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Draft lesson. Section 7 proves the bounded interval support comparisons; the triangulation branch uses the separate hypotheses of Section 4.

Attachment maps from diagrams to costalks

Written with GPT-6.1 Sol (OpenAI). New text: CC0.

Pointwise abelian structure lets us take a kernel in a diagram without losing its attachment maps. This becomes concrete for a sheaf on an interval: the same two arrows determine restrictions of sections, the costalk and the two directional tests. A smart cutoff changes a boundary module by its kernel, so it changes those arrows too. We carry one complex through all these operations and calculate the resulting comparison maps.

Use the abelian-diagram proof in Pointwise abelian structure and natural splittings, Sections 1–2. Its pointwise limit construction and the induced kernels, cokernels and cohomology maps are proved there. The internal cohomology provider is Exact complexes, signs and lifting covers, Proposition 2.1.

For the sheaf interpretation, use the written proofs of interval gluing, Theorems 1–3 and Proposition 4, directional fibres, Proposition 1, and truncation triangles and abelian hearts. These are specific programme providers; their sheaf and derived-category prerequisites are retained. The calculation below supplies the actual module maps and comparisons. All indices and coefficient modules are small in a fixed universe.

1. Fix the complex and both attachment maps

On \(I=(-1,1)\), with distinguished point \(0\), set \[ \begin{gathered} V^\bullet=[\,\mathbb Z^2\xrightarrow{(2,0)}\mathbb Z\,],\\ L^\bullet=R^\bullet=\mathbb Z[0], \end{gathered} \tag{1.1} \] where \(V\) is in degrees \(0,1\). Define \[ \begin{gathered} r_-^0(x,y)=y,\\ r_+^0(x,y)=x+y,\\ r_-^1=r_+^1=0. \end{gathered} \tag{1.2} \] Both are cochain maps: their targets have zero degree-one differential and zero degree-one module. The diagram is \(L^\bullet\leftarrow V^\bullet\to R^\bullet\), with the two arrows pointing away from the singular point.

In degree zero, a section of the associated sheaf on a connected neighborhood of zero has value \((x,y)\) at zero, value \(y\) on its negative half and value \(x+y\) on its positive half. In degree one only the value \(z\in\mathbb Z\) at zero remains; both nearby values are zero. Restrictions keep these values on the relevant components. The differential takes \((x,y)\) to \(2x\) at zero and to zero on either side. This gives the same sheaf complex as the interval gluing provider, including its maps.

Consequently its global section complex is \(V^\bullet\). The cited interval acyclicity proof also identifies derived global sections with this complex. Direct calculation gives \[ \begin{gathered} H^0R\Gamma(I;F)=\mathbb Z,\\ H^1R\Gamma(I;F)=\mathbb Z/2, \end{gathered} \tag{1.3} \] where \(H^0\) is the kernel generated by \((0,1)\). The degree-zero attachment maps on this kernel are both the identity of \(\mathbb Z\).

2. One fibre convention for three tests

For a cochain map \(u:A^\bullet\to B^\bullet\), use the fibre complex \[ \begin{gathered} \operatorname{Fib}(u)^n=A^n\oplus B^{n-1},\\ d(a,b)=(d_Aa,u(a)-d_Bb). \end{gathered} \tag{2.1} \] Its differential squares to zero since \(u d_A=d_Bu\). It is isomorphic to \(\operatorname{Cone}(u)[-1]\): in the usual shifted-cone coordinates \((b,a)\), the comparison is \((a,b)\mapsto(-b,a)\). Thus signs and shifts are specified.

The localization providers identify the costalk \(C=Ri^!F\) and the two directional tests with \[ \begin{gathered} C=\operatorname{Fib}(r_-,r_+),\\ (r_-,r_+):V^\bullet\longrightarrow L^\bullet\oplus R^\bullet,\\ J_+=\operatorname{Fib}(r_-),\\ J_-=\operatorname{Fib}(r_+). \end{gathered} \tag{2.2} \] The support \(t\geq0\) has negative open complement, hence uses \(r_-\); the support \(t\leq0\) uses \(r_+\). The arrows in (1.2) are retained exactly.

Here the costalk is the two-term complex in degrees \(0,1\) \[ \begin{gathered} \mathbb Z^2\longrightarrow\mathbb Z^3,\\ (x,y)\longmapsto(2x,y,x+y). \end{gathered} \tag{2.3} \] The first target coordinate is the degree-one central value; the other two are the degree-zero side values in the fibre. The differential is injective. The map \[ q(z,l,r)=z+2l-2r \tag{2.4} \] annihilates it and is surjective. If \(q(z,l,r)=0\), then \(z=2(r-l)\), so the tuple is exactly the image of \(x=r-l,\ y=l\). Thus its kernel is exactly the differential image. This proves \[ H^0C=0,\qquad H^1C=\mathbb Z. \tag{2.5} \] The quotient isomorphism is the specified \(q\), not an unspecified choice of a free generator.

The positive test has differential \((x,y)\mapsto(2x,y)\); the negative test has differential \((x,y)\mapsto(2x,x+y)\). Both are injective. Their cokernels are \(\mathbb Z/2\), identified in each case by the first coordinate modulo \(2\). For the second differential, an even first coordinate \(z=2x\) allows the second coordinate to be fitted by \(y=l-x\), proving the kernel description. Therefore \[ H^0J_\pm=0,\qquad H^1J_\pm=\mathbb Z/2. \tag{2.6} \]

Forgetting the right, respectively left, fibre coordinate gives actual cochain maps \(C\to J_+\) and \(C\to J_-\). Under (2.4) and the specified directional identifications, both induced maps are \[ \mathbb Z\longrightarrow\mathbb Z/2,\qquad a\longmapsto\overline a, \tag{2.7} \] because \(z\equiv q(z,l,r)\pmod2\). The directional tests detect torsion quotients of a torsion-free costalk. Since both are nonzero, the cited monotone-test theorem puts both nonzero cotangent directions at zero in microsupport. This statement uses its neighborhood test, rather than inference from a single isolated test function.

3. A smart cutoff changes the diagram

The lower cutoff \(\tau^{\leq0}F\) uses the pointwise kernel of the degree-zero differential. Its central module is \(\ker(2,0)=\mathbb Z(0,1)\), and both side modules stay \(\mathbb Z\). The attachment maps are the restrictions of (1.2), so both are identities. The cutoff is the constant sheaf \(\mathbb Z_I\) in degree zero.

The upper cutoff \(\tau^{\geq1}F\) has central module \(\mathbb Z/2\) in degree one and zero side modules. It is the skyscraper \(i_*(\mathbb Z/2)[-1]\). The lower comparison is \((0,y)\mapsto(0,y)\) at the center and the identity on sides. The upper comparison is reduction modulo \(2\) in central degree one and zero in degree zero. These are maps of diagrams and of complexes.

The costalk of the lower cutoff is the fibre of the diagonal \(\mathbb Z\to\mathbb Z^2\); its degree-one quotient is identified by \((l,r)\mapsto r-l\). Its induced map into \(H^1C\) sends the class of \((l,r)\) to \[ q(0,l,r)=-2(r-l). \tag{3.1} \] The upper comparison gives \(H^1C\to\mathbb Z/2\) by \(q(z,l,r)\mapsto\overline z\), which is reduction of \(q\). Thus the costalk maps of the truncation triangle give the nontrivial exact sequence \[ 0\longrightarrow\mathbb Z \xrightarrow{-2}\mathbb Z \longrightarrow\mathbb Z/2\longrightarrow0. \tag{3.2} \] This sequence does not split: any map from \(\mathbb Z/2\) to the middle torsion-free group is zero. Its sign records the chosen quotient orientation \(r-l\); negating that generator changes \(-2\) to \(2\), without changing exactness.

There is also a direct failure of commuting a lower cutoff with the costalk: \[ \begin{gathered} Ri^!(\tau^{\leq0}F)=\mathbb Z[-1],\\ \tau^{\leq0}(Ri^!F)=0. \end{gathered} \tag{3.3} \] The first has degree-one cohomology \(\mathbb Z\); the second vanishes because (2.5) has only degree-one cohomology. Pointwise exactness of evaluation justifies the smart cutoff of diagrams. It does not make a costalk functor preserve the lower half of the standard t-structure.

Deleting degree one instead would leave \(\mathbb Z^2\) at the center with the original two maps, giving the wrong degree-zero cohomology. The kernel, its inclusion and its restricted attachments are all necessary.

4. The face-poset interface and its finite-star condition

For a small face poset \(S\), a diagram assigns \(M_\sigma\) to each simplex and maps \(r_{\sigma\tau}:M_\sigma\to M_\tau\) for each face inclusion \(\sigma\leq\tau\), with the identity and composition equations. The pointwise course theorem proves that this diagram category is abelian and that kernels, cokernels and cohomology use the induced maps at every face. This assertion itself requires no local finiteness.

The sheaf construction and its exact equivalence are supplied by Constructible sheaves on a triangulation, Theorem 1. It sends a face element \(m\in M_\sigma\) to the germ with value \(r_{\sigma\tau}m\) in every nearby coface. The identity \(r_{\tau\upsilon}r_{\sigma\tau}=r_{\sigma\upsilon}\) is what makes these local sections agree. On an open star \(U_\sigma\), a section is determined by \(m\), and the restriction \(U_\tau\subset U_\sigma\) is exactly \(r_{\sigma\tau}\). Thus the interval's two outgoing maps are the local one-vertex, two-edge instance of this same convention.

The closed-simplex resolution in that provider uses the right-adjoint diagrams \(I_\tau(J)\), whose value at \(\rho\) is \(J\) if \(\rho\leq\tau\) and zero otherwise, with identity transitions inside that closed simplex. If a point lies in a coface \(\rho\) of \(\sigma\), a nonzero restriction from the factor indexed by \(\tau\) requires \(\rho\leq\tau\). Hence \(\sigma\leq\tau\). For a locally finite complex, choose a vertex of \(\sigma\); every coface contains it, and only finitely many simplices contain that vertex. Consequently only finitely many factors restrict nontrivially to \(U_\sigma\). This is the exact finite-product step in the closed-simplex acyclic resolution and Theorem 3. No arbitrary product has been interchanged with stalks or derived sections.

The bounded derived-equivalence provider, Theorem 5 retains finite global dimension of the coefficient ring and a uniform finite dimension bound for the triangulation. Its finite-coefficient section also retains Noetherianity where kernels must be finitely generated. Local finiteness, uniform dimension and coefficient finiteness are distinct hypotheses. We use the written provider at those scopes; the elementary interval calculation above does not prove a derived equivalence for arbitrary stratifications.

5. Graded exercises with full solutions

Exercise 1 (foundation: remove the twist). Replace \(r_+^0(x,y)=x+y\) by \(r_+^0(x,y)=y\), keeping the same central differential. Calculate the costalk and its two directional quotient maps.

Solution. The fibre differential is \((x,y)\mapsto(2x,y,y)\). The quotient is identified by \((z,l,r)\mapsto(\overline z,r-l)\) with \(\mathbb Z/2\oplus\mathbb Z\): if both coordinates vanish, \(z=2x\) and \(r=l=y\), exactly a differential image; arbitrary residue and difference have representatives. Its degree-zero kernel remains zero. Each directional fibre still has cokernel \(\mathbb Z/2\), and both comparison maps project to the torsion summand. The modules and central differential are unchanged, but changing the attachment map changes the costalk from a free group to a group with torsion.

Exercise 2 (intermediate: a general twisting coefficient). Replace the central differential by \((m,0)\), with \(m\geq2\), and put \(r_-^0=y,\ r_+^0=ax+y\), for any integer \(a\). Calculate the costalk cohomology.

Solution. The differential is \((x,y)\mapsto(mx,y,ax+y)\), so it is injective. Changing target coordinates to \((z,l,r-l)\) changes its image to the span of \((m,0,a)\) and \((0,1,0)\). The cokernel is therefore \(\mathbb Z^2/\mathbb Z(m,a)\). Put \(g=\gcd(m,a)>0\). Bezout coefficients give an integral change of basis taking the vector \((m,a)\) to \((g,0)\): if \(um+va=g\), the matrix with rows \((u,v)\) and \((-a/g,m/g)\) has determinant \(1\). Hence \(H^1C\simeq\mathbb Z/g\oplus\mathbb Z\), and \(H^0C=0\). This includes \(a=0\), where \(g=m\), and \(a=1\), where the quotient is free. Both directional quotients remain \(\mathbb Z/m\), since each differential \((mx,y)\) or \((mx,ax+y)\) has cokernel identified by the first coordinate modulo \(m\).

Exercise 3 (advanced: a cutoff in the diagram category). Give the cohomology diagrams \(H^0F,H^1F\) and prove that evaluating a smart cutoff at each of the three vertices gives the smart cutoff of the evaluated complex. Does this imply (3.3) can be replaced by an isomorphism?

Solution. At the center \(H^0F=\mathbb Z(0,1)\), and at either side it is \(\mathbb Z\); the induced arrows are identities. The degree-one diagram is \(\mathbb Z/2\) at the center and zero elsewhere, with zero arrows. The linked pointwise proof gives both the kernel of the boundary differential and the quotient by an incoming image as diagrams, including their unique induced transition maps. Thus evaluation preserves exactly the kernel or quotient used in either smart cutoff, with the same comparison maps. In this example the inclusions and projections are those of Section 3. Evaluation is an exact functor on the diagram category; the costalk is a derived fibre of a separate map. Its computed outputs in (3.3) have different cohomology, so the evaluation assertion cannot give the proposed costalk isomorphism.

Exercise 4 (expert: an infinite star). Let a graph have one vertex \(v\) incident to infinitely many edges \(e_n\). Explain exactly which finite-star argument fails, while retaining abelianity of its small diagram category.

Solution. The face poset is small when the displayed edge family is small, so the licensed pointwise theorem still supplies kernels, cokernels and abelianity. But \(v\) has infinitely many cofaces. Each closed-edge factor \(I_{e_n}(J_n)\), with nonzero \(J_n\), has nonzero restriction to the star of \(v\). Thus the product used in the resolution does not reduce there to a finite product. The finite-product acyclicity argument in Section 4 cannot be applied. This identifies a missing hypothesis in that proof; it does not claim that every such infinite-star sheaf has nonzero higher sections, or that a different argument could never establish an acyclicity result.

6. References and component licences

7. Why the fibre comparisons are the actual support maps

The three fibre complexes in Section 2 must compute the sheaf operations together with their maps. We prove this for a bounded complex that is already given by a diagram of modules. This statement requires no finite global dimension of the coefficient ring; such a hypothesis belongs to a separate assertion that every constructible derived object admits a bounded diagram representative.

Direct proofs of the constant-coefficient interval prerequisite are in The cohomological homotopy theorem, Corollary 6.1 and its contractible-space consequence. For enough injectives use Injective objects, Sections 3–5; the bounded-below complex construction is Injective replacements, Theorem 2.1. Let \(k\) be any commutative ring and let \(F^\bullet\) be the sheaf complex constructed term by term from a bounded diagram \(L^\bullet\leftarrow V^\bullet\to R^\bullet\). Use Constructible gluing on an interval, Theorem 1 and Proposition 2 for that exact construction and the acyclicity of its degree-zero sheaves. Constant-module acyclicity on all the intervals used there is proved in Extending sections on convex sets, Constant coefficients. For the injective embeddings and bounded-below resolution construction use Sheaves of modules and their derived categories, Sections 5–6.

Proposition 7.1. Derived sections on the whole interval are naturally \(V^\bullet\). At zero, the local support objects for \(\{0\}\), \(\{t\geq0\}\) and \(\{t\leq0\}\) are respectively \[ \operatorname{Fib}(V^\bullet\xrightarrow{(r_-,r_+)}L^\bullet\oplus R^\bullet),\qquad \operatorname{Fib}(r_-),\qquad \operatorname{Fib}(r_+). \tag{7.1} \] Under these comparisons, inclusion of point support into either half-interval support is identity on \(V^\bullet\) and projection to the indicated side coordinate of the fibre. The comparisons are natural for cochain maps of the original diagrams and remain so after passing to the derived category.

Proof. Choose a bounded-below injective resolution \(F^\bullet\to I^\bullet\). Injective sheaves are flabby: for opens \(W\subset U\), the inclusion of the open extensions of \(k_W\) into \(k_U\) is a monomorphism, and applying \(\operatorname{Hom}(-,I^n)\) makes the restriction \(I^n(U)\to I^n(W)\) surjective. The extension-by-zero adjunction and its exactness also show that restricting an injective sheaf to an open set remains injective. These statements are proved in Sections 5–6 of the linked resolution lesson.

For each connected neighborhood \(U\) of zero, put \(U_-=U\cap(-1,0)\) and \(U_+=U\cap(0,1)\). Take the filtered colimits, over such shrinking neighborhoods, of the section complexes: \[ A=\mathop{\mathrm{colim}}_{U\ni0}\Gamma(U;I^\bullet),\qquad B_\pm=\mathop{\mathrm{colim}}_{U\ni0}\Gamma(U_\pm;I^\bullet). \tag{7.2} \] The resolution map gives a commuting diagram from \(V^\bullet\to L^\bullet\oplus R^\bullet\) to \(A\to B_-\oplus B_+\). Its source comparison is a quasi-isomorphism because \(A\) is the stalk of the resolution at zero and stalks are exact. Each side comparison is a quasi-isomorphism too. Indeed, restriction of the resolution computes derived sections on \(U_\pm\); the terms of \(F^\bullet|_{U_\pm}\) are constant-module sheaves and have no higher sections there. The bounded acyclic-resolution argument therefore computes these derived sections by \(L^\bullet\) or \(R^\bullet\). Filtered colimits of modules preserve this conclusion: a cycle, boundary, or equality between two representatives is witnessed at a common later index. The comparisons also identify the two restriction maps with the given \(r_-\) and \(r_+\).

For completeness, the bounded acyclic-resolution argument just used is finite. If a complex has one term acyclic for sections, its termwise section complex computes its derived sections by that hypothesis. Remove the lowest term of a longer bounded complex with the degreewise split sequence consisting of its upper brutal truncation, the whole complex, and its lowest term placed in its degree. Sections preserve that split sequence. The natural comparison with derived sections gives a map between their cohomology exact sequences. Induction and exactness make the middle comparison an isomorphism in every degree. This proves the assertion without a convergence claim for an infinite filtration. The same argument applied to Proposition 2's acyclic diagram sheaves proves \(R\Gamma(I;F^\bullet)\simeq V^\bullet\), with its actual restriction maps.

Flabbiness makes \(A^n\to B_-^n\oplus B_+^n\) surjective: sections on the two disjoint halves form one section on their union and extend across zero. The kernels of this map, of \(A\to B_-\), and of \(A\to B_+\) are exactly the filtered section complexes supported, respectively, at zero, in the positive half, and in the negative half. Thus they compute the three derived local support objects at zero. For point support this is also the costalk: a sheaf map from the skyscraper \(i_*M\) into a sheaf \(G\) is precisely a module map from \(M\) to its point-supported germs. Such a germ extends over any larger neighborhood by zero, so these maps are inverse constructions. This proves the underived adjunction defining \(i^!\); the injective resolution computes its right derived functor.

Here is a direct complex comparison, including its sign. If \(u:A\to B\) is any termwise surjective cochain map and \(K=\ker u\), the inclusion \[ K\longrightarrow\operatorname{Fib}(u),\qquad x\longmapsto(x,0) \tag{7.3} \] is a quasi-isomorphism. Its quotient has terms \(B^n\oplus B^{n-1}\), differential \((z,b)\mapsto(d_Bz,z-d_Bb)\), and contracting homotopy \((z,b)\mapsto(b,0)\). Direct substitution gives \(dh+hd=1\). Hence that quotient is acyclic, proving (7.3). The fibre differential is exactly (2.1). A commuting square of maps induces the corresponding coordinate map of fibres; if both vertical maps are quasi-isomorphisms, so is the fibre map, by the cohomology exact sequence of \(0\to B[-1]\to\operatorname{Fib}(u)\to A\to0\).

Apply (7.3) to the three surjective restriction maps in the resolution model and then use the quasi-isomorphisms in (7.2). This gives (7.1). Point-supported sections lie in either half-supported kernel. Their inclusions commute with (7.3), while the maps between the resolution fibres are \[ (a,b_-,b_+)\longmapsto(a,b_-),\qquad (a,b_-,b_+)\longmapsto(a,b_+). \tag{7.4} \] This proves the claimed comparison arrows, rather than merely identifying their domains and codomains. In cone coordinates the fibre comparison is \((a,b)\mapsto(-b,a)\) into \(\operatorname{Cone}(u)[-1]\); substituting in the two differentials verifies this sign. In particular, no independent sign change of one directional arrow is permitted after fixing these coordinates.

Finally, maps of diagrams give commuting restriction squares on resolution models. A different injective resolution is related by a comparison homotopy equivalence; homotopies induce homotopies of all restriction and fibre comparisons. Thus the induced maps in the derived category do not depend on the model, and they respect compositions and identities. Quasi-isomorphisms of bounded diagrams are taken to quasi-isomorphisms of their three evaluated complexes and hence of these fibres. The natural comparisons therefore descend through their inverses as well. This proves the last assertion. \(\square\)

For the integral example, (7.4) is precisely the pair of projections used in (2.7). The truncated central kernel is included by \(y\mapsto(0,y)\). Its fibre class is sent, under the quotient \(q(z,l,r)=z+2l-2r\), to the integer \(-2(r-l)\) in Section 3. Thus the sign in (3.2) is fixed by the actual support comparison and the displayed quotient generators.