finite at the retained finite label corner; in the minimal model it is actual represented mapping cohomology, including its degree-zero maps. The number of cross letters on a path is bounded by its finite directed level range. For a homogeneous labelled path term \(w\), let \(p(w)\) be its number of parameter-bar letters and \(r(w)\) its number of cross letters. Define \[ P(w)=\sum_{\text{positive core factors }c}|c| +\sum_{\text{central factors }m}|m| =|w|+p(w)+r(w). \tag{C.49.1} \] Here a cross letter is \(sm\) of degree \(|m|-1\). The positive core factor degree is already its bar degree. Thus \(P\) is nonnegative, even though the total cochain degree \(|w|\) need not be. Parameter heights serve only to bound the number \(p\); they are not included in this filtration and need not be renumbered under a face inclusion. **Theorem C.49.1.1.** On a selected finite system, the subspace \(P\le D\) is finite-dimensional and preserved by every degree-zero common-core path coalgebra coordinate which retains the endpoint and string labels, is over \(g^{\otimes n}\otimes1_{D_I}\) on each indexed pure core, and has identity bare central leading maps. Such coordinates and their inverses never increase \(P\). Their full differential intertwining equations on \(P\le D-1\) use outputs only through \(P\le D\). Full parameter faces preserve \(P\) exactly, including faces that skip an intermediate vertex. **Proof.** The number of positive core letters is at most \(D\), and each central degree is at most \(D\). Their fixed-degree spaces are finite on the finite retained label pools. The cross count has a finite directed bound \(R\). Each of the at most \(R+1\) surrounding core runs has a finite index-height bound, so the total parameter count is bounded, for example by \((R+1)H\) when every selected index height is at most \(H\). There are finitely many index strings and coefficient labels at this stage. Hence only finitely many basis terms satisfy \(P\le D\). Use the full consecutive-cut reconstruction of C.40.1 for a path coordinate. A term in its output partitions the input cross string into consecutive nonempty blocks. Each block produces one cross central output. Consequently the output cross count \(r'\) is at most the input count \(r\). Outside and intermediate indexed core runs are obtained by cutting the input runs and applying their prescribed pure-core maps. Those maps apply the common \(g\) to the positive factors and the identity to each \(D_I\). They create no parameter letters. A central Taylor operation consumes its entire selected block of surrounding input runs and emits a bare central letter, hence emits no new parameter run. Thus \(p'\le p\) too. These assertions include a cut immediately beside an empty run: its coaugmentation has no parameter letters. Every such output term has total degree \(|w|\). Therefore \[ P(w')=|w|+p'+r'\le |w|+p+r=P(w). \tag{C.49.2} \] No sign affects this degree equation. A pure core coordinate likewise retains its positive bar degree and exactly retains its parameter letters. This proves preservation of the whole window, with its complete outside coactions. The finite triangular inverse retains the same prescribed pure cores and bare central leading maps, by C.40.3 and C.42.3. Its central nonleading terms also partition crosses and consume surrounding runs, so the same two count inequalities prove it never increases \(P\). This does not assume invertibility of an arbitrary chain projection from transfer. The actual path differential has its full core differentials, one-cross module terms and operations merging several crosses. A core \(C\) differential has degree one and no parameter or cross count change. A pure index bar differential composes two adjacent index arrows, reducing \(p\) by one. A module or several-cross Taylor operation consumes its selected parameter inputs, outputs one bare cross, and retains all outside runs. It therefore never increases \(p\) or \(r\). Every output term has total degree \(|w|+1\), giving \[ P(dw\text{ term})\le P(w)+1. \tag{C.49.3} \] Thus applying a coordinate before or after a differential to an input in \(P\le D-1\) uses only the retained top window \(P\le D\). It is unnecessary to impose a truncated top-row differential equation whose output has been discarded. A full face embedding includes a subset of parameter vertices and sends each nonidentity source arrow to that same nonidentity ambient arrow. It retains its number of bar letters, even when a face skips a vertex: a direct arrow over the skipped vertex is still one indexed bar letter. It leaves the positive factors and central mapping degree unchanged, so preserves \(P\) exactly. Fixed cell representatives restrict to those faces by C.48.2, and their assigned cohomology contractions can be chosen once on their identical central mapping blocks. The module perturbation sums in C.45.2 then restrict to those same faces, since every Taylor term retains its index labels. Thus these are the actual face restrictions on the finite windows. ∎ #### C.49.2. Exact tensor and grafting equation domains **Theorem C.49.2.1.** In the shifted product module of C.41.2, \(P\) is the sum of the factor \(P\)'s. The actual core shuffle, noncollapsing native index graft, face restrictions and coefficient coaugmentation insertions do not increase that sum. A central path operation merging \(r\ge2\) crosses and consuming \(q\) parameter letters has output \(P\) equal to input \(P+2-r-q\), hence never larger than the input. This supplies finite operation domains for these actual maps on the compatible native grafting domains of C.47.3. **Proof.** Undo each cross suspension by \(K(Z)=Z[-1]\), take the ordinary tensor of the modules, and restore one total cross suspension, as in C.41.4–C.41.6. The unsuspended central output is the tensor of the unsuspended central factors, with degree their sum. Positive surrounding core degrees add too. Parameter shuffle interleavings retain the total number of parameter letters. It follows directly from the definition that \(P_{\rm product}=\sum P_i\), with no correction depending on the number of tensor factors. The positive core shuffle preserves total cochain degree, and the index shuffle preserves the number of its suspended index letters. The graft functors of C.47.3 are noncollapsing on their actual compatible full subcategory, so their bar maps retain every nonidentity index letter. Neither map creates a new central degree. Their outputs thus lie in the sum window. A coaugmentation insertion has degree zero and creates no bar letter or cross mapping degree, so it contributes zero to \(P\). Faces were checked in C.49.1. For a merging path Taylor operation, the total bar degree increases by one, the cross count changes from \(r\) to one, and the selected \(q\) parameter inputs disappear into its bare central output. Its outside core runs retain their counts. Therefore \[ P_{\rm out}-P_{\rm in} =1+(1-r)-q=2-r-q\le0. \tag{C.49.4} \] This applies to the full higher operation, not only to binary represented evaluation. It does not erase the actual Taylor coefficients: they must still be supplied by the native cell/path construction. It proves their finite domain bound when those coefficients are present. For a product mark impose its equation on every tuple of factor windows \(P_i\le D_i\) with \(\sum_iD_i\le D\), retaining the full tensor domain and the target sum window. Bilinearity and decomposition into homogeneous \(P\)-pieces show that the union of these domain pieces spans every tensor input of sum at most \(D\). For a graft mark, restrict this domain to C.47.3's compatible index tuples before applying the actual noncollapsing bar graft. A differential comparison uses the same domain with sum at most \(D-1\), since its output can raise \(P\) by one. These are complete domains of the stated maps; no incompatible outer/inner unit state is inserted and no output above the retained top window is silently deleted. ∎ #### C.49.3. The actual finite marked coordinate groups **Theorem C.49.3.1.** For a finite selected face/path/product system whose full actual operation maps have been provided, impose common-core coalgebra and bicomodule equations, fixed leading entries, all face restrictions and covariance of its provided product/graft maps on the domains of C.49.1–C.49.2. These equations define a reduced closed subgroup of the finite unipotent coordinate group. Composition, inverse and smaller full-window restrictions are the actual coordinate operations. Given two sets of full differentials and actual marked maps on those frames, their nonempty simultaneous intertwining scheme is a torsor under the target stabilizer. Nonemptiness is an additional assertion to be proved using the whole calibrated native comparison, not an assumption supplied by this proposition. **Proof.** Order each finite cofree-path basis by total primitive length, retaining the endpoint, string, parameter and mapping labels. C.40.3's iterated-cut injection proves that a coordinate with identity primitive and bare cross leading maps is unitriangular. C.49.1 proves its action and its finite inverse stay inside \(P\le D\). On every pure state the prescribed coordinate is the true tensor power \(g^{\otimes n}\), and on a parameter factor it is the identity. These restrictions are preserved by composition and inverse, since \(h^{\otimes n}g^{\otimes n}=(hg)^{\otimes n}\). The coalgebra equations, prescribed leading entries and face restrictions are finite polynomial matrix equations. Each provided degree-zero operation map has its complete finite domain from C.49.2, so its covariance equation is a finite polynomial as well. Applying first one coordinate and then another preserves that domain and their tensor sum window by C.49.1. Substitution therefore proves covariance for their product. Substitution by their inverses, which preserve the same domains, proves it for the inverse. The face equations have the same substitution argument. The equations thus cut out a closed subgroup of the finite unitriangular matrix group; taking its reduced structure over characteristic zero is the convention used in C.30.3. This proves the group assertion without defining a product by maps that would insert an extra transferred \(IP\). For differential equations use only inputs through \(D-1\), with outputs through \(D\), as proved in C.49.1. A quotient of two simultaneous solutions retains all coordinate restrictions and commutes with the target differentials and target marks on exactly those full domains, by substitution with their finite inverses. Conversely the target stabilizer sends a solution to a solution on those same domains. With a chosen solution these two constructions are inverse morphisms of the reduced affine schemes; their formulas are matrix product and finite unitriangular inverse. This is the torsor assertion. Smaller windows are preserved by C.49.1; smaller full endpoint/string corners use only the original cuts and intermediate labels. A smaller full face family uses the identical parameter letters and central mapping blocks, as checked there. The operations themselves restrict with their whole domains by C.49.2. Hence composition, inverse and the defining equations all restrict to the actual smaller system. The conclusion remains conditional on the supplied entire native operation presentation and its simultaneous point; no family of independent edge comparisons is substituted for either of them. ∎ #### C.49.4. Actual face-compatible comparison coordinates Write \(C=B\mathcal M\), \(A_m=C^{\otimes m}\), \(U_m=(\Omega C)^{\otimes m}\), \(Q_m=\Omega B U_m\). The actual native cell representatives of C.48 are \[ \widetilde F_\alpha:R_{n,I}\longrightarrow \operatorname{Add}(Q_m),\qquad R_{n,I}=\Omega B(U_n\otimes kI), \tag{C.49.5} \] with the finite directed parameter category \(I\) retained. For a simplex one may take \(I=[p]\); the full prism uses its standard subdivided \([p]\times[1]\) parameter. Its cell and face labels are never erased. The canonical target and input coalgebra maps are \[ \ell_m=\eta_{B U_m}\psi_m:A_m\longrightarrow BQ_m, \qquad a_{n,I}=\eta_{B(U_n\otimes kI)}\psi_{n,I}: A_n\otimes D_I\longrightarrow BR_{n,I}. \tag{C.49.6} \] These are exactly the maps of C.43.1 and C.46.3; both strictly commute with the indicated tensor powers of one core map. For a represented cell graph put \[ M_\alpha(y,x,i)= \operatorname{Hom}_{Q_m}(\widetilde F_{\alpha,i}x,y), \qquad Z_\alpha=A_m\otimes sM_\alpha\otimes(A_n\otimes D_I), \tag{C.49.7} \] with its full pulled-back graph differential, not merely its central cohomology. The full graph construction is C.43.2, with target \(Q_m\). The target is augmented and positive, as required there. The maps \(a,\ell\) retain the genuine scalar outputs of the cell functor in its right module actions. A parameter face is a genuine embedding of these cofree modules. For a fixed target object and source parameter vertex, the central complex \(M_\alpha\) depends only on the actual vertex functor and its fixed object image. It is the identical central complex wherever that vertex appears as a face of a larger cell. Its mappings are nonnegative and degreewise finite at a finite corner, since \(Q_m\) is a positive core and its images are finite perverse IC sums. Its cohomology is the original nonnegative even represented IC mapping cohomology, by C.33.1 and graph recovery. Choose its coefficient-linear central contraction once on each such actual block, retaining it at every occurrence. The full supplied inverse prism gives the following canonical comparison. Let the calibrated core map over \(E\) be \(g:C_E\to C'_E\), as in C.48.1. Its target state map is \(\widehat q_m=\Omega B((\Omega g)^{\otimes m})\); its parameter input map is \(\Omega B((\Omega g)^{\otimes n}\otimes1_{kI})\). The full comparison prism of C.48.3 gives the enhanced comparison of the native functor families along precisely these maps. Use its supplied coherent inverse, with direction \[ \beta_\alpha: \widetilde F'_\alpha\,\Omega B((\Omega g)^{\otimes n}\otimes1) \ \Longrightarrow\ \widehat q_m\,\widetilde F_\alpha. \tag{C.49.8} \] The inverse exists as part of C.29's actual enhanced equivalence and its coherent inverse, transported and lifted as in C.48. It is not obtained by formally reversing the interval coalgebra of C.36. **Theorem C.49.4.1.** There are actual chain bicomodule comparison maps \[ T_\alpha^{\rm raw}:Z_{\alpha,E}\longrightarrow Z'_{\alpha,E} \tag{C.49.9} \] over \(g^{\otimes m}\) on the left and \(g^{\otimes n}\otimes1_{D_I}\) on the right, simultaneously compatible with all parameter faces. Transfer the full modules using the fixed central contractions and C.45.2's consumed-letter perturbation. Their minimal-central comparisons \(T_\alpha=P'_\alpha T_\alpha^{\rm raw}I_\alpha\) are actual bicomodule isomorphisms with identity bare central component. Together with the single \(g\), they give an actual extension-field point of every finite face/module/differential coordinate system of this family. **Proof.** On a bare represented mapping \(m:\widetilde F_{\alpha,i}x\to y\), the map is the actual composition \[ m\longmapsto \widehat q_m(m)\,\beta_{\alpha,x,i}. \tag{C.49.10} \] This has the required type \(\widetilde F'_{\alpha,i}(q_nx)\to\widehat q_m y\). For the higher input Taylor maps use the complete source-bar components of the given natural transformation \(\beta_\alpha\), followed by that same represented target composition. More explicitly, in the ordinary unsuspended cofree module the central Taylor map on a source bar word is obtained by applying that component of \(\beta\) and composing it after the transformed represented arrow. Expand any functor component on the source word into its consecutive nonempty blocks. Apply the target composition to those outputs, the transformation component and the represented arrow. These are the consecutive-partition evaluation formulas of C.33.2, with their tensor signs fixed by moving the homogeneous component maps past their preceding suspended inputs. The strict state functor \(\widehat q_m\) handles the target core action. Restore the one cross suspension by the exact identification C.41.6. Reconstruct the entire bicomodule map from this central Taylor map by C.39.5's outside coactions, using \(g^{\otimes m}\) and \(g^{\otimes n}\otimes1\) in the two outside slots. The strict natural squares of \(\ell_m,a_{n,I}\) show that these are exactly its input and output core maps, without choosing a naturality property for a contraction inclusion. Coassociativity gives both coaction equations. The chain equation can be checked at its central Taylor projection. The terms applying a source differential or source composition, or splitting a source word into consecutive functor/transformation blocks, are the normalized natural-transformation equation of the supplied whole prism. Its remaining target-composition terms cancel with the represented left and right module actions. A target unary term is the differential of the displayed composed represented map; its Leibniz signs are exactly the two adjacent unary terms. These exhaust the central terms. Thus that projection is zero. The two coaction equations reconstruct every other term, giving the full chain equation. This is the natural-transformation part of the complete graph proof C.33.2, used on the actual supplied \(\beta\) rather than on separately invented homotopies. The native comparison prisms and their inverse families agree on all faces by C.48.3, and the evaluation formulas use only their original ordered labels and the two canonical core maps. Their restrictions therefore give the identical \(T^{\rm raw}\) on every face. This proves the simultaneous raw assertion. Now extend each previously fixed central contraction to the two cofree cores and perform the full module perturbation of C.41.2 and C.45.2. Its correction terms consume a nonempty surrounding core word, so all sums are finite wordwise even with the finite negative-degree index bars. On a full face the central contraction is the same chosen contraction, and the actual module action is the restriction of the larger cell's action. Every term of the perturbation therefore restricts to that same face term. The full inclusions \(I_\alpha\), projections \(P_\alpha\) and transferred module differentials are consequently face-compatible. Their composite \(P'_\alpha T^{\rm raw}_\alpha I_\alpha\) is a chain bicomodule map with the stated core maps. Its bare central component is the cohomology map of the displayed represented composition. C.48.1 calibrates the full strong comparison against its actual rational cohomological tensor diagram. Hence this cohomology map, including its object/tensor comparison, is the identity in the fixed represented cohomology frames. This uses that calibrated whole diagram, not just an isomorphism of dimensions. Thus the bare central component of \(T_\alpha\) is exactly the identity. Factor out its specified invertible core maps. Every nonleading central component consumes at least one surrounding original bar letter. The difference from the identity consequently lowers surrounding word length; the finite recursive inverse of C.41.2 exists on every word. It preserves both coactions, the chain equation and all face restrictions by the original identities. No transferred projection is declared to be an inverse. This proves the actual isomorphism assertion. Finally use the nonnegative \(P\)-windows in C.49.1–C.49.3. Every coordinate and its inverse preserves these windows. The full chain equation is imposed on inputs through \(D-1\), retaining outputs through \(D\); faces preserve \(P\) exactly. All these equations hold for the actual just-constructed maps before truncation, so their finite restrictions are simultaneous extension-field points. This proves the claimed nonemptiness for the **face/module/differential** system. The additional whole nested-path, tensor and grafting marks have not been imposed in this proposition, and its point must not be substituted for their simultaneous marked point. ∎ #### C.49.5. Keep the parameter tag in the target Let \(I\) be a finite directed parameter category, with no nonidentity endomorphisms and finitely many arrows. Its relative coefficient category has the distinct object identities. A native resolved cell has a normalized unital enhanced functor family with fixed input type \(U_n\), output type \(U_m\), and parameter \(I\), as recorded in C.48. Its full mixed-input components define a normalized unital \(A_\infty\) functor \(F:U_n\otimes kI\to\operatorname{Add}(U_m)\). Retain the parameter vertex in each output object. On a mixed composable word with parameter arrows \(a_1:i_0\to i_1,\ldots,a_r:i_{r-1}\to i_r\), tag its component by their actual composite: \[ F_I(\text{mixed word}) =F(\text{mixed word})\otimes(a_r\cdots a_1). \tag{C.49.11} \] For parallel arrows the composite is that of the actual ordered arrows, not a replacement determined only by endpoints. This formula is extended linearly on each specified labelled source-word block. **Theorem C.49.5.1.** The tagged components give a normalized unital \(A_\infty\) functor \[ F_I:U_n\otimes kI\longrightarrow \operatorname{Add}(U_m\otimes kI) \tag{C.49.12} \] over the identity on parameter objects and arrows. It preserves the relative augmentation. Consequently its genuine bar map and canonical strict representative are \[ \begin{aligned} B F_I &:B(U_n\otimes kI)\longrightarrow B(U_m\otimes kI),\\ \Omega B F_I &:Q_{n,I}\longrightarrow\operatorname{Add}(Q_{m,I}),\\ Q_{r,I}&=\Omega B(U_r\otimes kI). \end{aligned} \tag{C.49.13} \] They retain the entire cell, its faces and all actual scalar outputs. They compose strictly when the underlying tagged functors are composed, and need no independent lift through a target counit. **Proof.** In the full functor equation on a mixed word, every term has the same total ordered parameter composite. Applying a source composition replaces two adjacent arrows by that composite, leaving the total unchanged. Composing two target blocks multiplies their two tags to that same arrow. The parameter category is concentrated in degree zero, so adding a tag introduces no new Koszul sign. The original entire mixed-input functor equation from C.48 thus proves the tagged equation term by term. Ordinary identity normalization and the object-unit maps are retained as well. This proves the normalized unital functor assertion, including every higher component, rather than just a strict functor on its first components. Consider a reduced input word. If one of its parameter arrows is nonidentity, its total parameter composite is nonidentity: height strictly increases at that step and never decreases thereafter. Its output, even with ordinary component a scalar identity, is therefore in the reduced mapping space of the tagged target. If every parameter arrow is an identity, every reduced ordinary input has degree at least two. On \(r\ge1\) such inputs the functor component has ordinary output degree at least \(2r+1-r=r+1>0\). It cannot have a scalar output. These two cases exhaust the reduced inputs, proving preservation of the relative augmentation. Finite additive multiplicity matrices use the full relative matrix coefficient category of C.38.1 and the same degree argument. The consecutive-partition bar formula is now a legitimate reduced bar map: every component just proved reduced can be used as a target bar letter. Its coproduct equation follows by cutting a consecutive partition; its chain equation is exactly the tagged full functor equation. Thus \(BF_I\) is an actual chain coalgebra map, with no scalar component thrown away. The cobar formula of C.38.1 sends its free generator \(tc\) to \(t(BF_I(c))\), extended multiplicatively. The chain and coproduct equations prove the two terms of the dg equation, respectively. This gives the displayed canonical strict representative. Composition of the bar maps is the original full functor composition by the same consecutive-block formula. Applying \(\Omega\) makes that composition strictly equal on free generators and hence on all morphisms. Face inclusions restrict the original parameter-word labels and commute with their actual compositions, so restrict these same formulas. A nonidentity index arrow with ordinary output one is retained as the reduced letter \(1\otimes a\), rather than sent to zero. This is exactly why the tagged target avoids the loss of unit data exhibited in C.42.4 and repaired for a collapsed target in C.43.2. ∎ The phrase “over the identity on parameter arrows” means the tag of each component is the specified input composite. It does not assert that its ordinary target mapping component is one, or that the pure input \(D_I\) maps to a tensor-unit copy of \(D_I\). Actual parameter comparison matrices between different output objects remain present. #### C.49.6. The whole nested common-core parameter graph Fix a finite directed family of native execution steps, and a common finite parameter category \(I\) for a cell or subdivided comparison prism. At each state of frontier arity \(n\), use \(Q_{n,I}\). For each step use the strict tagged representative of C.49.5. Keep every composable string of steps as its own label and use their literal strict composite as its functor. For a string \(\sigma:i\to j\), put \[ M_\sigma(y,x)=\operatorname{Hom}_{Q_{n_j,I}}(F_\sigma x,y). \tag{C.49.14} \] For composable strings \(\sigma:i\to j\), \(\tau:j\to k\), the cross product is \[ M_\tau(z,y)\otimes M_\sigma(y,x)\longrightarrow M_{\tau\sigma}(z,x),\qquad m\otimes v\longmapsto m\,F_\tau(v). \tag{C.49.15} \] All source, intermediate, target and parameter labels are retained. **Theorem C.49.6.1.** These data give one actual directed dg graph for the whole nested parameter family. They include all several-cross evaluation products and their dg identities, are strictly compatible with the parameter faces, and retain the original native enhanced functors and their literal composites under the canonical bar-cobar equivalences. Thus a supplied entire native comparison prism gives its entire nested graph, rather than a set of independent one-cross graphs. The canonical common-core pullback has pure cores \(C^{\otimes n}\otimes D_I\), retains the entire graph differential and restricts to the same parameter faces. Its cross central cohomology is nonnegative, even and degreewise finite at finite corners. **Proof.** The edge representatives are strict dg functors by C.49.5. The two possible products of three cross mappings are consequently \[ (mF_\tau(v))F_{\tau\sigma}(w) =mF_\tau(vF_\sigma(w)), \tag{C.49.16} \] after inserting the appropriate three consecutive string labels and their strict composite. Both sides are target composition of the same three transformed maps. The equality follows from strict functoriality and target associativity. Inserting an intermediate pure-core arrow is checked by the same equality; pure-core compositions are the original dg ones. These are every possible composable triple. Units act by the represented identities. The dg Leibniz rule is target Leibniz and the strict dg equation for the applied functor. This proves the entire dg graph, with its genuine two-cross and longer products, as in the full proof C.38.1. The canonical comparison from an augmented unital category to its bar-cobar replacement has the full consecutive-block formula. By its generator formula it commutes with the augmented tagged functor maps. Its finite-height quasi-equivalence and recovery proof is C.42.2 together with C.43.2; all parameter arrows remain directed and their possible bar length is bounded. Thus each edge functor represents the given entire native parameter cell. Naturality and strict composition of the canonical representatives recover each literal nested composite and the displayed represented evaluation product. No separate formality of a string corner has been assumed. Every parameter face restricts the original tagged functor and its canonical strictification. It therefore restricts each string functor, its represented mapping complex and its evaluation product exactly. Applying this to the full prism rather than merely to its boundary gives one nested dg graph whose two endpoints are the corresponding nested endpoint graphs and whose other faces are their actual comparison faces. This is an actual graph construction over the supplied prism. For the asserted common-core model take the full relative bar of this actual dg graph. On the pure state core use the actual map \[ f_{n,I}=\eta_{B(U_n\otimes kI)}\psi_{n,I}: C^{\otimes n}\otimes D_I\longrightarrow BQ_{n,I} \tag{C.49.17} \] from C.43.1. Change every pure run simultaneously by C.40.1's explicit consecutive-cut formula. In detail, on a path retain each original central cross letter and apply \(f_{n,I}\) in every intervening core slot before its original central graph operation. Reconstruct the result by splitting the boundary core runs. Its pure differentials are the given common-core differentials. The resulting whole-path coderivation is a coderivation by that same cut formula. The comparison to the original graph bar commutes with differentials at its central projection; coderivation uniqueness proves the full equation. Its square has zero pure part and zero central projection by the original square-zero identity, and is therefore zero too. This is the full proof of C.40.1, valid with the retained central unary differential and finite index cores as in C.43.2. It does not assume the comparison map is injective. Every several-cross evaluation and higher transferred native cell action is retained in this construction. There are no backward parameter mappings in \(Q_{n,I}\), by its endpoint labels. Its counit is a quasi-isomorphism to \(U_n\otimes kI\) by C.42.2. At a selected pair of parameter objects, its mapping cohomology is therefore the mapping cohomology in \(U_n\) tensored with the finite space of the corresponding actual index arrows. For an actual step image, the ordinary object is a finite perverse IC sum. Consequently every central cross mapping cohomology is nonnegative, even and degreewise finite by C.33.1; finite additive matrices preserve these properties. First transfer each actual cofree one-cross module using its chosen central contraction, and then transfer the several-cross terms by C.43.3. Core input words and cross counts make the respective perturbation sums finite. The resulting whole-path model retains those exact one-cross choices and has the stated central cohomology. Its canonical common-core pullback uses the same consecutive-cut formula and preserves those central spaces. The tagged target's face mapping complexes are actual subcomplexes, so compatible central contractions need a construction. For a fixed pair of parameter endpoints use the canonical counit on mappings \(\epsilon_{n,I}:Q_{n,I}\to U_n\otimes kI\). We construct a normalized contraction with exactly this projection, compatible with every full parameter face. Write a cobar word as nonempty blocks of original reduced \(U_n\otimes kI\)-letters, and let \(r\) be their total number. First retain the original unary differential and the differential inserting block cuts, but omit internal binary bar compositions. At \(r\ge2\) the cut contraction of C.46.3 removes the first boundary when it is present, with coefficient \((-1)^{|u_1|}\), merging the first one-letter block with its next block; it is zero when that boundary is absent. At \(r=0,1\) it is zero. Call it \(H_0\). Its cut identity and unary sign cancellation give a normalized contraction onto the scalar and one-letter complex, which is the original mapping complex of \(U_n\otimes kI\). Its inclusion \(I_0\) sends a reduced mapping to its single-letter block. Now restore the internal binary bar-composition term \(\delta\). It strictly lowers \(r\), while \(H_0\) preserves \(r\). Thus C.35.1's perturbation formulas are finite on each word and give \[ H_\epsilon=H_0(1+\delta H_0)^{-1},\qquad I_\epsilon=I_0. \tag{C.49.18} \] The inclusion is unchanged because \(\delta I_0=0\). Every nonzero output of \(H_0\) contains a block with at least two original letters. The canonical counit kills a word containing such a block, so \(\epsilon_{n,I}H_\epsilon=0\). Also \(\epsilon_{n,I}I_0=1\). Applying the counit to the perturbed homotopy identity shows that its perturbed projection is exactly \(\epsilon_{n,I}\), not an independently selected projection. Therefore \[ dH_\epsilon+H_\epsilon d=1-I_0\epsilon_{n,I},\qquad H_\epsilon^2=H_\epsilon I_0=\epsilon_{n,I}H_\epsilon=0. \tag{C.49.19} \] This proof retains the complete original unary differential and every internal composition. It uses decreasing original length, so also applies to the degree-zero nonidentity index arrows and their negative bar degrees. The cut homotopy keeps every original parameter vertex and arrow label; internal composition only removes intermediate vertices. Consequently \(H_0,\delta,I_0,\epsilon\) and their displayed finite formulas commute with every full parameter face inclusion. On the target \(U_n\otimes kI\), the mapping block is the original \(U_n\) mapping complex tensored with the finite space of actual index arrows. A full face retains that same arrow space between its retained endpoints. Choose its mapping-cohomology contractions once on these ordinary blocks, with the fixed finite multiplicity matrices, and tensor them with those arrow spaces. Compose them with the counit contraction by C.45.3's explicit normalized SDR composition formula. This gives the required central contractions on the actual tagged mapping complexes, compatible with all faces, although the face complexes themselves can be proper subcomplexes. The whole one-cross perturbation and the subsequent several-cross transfer now use these actual compatible contractions. Their formulas consume only the original words and retain the parameter, endpoint and string labels. They therefore restrict to the same face maps, proving the common-core family assertion. The cross block of a literal composite at the end of a native flattening cell can also be shared with that same string block in its nested path graph: canonical bar strictification commutes exactly with that composite by C.49.5, and the just-constructed target contractions are the same on its actual common face. No independent minimal model is substituted at their intersection. Extracting its endpoint coordinate isomorphism with prescribed product projections remains a separate transfer step, not an assertion that its individual edge maps suffice. ∎ #### C.49.7. Solved exercises **Exercise C.49.7.1 (advanced).** Let the positive left core have one degree-one primitive letter \(c\), and let the right index bar have the degree-minus-one arrow \(a:0\to1\). The central letters \(u_i\) have unsuspended degree zero at parameter vertex \(i\). Define a central coordinate correction that consumes one \(c\) and the arrow \(a\), sending their crossed input to \(su_0\). Write its entire cofree extension on \(c^k\otimes su_1\otimes a\), find its inverse, and calculate the total degree and \(P\) before and after its nonleading term. **Solution.** With deconcatenation on the left core, the full correction is \[ S(c^k\otimes su_1\otimes a) =c^{k-1}\otimes su_0\otimes1_0\quad(k\ge1), \tag{C.49.EX1} \] and it is zero on inputs with no \(a\), no \(c\), or other index endpoints. Cutting any remaining left prefix gives the same formula on the suffix, proving its left coaction equation. The right coaction of its output is that output tensored with \(1_0\); on its input, the only nonzero corrected term is the part whose whole \(a\) stays in the module. This proves its right coaction equation too. Its output has no index arrow, so \(S^2=0\), and the coordinate \(1+S\) has actual inverse \(1-S\). The input's total degree is \(k-2\), from \(k\) positive core letters, one cross suspension and one index suspension. The nonleading output has degree \((k-1)-1=k-2\) too. Its \(P\) changes from \(k\) to \(k-1\). Thus a genuine degree-zero invertible cofree coordinate can strictly lower \(P\); it need not preserve its homogeneous pieces. Both it and its inverse nevertheless preserve each full window \(P\le D\), as required in Theorem C.49.1.1. **Exercise C.49.7.2 (intermediate).** In the category \(0<1<2\), let \(a:0\to1,b:1\to2,c:0\to2\), with \(ba=c\). The collapsed ordinary functor sends each arrow to the scalar one. Describe the tagged version and check its two-arrow cobar equation. Explain why its finite negative-degree index words do not spoil the \(P\)-windows. **Solution.** The tagged arrow images are \(1\otimes a,1\otimes b,1\otimes c\). Each is a nonidentity arrow between distinct parameter vertices, even though its ordinary component is one. Hence each belongs to the reduced relative target. The actual reduced bar map retains \(sa,sb,sc\) and the two-letter word \(sb|sa\); it is not the map that projects those outputs to zero. For the generator \(w=t(sb|sa)\), the sign convention C.39.15 gives \[ d w=-t(sc)+t(sb)t(sa). \tag{C.49.EX2} \] Its tagged counit image is \(-1\otimes c+(1\otimes b)(1\otimes a)=0\), exactly the actual relation \(ba=c\). The one-letter generators have degree zero, while \(w\) has degree minus one. An index-bar word \(v\) of length \(p\) has total degree \(-p\), so its \(P=|v|+p\) is zero before adding any cross. These degree-zero \(P\)-terms remain finite because the directed index category allows at most two nonidentity arrows on a string. A full face that skips vertex1 retains the direct arrow \(c\) as one arrow, and therefore retains \(P\) exactly. This is the independent finite index bound used in Theorem C.49.1.1. ![The nonnegative mapping-degree filtration retains complete finite differential and product domains; actual scalar unit outputs keep their parameter tags; the canonical tagged representatives form the entire nested common-core prism graph](assets/native-prism-degree-windows.png) Theorems C.49.1.1–C.49.3.1 prove the invariant finite windows, complete operation domains and coordinate-group laws. Theorem C.49.4.1 constructs the actual face-compatible cell coordinates. Theorems C.49.5.1–C.49.6.1 retain the unit tags and construct the whole nested prism graph and common-core pullback. Exercises C.49.7.1–C.49.7.2 compute the strict filtration drop and tagged cobar relation. The canonical inputs and full native prism data were proved in C.43.1.1–C.43.2.1 and C.48.1.1–C.48.3.1. Editable SVG source. ### C.50. Whole prism endpoint recovery with exact core powers #### C.50.1. Recover every represented action together Let \(A,B\) be unital dg categories over a field, with \(A\) augmented over its specified split coefficient identities as in the actual directed graph. Let \(M(y,x)\) be an actual \((B,A)\)-bimodule, covariant in \(y\in B\) and contravariant in \(x\in A\). All its higher actions are retained. For each selected \(x\), suppose the left \(B\)-module \(M_x\) is quasi-isomorphic to the represented module \(R_x=B(b_x,-)\). No assumption is made that an arbitrary natural transformation space is contractible. The selected representing objects here have the split nonzero coefficient-identity cycles of C.30.1, as do the actual finite IC sums. **Theorem C.50.1.1.** Such data recover a unital enhanced functor \(F:A\to B\), with objects \(F(x)=b_x\), whose represented bimodule is equivalent to the original whole \(M\). The construction uses its entire right action; it does not infer a functor from independently chosen maps on the individual generator arrows. **Proof.** Work in the dg category of unital \(A_\infty\) left \(B\)-modules and their normalized bar module maps. The entire right \(A\)-action of \(M\), expanded on consecutive source blocks, is a unital enhanced functor from \(A^{\rm op}\) to that module category. The coefficient of its source word is precisely the right action Taylor map on that word; the bimodule equation is exactly its functor equation. This is the same action/evaluation correspondence proved in C.33.2. A module quasi-isomorphism can be inverted in this actual dg category. Here are the chain and bar details needed, rather than an assertion about inverses of arbitrary functors. Replace a given central quasi-isomorphism \(r:R\to N\) by the mapping-cylinder factorization \[ R\xrightarrow{i}L=R\oplus\operatorname{Cone}(1_N)[-1] \xrightarrow{p}N, \tag{C.50.1} \] where in degree \(q\) write an element as \((v,a,b)\in R^q\oplus N^{q-1}\oplus N^q\), and set \[ d_L(v,a,b)=(dv,-da-b,db),\quad i(v)=(v,0,0),\quad p(v,a,b)=r(v)+b. \tag{C.50.2} \] These are chain maps and \(p\) is graded-surjective. The retraction \(q(v,a,b)=v\) has contractible kernel: on its last two summands \(h(a,b)=(0,-a)\) gives \(dh+hd=1\) and \(h^2=0\). Both \(i\) and \(p\) are quasi-isomorphisms, so the kernel of the surjection \(p\) is acyclic and has a split contraction by C.30.1's explicit decomposition. The mapping cylinder is formed in the module category with the usual shifted action signs; its two actions and these maps are genuine module actions/maps. On the cofree bar modules extend a central kernel homotopy by the identity on surrounding cores. Separate off the core and unary differential. Every remaining action term consumes at least one surrounding input letter. The finite-on-each-word perturbation formula of C.45.2 contracts the full cofree kernel, with all actions retained. Its fixed-projection section and homotopy therefore give inverse module maps and their actual module homotopies for the surjective factor \(p\). The explicit cylinder contraction gives them for \(i\). Their composite inverts \(r\). When the original quasi-isomorphism is already a bar module map with higher components, apply the same construction to its cofree mapping cylinder; the leading surjective factor has that same central kernel and all higher terms still consume surrounding letters. The identical perturbation proof then applies. Thus each stipulated module equivalence is an actual homotopy equivalence in this dg module category. Transport the entire right \(A\)-action along those objectwise module homotopy equivalences. Write \(r_x:R_x\to M_x\), \(s_x:M_x\to R_x\), and retain the module homotopy \(1-r_xs_x=d h_x+h_xd\). For a reduced source word partition it into nonempty consecutive blocks. Apply the original whole right-action component to each block, insert \(h\) between consecutive outputs, and compose with \(r\) on the incoming end and \(s\) on the outgoing end. Sum over these partitions with the suspended tensor evaluation signs. This is the consecutive-word transfer formula of A.3.1.1 and C.35.1 in the dg module category. It has the correct component degree: a term with \(v\) action blocks has their total degree plus \(v-1\) homotopies of degree minus one. At each fixed source word the sum is finite. Commuting the differential through it replaces an internal \(h\) by \(1-rs\). The \(1\) terms cancel the adjacent source-action terms, and the \(rs\) terms are exactly the two target functor-composition blocks. The remaining terms are the original action functor equation. This is the full bar telescope and proves the functor equation into the **existing** dg category of the represented modules. On the coefficient identities set the first component to \(1_{R_x}\) and every higher component with an identity input to zero. The source is augmented, so operations on reduced source words never produce a coefficient identity. This assignment changes none of the just-proved reduced-word equations. In a word containing an identity, its adjacent source composition cancels the corresponding target composition with the assigned true identity, and every other term is zero or the shorter-word equation. Thus the result is actually unital even when an initially chosen \(s_xr_x\) was only homotopic to one. The same partition construction gives its action equivalence to the original \(M\)-action. Here normalization at the ordinary object identities is part of the contraction choice. For clarity, the elementary Hom contraction associated with an object retract \(iq=E\) need not annihilate the identity: its inclusion sends \(1_R\) to \(E\), rather than to \(1_L\). Keep its projection fixed. Choose its chain inclusion to send the specified unit to \(1_L\), using the split coefficient-identity summand, and correct its kernel homotopy by \(H^{u}=H(1-I^{u}P)\). On the kernel of \(P\), the old \(H\) already contracts to zero and squares to zero. Hence \(dH^u+H^u d=1-I^uP\), \((H^u)^2=PH^u=H^uI^u=0\), and \(H^u(1_L)=0\). This is the fixed-projection construction proved in C.45.2. Thus the transfer really is normalized at the units. In computing it, full unital output components are retained; no reduced augmentation of the dg category of module maps is assumed. The consecutive formulas can be computed on unreduced words and then normalized, with each fixed input word using finitely many terms, as in C.43.2. Finally this represented module subcategory recovers \(B\), not a larger category of unrelated transformations. Its mapping complex from \(R_y\) to \(R_x\) is quasi-isomorphic to \(B(b_x,b_y)\) by evaluation at the identity of \(b_y\). Yoneda precomposition is its section. The full normalized bar contraction is insertion of that identity in the incoming represented-module argument, with its graded insertion sign. Its differential pairs all other adjacent bar terms; the two unpaired terms are the original module map and Yoneda applied to its identity evaluation. This is the complete evaluation/contraction proof C.33.2, or equivalently the represented instance of the explicit normalized bar contraction in C.39.3. It retains every source word, not just the bare central map. Taking the opposite module subcategory changes this contravariant represented-module direction into the required forward direction \(x\to y\) in \(B\). Choose its normalized contraction so that the homotopy vanishes on the Yoneda section. Transfer through it. The transferred binary operation is the original composition in \(B\), because the Yoneda section preserves composition exactly. Every higher transferred tree on pure Yoneda inputs contains a homotopy applied to such a composed section value and vanishes. Thus the recovered target is exactly the given dg \(B\). Composing the transported entire action with this actual Yoneda inverse gives \(F\). Its represented action is the given \(M\) up to the constructed module equivalence. This proves the whole-functor assertion. ∎ #### C.50.2. The actual whole prism endpoint comparison Let \(\mathcal H\) be C.49.6's actual entire nested dg graph for a calibrated comparison prism. The ordinary comparison parameter is \(0<1\); other native cell parameters are retained separately. Let \(A,B\) be its complete endpoint graphs. Let \(M(y,x)=\operatorname{Hom}_{\mathcal H}(x_0,y_1)\), retaining every string grade and every several-cross action. This is an actual bimodule between those whole endpoint categories. At each object \(x\) the parameter arrow supplies a closed degree-zero map \(e_x:x_0\to (q x)_1\). In the raw state core it is the canonical one-letter lift of the actual identity-valued parameter arrow, tagged by \(0\to1\); in finite additive images it has the corresponding fixed matrix identity. Composing with it gives \[ B(qx,y)\longrightarrow M(y,x). \tag{C.50.3} \] **Theorem C.50.2.1.** These maps are quasi-isomorphisms of left \(B\)-modules. Consequently the complete calibrated native prism recovers an enhanced equivalence of its whole endpoint graphs, with the actual leading graded graph comparison. Every several-cross functor component is included. This statement does not yet assert an exactly prescribed common-core bar coordinate or exact product projection normalization. The objectwise kernel and action-transfer choices can be made compatibly with the actual finite full parameter faces and shared string corners. **Proof.** On a fixed string summand the target complex is a represented mapping complex in the final indexed state. The source is its endpoint1 version. Composition with \(e_x\) uses that string's actual strict functor action on the canonical parameter arrow. Under the canonical counits to \(U_n\otimes k[1]\), and the original represented-module equivalences, it is exactly precomposition by the entire string's supplied enhanced comparison. The latter is an equivalence: it is the original full comparison of C.29, calibrated in C.48, and all nested strings use its actual composite. Precomposition by an equivalence is a quasi-isomorphism of mapping complexes. Concretely compose with the supplied coherent inverse; its unit and counit homotopies give the two chain homotopies of the mapping composites to the identity. The canonical counits and graph recovery are quasi-equivalences by C.42.2 and C.49.6, so the raw composition map is a quasi-isomorphism too. The arrow \(e_x\) is an actual arrow in the whole dg graph. Thus postcomposition by any target graph morphism commutes with its precomposition exactly. This proves the full left-module equation, including all allowed target string summands. Its right source actions are the actual graph products of \(\mathcal H\), with their whole dg and bar identities. C.50.1.1 therefore applies to this actual whole bimodule and recovers a functor from \(A\) to \(B\). Its first cohomology map is the original calibrated graph comparison because the representing map on each summand was just identified with that comparison. This first map is an isomorphism on every endpoint/string block and is essentially surjective on the stipulated generator objects, so graph and perfect-envelope recovery give the enhanced equivalence. Here is the compatibility argument for the choices, including the case of proper face subcomplexes. The corrected canonical counit contraction in C.49.6 commutes with all full parameter faces. The cylinder kernel for a fixed central comparison map is graded-isomorphic to the direct sum of its source and shifted target: in \(p(v,a,b)=r(v)+b=0\), eliminate \(b\) as \(-r(v)\). Each face keeps exactly the corresponding original word labels in \(v,a\). Thus face kernel intersections are the kernels of the common faces, and their graded subspace sums have the distributive word-support intersection property. All these kernel complexes are acyclic, since the comparison map on every face is the same proved quasi-isomorphism for that face. The sum of the previously assigned proper-face kernel complexes is acyclic as well. To prove this explicitly, for two subcomplexes use \((K_1+K_2)/K_1=K_2/(K_1\cap K_2)\). A quotient of two acyclic complexes is acyclic: lift a quotient cycle, lift a primitive of its differential in the subcomplex, subtract it, and then lift a primitive of the resulting ambient cycle. Induction, using the word-support distributivity and the acyclicity of all finite intersections, proves the claim for a finite sum. The quotient of the full kernel by this sum is therefore acyclic and split contractible over the field. Existing compatible normalized kernel contractions agree on intersections and preserve each face, so define one contraction on this sum. Extend it to the full kernel using an actual chain splitting of its contractible quotient. For completeness a graded section \(s\) with defect \(a=ds-sd\) is corrected to the chain section \(s+a h_Q\), where \(h_Q\) contracts the quotient: the Hom differential of \(a h_Q\) is \(-a\). In the resulting direct-sum decomposition extend the old contraction by the quotient contraction. The square-zero side condition and all restrictions then hold on each summand. This constructs compatible central kernel homotopies by induction over the face vertex sets, without calling different tagged face mapping complexes identical. The cofree module perturbation and the subsequent entire-action transfer consume only the input words and retain their endpoint/string and parameter labels. The compatible central choices therefore give the same full module maps on each face. The Yoneda identity-insertion homotopy has that same restriction property. Thus the recovered whole endpoint functors and their original action equivalences can be chosen face-compatible. A nested corner exhaustion retains the assigned words and extends these same choices rather than reconstructing each finite graph independently. The recovery never replaced the right \(A\)-action by a set of individual edge maps: that entire action was the source of the higher transferred functor components. In particular the construction includes all several-cross components required by the full endpoint functor equation. Its exact restrictions to a chosen prescribed core coordinate and prescribed product comparison are extra relative requirements, addressed in the next step rather than inferred from its invertible first cohomology map. ∎ #### C.50.3. Relative replacement by the exact core maps For a dg category \(B\), define its dg path category as follows. An object is a closed degree-zero homotopy equivalence \(u:b_0\to b_1\). A degree-\(r\) map from \(u\) to \(v:c_0\to c_1\) is a triple \[ \begin{aligned} (a,b,h)\in{}&B(b_0,c_0)^r\\ &\oplus B(b_1,c_1)^r\\ &\oplus B(b_0,c_1)^{r-1}. \end{aligned} \tag{C.50.4} \] Set \[ \begin{aligned} D(a,b,h)={}&(\delta a,\delta b,\\ &-\delta h+va-bu), \end{aligned} \tag{C.50.5} \] For composable maps \((a_1,b_1,h_1)\) of degree \(r\) and \((a_2,b_2,h_2)\) of degree \(s\), set \[ \begin{aligned} &(a_2,b_2,h_2)(a_1,b_1,h_1)\\ &\quad=(a_2a_1,b_2b_1,\\ &\qquad h_2a_1+(-1)^s b_2h_1). \end{aligned} \tag{C.50.6} \] Here \(\delta\) denotes the original mapping differential. Identities are \((1,1,0)\). Both endpoint projections \(\pi_0,\pi_1\) are strict dg functors. **Theorem C.50.3.1.** Let \(A\) be the actual augmented whole dg path graph and \(A_i\) its pure state subcategories. Suppose an actual enhanced endpoint comparison \(F:A\to B\) is given, together with its supplied coherent equivalences on \(A_i\) to the desired actual pure core functors \(q_i\). There is an enhanced comparison \(\widehat F:A\to B\) equivalent to \(F\), restricting **exactly** to every \(q_i\). The construction retains a previously assigned compatible replacement on any full face/path subfamily whose bar generator collection is differential-stable. It therefore applies successively to the native face family and its pure maps \(q_i=\Omega B((\Omega g)^{\otimes n_i}\otimes1_{kI})\). After the actual common-core pullback and module-first graph transfer, the calibrated native comparison consequently gives a face-compatible whole-path chain coalgebra isomorphism over the exact core powers \(g^{\otimes n_i}\otimes1_{D_I}\), with identity bare central maps. **Proof.** The path formulas define a dg category. In the square of \(D\), the mixed terms in the third slot are \(-\delta(va-bu)+v\delta a-(\delta b)u=0\), since \(\delta u=\delta v=0\). Its composition is associative: the third slot of either threefold composition is the sum of its last homotopy times the first two maps, its middle homotopy with the sign of the final degree, and its first homotopy with the sign of the two final degrees. The two degrees add in the same order. For Leibniz, expanding the third slot gives \[ \begin{aligned} &(-\delta h_2+wa_2-b_2v)a_1\\ &\quad+(-1)^s h_2\delta a_1\\ &\quad+(-1)^{s+1}(\delta b_2)h_1\\ &\quad+b_2(-\delta h_1+va_1-b_1u), \end{aligned} \tag{C.50.7} \] where \(w\) is the third object's equivalence. These are precisely the third slots of \(D(a_2,b_2,h_2)(a_1,b_1,h_1)+(-1)^s(a_2,b_2,h_2)D(a_1,b_1,h_1)\). The first two slots are ordinary dg composition. This proves every dg identity and both endpoint functor assertions. For any fixed choices of path objects, \(\pi_0\) is surjective on graded mapping complexes. Its kernel is the shifted cone of \[ B(b_1,c_1)\longrightarrow B(b_0,c_1),\qquad b\longmapsto bu, \tag{C.50.8} \] with the harmless displayed minus sign in the differential. This map is a homotopy equivalence: precomposition by the chosen homotopy inverse of \(u\), and precomposition by its two homotopies, give its inverse and chain homotopies. Its cone is contractible. One may also split that acyclic cone over the field by C.30.1. Thus \(\pi_0\) is a surjective quasi-isomorphism on each fixed mapping complex, with an actual split kernel contraction. Choose for each source object the supplied closed homotopy equivalence \(u_x:F(x)\to q_i(x)\). It is part of the original enhanced comparison on its state. That entire supplied natural equivalence records a dg functor from \(\Omega B A_i\) into the path category with first projection \(F^\sharp|_{\Omega BA_i}\) and second projection \(q_i^\sharp\). Explicitly its first and second components on a free generator are the two prescribed functor values; its third component is the natural-equivalence component on that source bar word. The third-slot dg equation just displayed is the full natural-transformation equation, including its two endpoint-composition terms. This proves the recording on every word; it is not an inference from the object components alone. The source \(\Omega BA\) is a free relative cobar. Its generator filtration has the length-preserving unary complex and smaller-length composition/splitting terms required in C.46.2. The prescribed pure-state generator union is differential-stable: its differentials, compositions and coproduct pieces keep their state and parameter labels. Apply that theorem's relative lifting proof to \(\pi_0:\operatorname{Path}(B)\to B\), the strict source functor \(F^\sharp\), the chosen object paths, and the already assigned pure state lift. The acyclic mapping kernels just proved are exactly its needed kernels. It extends to a full dg functor \(H:\Omega BA\to\operatorname{Path}(B)\) without changing those prescribed generator images. Take the second projection of \(H\) and precompose with the canonical unital comparison \(A\to\Omega BA\). Its restriction to each \(A_i\) is exactly \(q_i\), since C.43.2 proves that this composite recovers all the prescribed normalized unital components, not just their first map. The path functor \(H\) supplies the enhanced equivalence to \(F\). A union of already assigned full faces or string corners is another differential-stable generator collection, by the same original label argument. The relative lifting correction vanishes there. Thus one can keep all earlier choices on overlaps and extend along the finite face family, then along a countable increasing family. For the native comparison, the coherent pure-state equivalences used here are the actual restrictions of the whole prism recovery and its represented action equivalence. C.48 identifies their desired endpoint state functors with the displayed canonical \(q_i\). The actual canonical common-core maps from C43 strictly commute with them. Pulling the full comparison to the common cores therefore has precisely \(g^{\otimes n_i}\otimes1_{D_I}\) on every state. Its remaining cross components come from the **whole** \(H\), including all several-cross generator values. This proves exact pure-core replacement without asserting anything about arbitrary natural-transformation spaces. We give the common-core graph argument explicitly. Write \(Q_A:L_C\to BA\), \(Q_B:L_D\to BB\) for the actual whole graph pullbacks of C.40.1 and corrected C.49.6. Let \(T=B\widehat F\). This bar map is augmented: a crossed input retains its different graph levels and cannot have scalar output; on a pure state its functor is exactly the prescribed augmented \(q_i\). Define \(G:L_C\to L_D\) by its pure maps \(g_i\) and by central cross Taylor map equal to the central projection of \(TQ_A\), and reconstruct it by the full consecutive-cut formula. Strict naturality of the canonical pure input maps gives identical pure maps in \[ Q_B G=TQ_A. \tag{C.50.9} \] Their central projections also agree by construction, so the full cofree path reconstruction proves this equality, including every several-cross component. The chain defect of \(G\) is a coderivation along \(G\). Its image under \(Q_B\) is zero by that equality and the chain equations for \(T,Q_A,Q_B\). The central projection of \(Q_B\) is the identity on each bare cross block. Hence the chain defect has zero central projection; its pure part is zero since all \(g_i\) are chain maps. C.40.1's coderivation uniqueness proves the full chain equation. No injectivity of \(Q_B\) has been assumed. Use corrected C.49.6's face-compatible central contractions and C.43.3's module-first entire graph transfer on both sides. The transferred map is \(P_D^\infty G I_C^\infty\). It remains a full chain coalgebra map with those exact pure core maps. Its bare central term is the calibrated mapping-cohomology comparison, hence the identity by C.48.1. Its core map has identity primitive term. The iterated-cut argument of C.40.3 therefore makes it unitriangular by total primitive length; its finite recursive inverse preserves coproduct and differential, and C.49.1 retains its finite \(P\)-windows. All pullback and transfer terms keep the existing labels and the compatible face contractions, so the maps and inverses restrict to the same faces and shared corners. This gives the asserted actual whole-path/face/differential extension-field coordinate. Product and grafting covariance have not been inferred from this argument. ∎ #### C.50.4. Solved exercises **Exercise C.50.4.1 (advanced).** Let \(L=ke\oplus kx\oplus ky\), with \(e,x\) in degree zero, \(y\) in degree one, and \(dx=y\). Let \(E\) project onto \(ke\), and let \(h(y)=x\), with \(h(e)=h(x)=0\). On the graded endomorphism complex put \[ P(f)=f_{ee},\qquad H(f)=hf+(-1)^{|f|}Efh. \tag{C.50.EX1} \] Explain why the ordinary Hom contraction is not normalized at the identity, and give a contraction with the same projection and inclusion \(I^u(\lambda)=\lambda1_L\). **Solution.** Since \(hE=Eh=0\), one has \(H(1_L)=h\ne0\), whereas the original inclusion of the scalar identity was \(E\). The Hom differential is \(Df=df-(-1)^{|f|}fd\). Its direct telescope gives \(DH+HD=1-I^0P\), with \(I^0(\lambda)=\lambda E\), and \(H^2=PH=0\). On \(\ker P\) this is a normalized contraction to zero. Set \[ H^u(f)=H\bigl(f-P(f)1_L\bigr). \tag{C.50.EX2} \] Then \(PI^u=1\), and the chain projection \(1-I^uP\) has range in \(\ker P\), giving \(DH^u+H^uD=1-I^uP\). Since \(H\) preserves this kernel and squares to zero, \((H^u)^2=PH^u=H^uI^u=0\). In particular \(H^u(1_L)=0\). These are contractions of mapping complexes; the projection \(P\) has not been asserted multiplicative. The higher comparison uses the entire transferred action, as in C.50.1.1. **Exercise C.50.4.2 (intermediate).** In the dg path category use a graded target algebra \(k[\theta]/(\theta^2)\), with \(|\theta|=1\) and zero differential, and take all three object paths to be the identity. Let \(x=(1,0,0)\) have degree zero and \(y=(0,\theta,0)\) have degree one. Check the signed Leibniz rule, explain why the factor \((-1)^s\) in the third composition slot is needed, and contract the kernel of the first projection when the object path is the identity. **Solution.** The path differential gives \(Dx=(0,0,1)\) and \(Dy=(0,0,-\theta)\), while \(yx=0\). With the sign in C.50.3.1, \[ (Dy)x=(0,0,-\theta),\qquad y(Dx)=(0,0,-\theta), \tag{C.50.EX3} \] so \(D(yx)=(Dy)x-y(Dx)=0\). Omitting \((-1)^s\) would make the second term \((0,0,\theta)\), leaving the nonzero defect \((0,0,-2\theta)\) over the characteristic-zero coefficient field. An element of the first-projection kernel is \((0,b,h)\), with differential \((0,\delta b,-\delta h-b)\). Define \[ K(0,b,h)=(0,-h,0). \tag{C.50.EX4} \] Then \(DK+KD=(0,b,h)\) and \(K^2=0\), directly retaining the original mapping differential \(\delta\). For a strictly invertible object path \(u\), replace \(-h\) by \(-hu^{-1}\). For a general homotopy equivalence, its actual inverse and two homotopies give an acyclic cone as proved in C.50.3.1. This kernel, rather than an arbitrary transformation space, is what makes the relative exact-core lift possible. ![The entire native prism bimodule recovers all endpoint actions; a Hom contraction is normalized at the true unit; the dg path category keeps the prescribed pure core maps and yields a whole common-core coordinate](assets/whole-prism-endpoint-recovery.png) Theorem C.50.1.1 proves recovery from the entire represented action. Theorem C.50.2.1 applies it to the actual whole tagged native prism with compatible faces. Theorem C.50.3.1 proves the graded path-category formulas,relative exact-core replacement and actual whole common-core path coordinate. Exercises C.50.4.1–C.50.4.2 compute the unit correction and essential Koszul sign. The full prism and corrected compatible counit contractions are proved in C.49.6.1;the relative generator and module constructions are C.46.2.1 and C.45.2.1. Editable SVG source. ### C.51. Correct product coordinates inside the whole graph An exact product comparison must retain the whole graph differential, including operations on several successive transitions. This section proves how actual product-square homotopies give such corrections. Work over a characteristic-zero field. The hypotheses concern actual maps and homotopies over the fixed pure core powers; agreement on cohomology alone is weaker. All contractions use the full cofree module differential. #### C.51.1. The homotopy after changing contraction choices All complexes below are full cofree one-cross bicomodules over their stated pure cores. Maps and homotopies satisfy both outside coaction equations. Source-to-target maps are over the same fixed core powers. Thus all their composites have the displayed types; the Hom differential is \(\partial f=d f-(-1)^{|f|}f d\). Use normalized contraction data \[ (i,p,h):Z\rightleftarrows N,\qquad (i',p',h'):Z'\rightleftarrows N', \] and product contraction data \[ (j,q,k):W\rightleftarrows P,\qquad (j',q',k'):W'\rightleftarrows P'. \] Here \(pi=1\), \(\partial h=1-ip\), and the corresponding identities hold for the other three contractions. Each projection followed by its homotopy, and each homotopy followed by its inclusion, is zero. Let \(\phi:Z\to W\), \(\phi':Z'\to W'\) be the actual fixed-core product projections, and let \(t:Z\to Z'\), \(u:W\to W'\) be chain comparisons. Suppose the raw square has a specified degree-minus-one homotopy \[ R:Z\longrightarrow W',\qquad \partial R=\phi't-u\phi. \] Set \[ T=p'ti,\quad U=q'uj,\quad J=q\phi i,\quad J'=q'\phi'i'. \] In the application \(J,J'\) are the actual identity-core isomorphisms between two minimal product models: C41 and C45 construct them, and their identity leading central maps make them invertible by the finite cofree word recursion. In a general contraction diagram their invertibility is an additional hypothesis, not a consequence of the symbols above. **Theorem C.51.1.1.** If \(J'\) is invertible, the desired block comparison is \(T^{\mathrm{prod}}=(J')^{-1}U J\). There is an explicit full bicomodule homotopy \[ \partial K=T^{\mathrm{prod}}-T, \] given by \[ \begin{aligned} L={}&q'R i-q'\phi'h'ti+q'u k\phi i,\\ K={}&-(J')^{-1}L. \end{aligned} \] The formula preserves any face or sector on which all its stated raw maps, homotopies and contractions have compatible restrictions. In particular it retains the actual raw homotopy \(R\). **Proof.** All chain maps in the formula have degree zero. The Hom Leibniz rule therefore gives \[ \begin{aligned} J'T &=q'\phi'i'p'ti\\ &=q'\phi'ti-\partial(q'\phi'h'ti),\\ UJ &=q'ujq\phi i\\ &=q'u\phi i-\partial(q'u k\phi i). \end{aligned} \] Subtract. The difference of the two first terms is \(q'(\phi't-u\phi)i=\partial(q'R i)\). Hence \(J'T-UJ=\partial L\). Since \(J'\) and its inverse are chain maps, \[ \partial K=-(J')^{-1}(J'T-UJ) =(J')^{-1}UJ-T. \] Every summand is a composite of actual bicomodule maps of the indicated degree and over the same pure core maps. It is therefore an actual bicomodule homotopy, including all consumed-core Taylor components. The identities were in the full mapping complexes and did not discard any unary or higher action term. Restriction commutes with each composite. If \(J'\) restricts to the same isomorphism on a face, its inverse restricts to its inverse there: restrict the two inverse identities and use uniqueness of an inverse. This proves the compatibility assertion. The same wordwise reasoning applies to an increasing compatible family of finite corners. No choice of new primitives on separate faces has been made. \(\square\) If the block contractions are the C45 product-normalized ones, then \(\phi i=j\), \(q'\phi'h'=0\), \(k\phi i=kj=0\), and \(J=J'=1\). For an exactly commuting raw square, \(R=0\), the displayed formula is zero and \(T=U\) exactly. This recovers the exact C45 square as a special case. With other contraction choices its two homotopy terms must be retained. #### C.51.2. Ordered tensor homotopies and the cross suspension First work in the unsuspended modules \(K(Z_i)=Z_i[-1]\) of C41. For each ordered factor suppose chain maps \(t_i,t_i'\), over the same pure core powers, have a specified bicomodule homotopy \[ \partial a_i=t_i'-t_i. \] The graded tensor convention evaluates a tensor of homogeneous maps with the sign obtained by moving each map past its preceding input factors. In particular a degree-minus-one map in slot \(i\) contributes \((-1)^{\sum_{j