Relative cohomology of classifying topoi

Written by Claude Opus 5.5 (Anthropic) on 23 October 2026. Self-checked by the writing AI. Public domain (CC0).

Let \(G\) be a group scheme over a scheme \(S\), flat and locally of finite presentation, and \(B\) its classifying topos. The Ext groups of the cotangent complex of the nerve \(N(G)\) into the nerve of a module are relative Ext groups of the adjoint co-Lie complex on \(B\) (Corollary 3.5 of The adjoint co-Lie complex of a group scheme): both are cocones of maps that correspond to each other, and two cocones of one map are isomorphic. Such an isomorphism is not unique, and two of them cannot be compared without more information. This lesson constructs these identifications canonically. Relative Ext groups on \(B\) are computed with the cocone of the unit of the coinduction adjunction on an injective resolution; on the nerve, with the cocone of the unit of the restriction to the level \(0\). For every distinguished triangle that relates an object of the topos of the nerve to the nerve of an object of \(B\) through the adjunction map of the level \(0\), this gives a canonical isomorphism of Ext groups, natural in the triangle and compatible with restriction along homomorphisms of group schemes. As an application, for a homomorphism \(u\colon F\to G\), the groups that classify the deformations of \(G\) together with an extension of \(u\) to a given deformation of \(F\) are relative Ext groups of the adjoint co-Lie complex on the classifying topos of \(G\) modulo that of \(F\). In the same way, for a category scheme acting on schemes, the groups that classify the equivariant deformations of the target of an equivariant morphism are relative Ext groups of the equivariant cotangent complex on the classifying topoi. For two deformations of the target of a homomorphism, the difference of the obstructions is the image of the difference of the deformations under the connecting map of the triangle that relates the adjoint co-Lie complexes of the two groups, with a sign; the proof is a relation between Kodaira–Spencer classes over a fibre product. Finally, the relative Ext groups are relative cohomology groups with coefficients in the internal Hom, by an isomorphism of complexes that is natural in the first argument and compatible with restriction, which gives the forms of these results with dual coefficients.

The lesson proves:

Prerequisites. The adjoint co-Lie complex of a group scheme (Conventions, Theorems 1.4 and 1.5, Lemma 1.8, Proposition 2.5, Theorem 3.3, Corollary 3.5, Lemma 3.7, Proposition 3.8, 3.9, Proposition 3.10, Theorem 3.11, Corollary 3.12); Deformations of flat group schemes and of homomorphisms (Theorems 2.1, 3.1 (3) and 3.3, Proposition 3.5); Deformations of diagrams of schemes (Propositions 1.2, 1.3 and 1.4); Deforming the target of a morphism of diagrams (Conventions, 2.1, Theorems 2.2 and 3.1); The arrow topos and relative Ext groups (Definition 3.1, Proposition 3.2, Exercise 5.3); Equivariant sheaves on simplicial topoi (Propositions 4.2 and 6.2); Cohomology of equivariant modules (Theorem 4.3, Proposition 5.1); Equivariant cotangent complexes ((0.1), Definition 1.1, Propositions 1.5, 2.5 and 3.2, Theorem 2.4, Lemma 3.1); Equivariant extensions and deformations (Definition 3.1); Fibred topoi and their total topoi (Proposition 3.1); Morphisms of fibred topoi and modules on total topoi (Proposition 4.2); Quasi-coherent and perfect complexes on big fppf sites (Theorem 3.1, Lemma 4.2). From the programme lesson The cotangent complex: functoriality maps compose (Theorem 2.2). From the AI-integrated Stacks project: adjoints and injectives [Stacks, Tag 015Z]; termwise split sequences and distinguished triangles [Tag 014I, Tag 014Q]; bounded below complexes of injectives are K-injective [Tag 070J]; derived Hom and triangles [Tag 08JC, Tag 0149]; consecutive maps of a triangle compose to zero [Tag 0146]; split triangles [Tag 05QT]; triangles with a zero vertex and the third isomorphism of triangles [Tag 05QR, Tag 014A]; composition of derived pullbacks [Tag 0D6D]; tensor products and inverse images [Tag 03EL]; internal Hom: its adjunction with the tensor product and its restriction to a localization [Tag 03EO, Tag 0E8H], internal Hom into a K-injective complex [Tag 0A94], the dual of a perfect complex [Tag 08JJ]; resolutions by a generating class [Tag 05T7].

Conventions

\(S\) is a scheme, \(T\) the big fppf topos of \(S\) with its ring \(\mathcal O\), and \(G\) a group scheme over \(S\), flat and locally of finite presentation, with the classifying topos \(B=B[G,S]^{\mathrm o}\) of the lesson on the adjoint co-Lie complex: a module of \(B\) is a module \(M\) of \(T\) with a right action \(m\) of \(G\). The forgetful functor \(U\colon\operatorname{Mod}(B)\to\operatorname{Mod}(T)\) is exact and has the right adjoint \(\operatorname{coind}\), with the unit \(\eta_E\colon E\to\operatorname{coind}UE\) and the counit \(\mathrm{cu}_L\colon U\operatorname{coind}L\to L\), and \(\mathrm{cu}_{UE}\circ U\eta_E=\mathrm{id}\) (Proposition 4.2 of the lesson on equivariant sheaves). The adjoint of a map \(f\colon E\to\operatorname{coind}L\) is \(\mathrm{cu}_L\circ Uf\). Both \(U\) and \(\operatorname{coind}\) carry injective modules to injective modules, \(B\) being the classifying topos of a good two-sided pseudo-category satisfying (I) and (E) (Proposition 6.2 of the lesson on equivariant sheaves; Proposition 5.1 of the lesson on cohomology of equivariant modules, as in the proof of Corollary 3.5 of the lesson on the adjoint co-Lie complex).

\(N=N(G)\), and \(\mathcal N\) is its fppf topos \(\operatorname{Top}(N_{\mathrm{fppf}})\), ringed by \(\mathcal O\). The nerve \(\operatorname{ner}\colon\operatorname{Mod}(B)\to\operatorname{Mod}(\mathcal N)\) is exact, and its level \(0\) is the underlying module: \(i_0^*\circ\operatorname{ner}=U\), where \(i_0^*\) is the restriction to the level \(0\) (Proposition 1.5 (2) of the lesson on equivariant cotangent complexes). The functor \(i_0^*\) is exact, with the exact left adjoint \(i_{0!}\), \((i_{0!}E)_n=\bigoplus_{k=0}^n\pi_n^*E\) (Proposition 4.2 (b) of the lesson on morphisms of fibred topoi), and with the right adjoint \(i_{0*}\), the direct image of modules along the morphism of ringed topoi from \(T\) to \(\mathcal N\) given by the level \(0\), whose inverse image is \(i_0^*\); its level \(0\) is the identity, by the formula \(i_*(L)_j=\prod_{f\colon i\to j}f_*L\) of Proposition 3.1 (1) of the lesson on fibred topoi. So \(i_0^*i_{0*}=\mathrm{id}\), the counit is the identity, and \(i_0^*\) of the unit \(\nu_K\colon K\to i_{0*}i_0^*K\) is the identity of \(i_0^*K\) by the triangle identity. Since \(i_0^*\) and \(i_{0*}\) have exact left adjoints, \(i_{0!}\) and \(i_0^*\), they carry injective modules to injective modules [Tag 015Z].

For a map of complexes \(f\colon A\to C\), the cocone \(\operatorname{cocone}(f)\) is the complex with the terms \(A^k\oplus C^{k-1}\) and the differential \((a,c)\mapsto(da,f(a)-dc)\). The sequence \(0\to C[-1]\to\operatorname{cocone}(f)\to A\to0\), with \(c\mapsto(0,c)\) and \((a,c)\mapsto a\), is termwise split, and its distinguished triangle is \(\operatorname{cocone}(f)\to A\xrightarrow{f}C\to\operatorname{cocone}(f)[1]\) [Tags 014I, 014Q]: with the splitting \(a\mapsto(a,0)\), the connecting map is \(a\mapsto f(a)\). A commutative square \(b\circ f=f'\circ a\) of maps of complexes induces the map \((a,b)\colon\operatorname{cocone}(f)\to\operatorname{cocone}(f')\). For complexes \(E\), \(\operatorname{Hom}^\bullet(E,\operatorname{cocone}(f))=\) \(\operatorname{cocone}(\operatorname{Hom}^\bullet(E,f))\), as one checks on the components of the differential. A bounded below complex of injective modules is K-injective [Tag 070J]: maps of complexes into it, up to homotopy, are the maps of the derived category, and \(\operatorname{Hom}^\bullet(E,J)\) computes \(R\operatorname{Hom}(E,J)\) for every complex \(E\). In a short exact sequence of complexes whose third term is acyclic, the first map is a quasi-isomorphism.

1. Relative Ext groups on the classifying topos

Construction 1.1. Let \(M\in D^+(B)\). A resolution of \(M\) is a bounded below complex \(J\) of injective modules of \(B\) with an isomorphism \(M\cong J\) in \(D(B)\); it exists, \(\operatorname{Mod}(B)\) having enough injectives (Proposition 6.2 (2) of the lesson on equivariant sheaves). Put \[ J^{\mathrm{rel}}=\operatorname{cocone}\bigl(\eta_J\colon J\to\operatorname{coind}UJ\bigr), \] a bounded below complex of injective modules, hence K-injective. For a complex \(E\) of modules of \(B\) put \[ \begin{aligned}&R\operatorname{Hom}_{B/S}(E,M)=\operatorname{Hom}^\bullet(E,J^{\mathrm{rel}}),\\&\operatorname{Ext}^j_{B/S}(E,M)=\operatorname{Hom}_{D(B)}(E,J^{\mathrm{rel}}[j]).\end{aligned} \]

Lemma 1.2.

  1. For every complex \(E\), \(R\operatorname{Hom}_{B/S}(E,M)=\) \(\operatorname{cocone}\bigl(\operatorname{Hom}^\bullet(E,\eta_J)\bigr)\), and under the adjunction \(\operatorname{Hom}^\bullet(E,\operatorname{coind}UJ)=\) \(\operatorname{Hom}^\bullet(UE,UJ)\) the map \(\operatorname{Hom}^\bullet(E,\eta_J)\) is \(f\mapsto Uf\). So there is a distinguished triangle \[ \begin{aligned}&R\operatorname{Hom}_{B/S}(E,M)\\&\quad\to R\operatorname{Hom}_B(E,M)\\&\quad\xrightarrow{\ U\ }R\operatorname{Hom}_T(UE,UM)\\&\quad\to R\operatorname{Hom}_{B/S}(E,M)[1],\end{aligned} \] natural in the complex \(E\), and \(R\operatorname{Hom}_{B/S}(E,M)\) is a cocone of the map induced by \(U\), as in Corollary 3.5 of the lesson on the adjoint co-Lie complex.
  2. For two resolutions \(J\), \(J'\) of \(M\), a map of complexes \(g\colon J\to J'\) representing the identity of \(M\) exists and is unique up to homotopy, and \((g,\operatorname{coind}Ug)\colon J^{\mathrm{rel}}\to J'^{\mathrm{rel}}\) is a quasi-isomorphism. The induced isomorphisms \(\operatorname{Ext}^j_{B/S}(E,M)\to\operatorname{Ext}^j_{B/S}(E,M)'\) do not depend on \(g\) and are compatible with the triangles of (1).
  3. The functor \(E\mapsto R\operatorname{Hom}_{B/S}(E,M)\) carries distinguished triangles of \(D(B)\) to distinguished triangles of \(D(\mathbf Z)\).

Proof. (1) The first statement is the identity of the Conventions. The adjunction sends \(\varphi\colon E\to\operatorname{coind}UJ\) (in any degree) to \(\mathrm{cu}\circ U\varphi\), and \(\mathrm{cu}\circ U(\eta_J\circ f)=\mathrm{cu}\circ U\eta_J\circ Uf=Uf\). The complexes \(J\), \(\operatorname{coind}UJ\) and \(UJ\) are bounded below complexes of injective modules (Conventions), so the three Hom complexes compute \(R\operatorname{Hom}_B(E,M)\), \(R\operatorname{Hom}_B(E,\operatorname{coind}UJ)\) and \(R\operatorname{Hom}_T(UE,UM)\), and the triangle of the cocone is the triangle of the statement. Naturality in \(E\) holds on the complexes.

(2) \(J'\) is K-injective, so the identity of \(M\), a map \(J\to J'\) in \(D(B)\), is represented by a map of complexes \(g\), unique up to homotopy. Naturality of \(\eta\) gives \(\eta_{J'}\circ g=\operatorname{coind}Ug\circ\eta_J\), so \((g,\operatorname{coind}Ug)\) is a map of cocones. It maps the sequence \(0\to(\operatorname{coind}UJ)[-1]\to J^{\mathrm{rel}}\to J\to0\) to the corresponding one for \(J'\); its outer components are quasi-isomorphisms, \(g\) being one and \(Ug\) being a quasi-isomorphism between bounded below complexes of injectives, hence a homotopy equivalence, which the additive functor \(\operatorname{coind}\) preserves. By the long exact cohomology sequences it is a quasi-isomorphism. If \(h\) is a homotopy from \(g\) to \(g_1\), with components \(h^k\colon J^k\to J'^{k-1}\), then \((x,y)\mapsto(h(x),-\operatorname{coind}Uh(y))\) is a homotopy between the two maps of cocones, by the naturality of \(\eta\) applied to the \(h^k\). Compatibility with the triangles holds on the complexes.

(3) \(\operatorname{Hom}^\bullet(-,J^{\mathrm{rel}})\) computes \(R\operatorname{Hom}(-,J^{\mathrm{rel}})\), which carries distinguished triangles to distinguished triangles [Tag 08JC]. \(\square\)

2. The comparison with the nerve

Construction 2.1. Let \(M\in D^+(B)\) with a resolution \(J\). Choose a bounded below complex \(I\) of injective modules of \(\mathcal N\) with a map of complexes \(\iota\colon\operatorname{ner}J\to I\) that is a quasi-isomorphism and injective in every degree (embed the terms successively into injective modules). Put \[ I^{\mathrm{rel}}=\operatorname{cocone}\bigl(\nu_I\colon I\to i_{0*}i_0^*I\bigr), \] a bounded below complex of injective modules, \(i_{0*}\) and \(i_0^*\) preserving injective modules. Let \(\theta\colon\operatorname{ner}(\operatorname{coind}UJ)\to i_{0*}i_0^*I\) be the map adjoint to \(i_0^*(\iota)\circ\mathrm{cu}_{UJ}\colon U\operatorname{coind}UJ\to UJ\to i_0^*I\), using \(i_0^*\operatorname{ner}=U\). Then \(\nu_I\circ\iota=\theta\circ\operatorname{ner}(\eta_J)\): both maps \(\operatorname{ner}J\to i_{0*}i_0^*I\) have the adjoint \(i_0^*(\iota)\), the first because \(i_0^*\nu_I=\mathrm{id}\), the second because \(\mathrm{cu}\circ U\eta_J=\mathrm{id}\). The nerve commutes with cocones, being exact, so \((\iota,\theta)\) is a map of complexes \[ c\colon\operatorname{ner}(J^{\mathrm{rel}})\longrightarrow I^{\mathrm{rel}}. \]

Lemma 2.2.

  1. \(i_0^*I^{\mathrm{rel}}\) is acyclic, and \(\operatorname{Hom}^\bullet(i_{0!}E,I^{\mathrm{rel}})\) is acyclic for every complex \(E\) of modules of \(T\).
  2. For a complex \(X\) of modules of \(\mathcal N\) with \(i_0^*X\) acyclic, the projection \(I^{\mathrm{rel}}\to I\) induces a quasi-isomorphism \(\operatorname{Hom}^\bullet(X,I^{\mathrm{rel}})\to\operatorname{Hom}^\bullet(X,I)\), which computes \(R\operatorname{Hom}(X,\operatorname{ner}M)\).
  3. For a bounded above complex \(\lambda\) of modules of \(B\), the map \[ \begin{aligned}&\operatorname{Hom}^\bullet(\lambda,J^{\mathrm{rel}})\\&\quad\longrightarrow\operatorname{Hom}^\bullet(\operatorname{ner}\lambda,I^{\mathrm{rel}}),\\&\qquad f\longmapsto c\circ\operatorname{ner}(f),\end{aligned} \] is a quasi-isomorphism.

Proof. (1) \(i_0^*\nu_I\) is the identity, so \(i_0^*I^{\mathrm{rel}}\) is the cocone of the identity of \(i_0^*I\), which is acyclic. By the adjunction, \(\operatorname{Hom}^\bullet(i_{0!}E,I^{\mathrm{rel}})=\operatorname{Hom}^\bullet(E,i_0^*I^{\mathrm{rel}})\), which computes \(R\operatorname{Hom}(E,i_0^*I^{\mathrm{rel}})=0\), \(i_0^*I^{\mathrm{rel}}\) being a bounded below complex of injective modules.

(2) By the Conventions, \(\operatorname{Hom}^\bullet(X,I^{\mathrm{rel}})\) is the cocone of \(\operatorname{Hom}^\bullet(X,\nu_I)\colon\operatorname{Hom}^\bullet(X,I)\to\) \(\operatorname{Hom}^\bullet(X,i_{0*}i_0^*I)\), and the target is \(\operatorname{Hom}^\bullet(i_0^*X,i_0^*I)\), acyclic as in (1). So the projection is a quasi-isomorphism, and \(I\) is a K-injective complex isomorphic to \(\operatorname{ner}M\).

(3) By the Conventions the map is the map of cocones of the commutative square formed by \(\operatorname{Hom}^\bullet(\lambda,\eta_J)\), \(\operatorname{Hom}^\bullet(\operatorname{ner}\lambda,\nu_I)\) and the maps \(f\mapsto\iota\circ\operatorname{ner}(f)\), \(f\mapsto\theta\circ\operatorname{ner}(f)\). The first of these is the comparison map \(R\operatorname{Hom}_B(\lambda,M)\to R\operatorname{Hom}(\operatorname{ner}\lambda,\operatorname{ner}M)\) induced by the nerve, a quasi-isomorphism by Theorem 4.3 (2) of the lesson on cohomology of equivariant modules. The second becomes, under the adjunctions \(\operatorname{Hom}^\bullet(\lambda,\operatorname{coind}UJ)=\operatorname{Hom}^\bullet(U\lambda,UJ)\) and \(\operatorname{Hom}^\bullet(\operatorname{ner}\lambda,i_{0*}i_0^*I)=\operatorname{Hom}^\bullet(U\lambda,i_0^*I)\), the composition with the quasi-isomorphism \(i_0^*\iota\colon UJ\to i_0^*I\) between bounded below complexes of injective modules: the adjoint of \(\theta\circ\operatorname{ner}(f)\) is \(i_0^*(\iota)\circ\mathrm{cu}\circ Uf\), and \(\mathrm{cu}\circ Uf\) is the adjoint of \(f\). So it is a quasi-isomorphism too. A map of short exact sequences of complexes whose outer components are quasi-isomorphisms is one in the middle. \(\square\)

Theorem 2.3 (the canonical identification). Let \(\lambda\in D^b(B)\), \(M\in D^+(B)\), and let \[ \begin{aligned}&X\xrightarrow{\ \beta\ }i_{0!}i_0^*\operatorname{ner}\lambda\\&\quad\xrightarrow{\ \epsilon\ }\operatorname{ner}\lambda\xrightarrow{\ \mathrm{at}\ }X[1]\end{aligned} \] be a distinguished triangle of \(D(\mathcal N)\) whose second map is the adjunction map. Then \(i_0^*X=0\), and for every \(j\) there is a unique isomorphism \[ \Theta\colon\operatorname{Ext}^{j+1}_{B/S}(\lambda,M)\longrightarrow\operatorname{Ext}^j(X,\operatorname{ner}M) \] such that, for every \(f\), \(\Theta(f)\) is the image under \(I^{\mathrm{rel}}\to I\) of the unique \(\psi\in\operatorname{Hom}(X,I^{\mathrm{rel}}[j])\) with \(\psi[1]\circ\mathrm{at}=c\circ\operatorname{ner}(f)\) in \(\operatorname{Hom}(\operatorname{ner}\lambda,I^{\mathrm{rel}}[j+1])\).

  1. \(\Theta\) does not depend on the choices of \(J\) and \(I\).
  2. (Naturality.) Let \(X'\to i_{0!}i_0^*\operatorname{ner}\lambda'\to\operatorname{ner}\lambda'\xrightarrow{\mathrm{at}'}\) \(X'[1]\) be a second such triangle, \(b\colon\lambda\to\lambda'\) and \(a\colon X\to X'\) maps with \(\mathrm{at}'\circ\operatorname{ner}(b)=a[1]\circ\mathrm{at}\). Then \(\Theta(f'\circ b)=\Theta'(f')\circ a\) for \(f'\in\operatorname{Ext}^{j+1}_{B/S}(\lambda',M)\).
  3. The image of \(\Theta(f)\) under \(\operatorname{Ext}^j(X,\operatorname{ner}M)\to\) \(\operatorname{Ext}^{j+1}(\operatorname{ner}\lambda,\operatorname{ner}M)\), \(\varphi\mapsto\varphi[1]\circ\mathrm{at}\), is the nerve of the image of \(f\) in \(\operatorname{Ext}^{j+1}_B(\lambda,M)\) under the map of Lemma 1.2 (1).

Proof. The unit \(E\to i_0^*i_{0!}E\) of the adjunction \(i_{0!}\dashv i_0^*\) is the identity, the level \(0\) of \(i_{0!}E\) being \(E\); by the triangle identity, \(i_0^*\epsilon\) composed with it is the identity, so \(i_0^*\epsilon\) is the identity of \(U\lambda\). Applying the exact functor \(i_0^*\) to the triangle gives \(i_0^*X=0\) [Tag 05QR]. By Lemma 2.2 (1), \(\operatorname{Hom}(i_{0!}i_0^*\operatorname{ner}\lambda,I^{\mathrm{rel}}[k])=0\) for all \(k\), so the long exact sequence of \(\operatorname{Hom}(-,I^{\mathrm{rel}}[j+1])\) for the triangle [Tag 0149] shows that \(\psi\mapsto\psi[1]\circ\mathrm{at}\) is a bijection \(\operatorname{Hom}(X,I^{\mathrm{rel}}[j])\to\) \(\operatorname{Hom}(\operatorname{ner}\lambda,I^{\mathrm{rel}}[j+1])\). By Lemma 2.2 (3), \(f\mapsto c\circ\operatorname{ner}(f)\) is a bijection from \(\operatorname{Ext}^{j+1}_{B/S}(\lambda,M)\), and by Lemma 2.2 (2), the projection is a bijection \(\operatorname{Hom}(X,I^{\mathrm{rel}}[j])\to\) \(\operatorname{Hom}(X,I[j])=\) \(\operatorname{Ext}^j(X,\operatorname{ner}M)\). So \(\Theta\) is the composite of three bijections.

(1) For other choices \(J'\), \(I'\), choose \(g\) as in Lemma 1.2 (2) and a map of complexes \(h\colon I\to I'\) with \(h\circ\iota=\iota'\circ\operatorname{ner}(g)\): it exists because \(\iota\) is injective in every degree and a quasi-isomorphism, so that \(\operatorname{Hom}^\bullet(I,I')\to\operatorname{Hom}^\bullet(\operatorname{ner}J,I')\) is surjective in every degree, \(I'\) having injective terms, and a quasi-isomorphism, \(I'\) being K-injective; a surjective quasi-isomorphism of complexes is surjective on cycles of degree \(0\). Then \((h,i_{0*}i_0^*h)\colon I^{\mathrm{rel}}\to I'^{\mathrm{rel}}\) is a map of cocones, by the naturality of \(\nu\), and \((h,i_{0*}i_0^*h)\circ c=c'\circ\operatorname{ner}(g,\operatorname{coind}Ug)\): on the first components this is the choice of \(h\); on the second, both maps \(\operatorname{ner}(\operatorname{coind}UJ)\to i_{0*}i_0^*I'\) have the adjoint \(i_0^*(h\iota)\circ\mathrm{cu}_{UJ}=i_0^*(\iota'\operatorname{ner}g)\circ\) \(\mathrm{cu}_{UJ}=i_0^*(\iota')\circ\mathrm{cu}_{UJ'}\circ\) \(U\operatorname{coind}Ug\), by the naturality of \(\mathrm{cu}\). Composing the characterization of \(\psi\) with this map gives that of \(\psi'\), and the projections to \(I\), \(I'\) correspond under \(h\), which represents the identity of \(\operatorname{ner}M\).

(2) Let \(\psi'\) correspond to \(f'\). Then \((\psi'\circ a)[1]\circ\mathrm{at}=\psi'[1]\circ\mathrm{at}'\circ\) \(\operatorname{ner}(b)=c\circ\operatorname{ner}(f'\circ b)\), so \(\psi'\circ a\) corresponds to \(f'\circ b\), by uniqueness.

(3) The composite of \(c\) with the projection \(I^{\mathrm{rel}}\to I\) is \(\iota\) composed with the nerve of the projection \(J^{\mathrm{rel}}\to J\). So \(\Theta(f)[1]\circ\mathrm{at}\) is \(\iota\) composed with the nerve of the image of \(f\) in \(\operatorname{Hom}(\lambda,J[j+1])=\operatorname{Ext}^{j+1}_B(\lambda,M)\). \(\square\)

Remark 2.4. The triangles of Corollary 3.5 (2) and of Corollary 3.12 (1) of the lesson on the adjoint co-Lie complex, and the triangle of the proof of Theorem 3.1 (3) of the lesson on deformations of flat group schemes, are of this kind after \(L\varepsilon^*\); so the isomorphisms of Ext groups stated there can be taken to be the isomorphisms \(\Theta\), which are canonical and natural.

3. Restriction along a homomorphism

Let \(u\colon F\to G\) be a homomorphism of group schemes over \(S\), flat and locally of finite presentation, \(B_F=B[F,S]^{\mathrm o}\), and \(u^*\colon\operatorname{Mod}(B)\to\operatorname{Mod}(B_F)\) the restriction along \(u\), exact, with \(U_Fu^*=U\) and \(\operatorname{ner}(u^*M)=N(u)^*\operatorname{ner}(M)\) for the morphism of nerves \(N(u)\colon N(F)\to N\) and the inverse image \(N(u)^*\) of the fppf topoi, also exact (Proposition 2.5 of the lesson on the adjoint co-Lie complex). Write \(\mathcal N_F\) for the fppf topos of \(N(F)\), and \(\operatorname{coind}_F\), \(\mathrm{cu}^F\), \(\eta^F\) for the coinduction of \(B_F\). The level \(0\) of \(N(u)\) is the identity of \(S\), so \(i_0^*N(u)^*=i_0^*\) and \(N(u)^*i_{0!}E=i_{0!}E\) for a module \(E\) of \(T\), compatibly with the adjunction maps, the levels \(\bigoplus_k\pi_n^*E\) pulling back to \(\bigoplus_k\pi_{F,n}^*E\). So \(N(u)^*\) carries a triangle as in Theorem 2.3 for \(\lambda\) to one for \(u^*\lambda\).

Construction 3.1. Let \(M\in D^+(B)\), with a resolution \(J\) and \(\iota\colon\operatorname{ner}J\to I\) as in Construction 2.1. Choose a resolution \(J_F\) of \(u^*M\), a map of complexes \(g\colon u^*J\to J_F\) representing the identity of \(u^*M\), and \(\iota_F\colon\operatorname{ner}J_F\to I_F\) as in Construction 2.1 for \(F\). Let \(\kappa\colon u^*\operatorname{coind}UJ\to\operatorname{coind}_FUJ_F\) be the map adjoint to \(Ug\circ\mathrm{cu}_{UJ}\colon U\operatorname{coind}UJ\to UJ\to UJ_F\). Then \(\kappa\circ u^*(\eta_J)=\eta^F_{J_F}\circ g\), both having the adjoint \(Ug\), so \[ \rho=(g,\kappa)\colon u^*(J^{\mathrm{rel}})\longrightarrow J_F^{\mathrm{rel}} \] is a map of complexes, \(u^*\) commuting with cocones. The restriction map is \[ \begin{aligned}&\operatorname{res}\colon R\operatorname{Hom}_{B/S}(E,M)\\&\quad\to R\operatorname{Hom}_{B_F/S}(u^*E,u^*M),\\&\qquad f\longmapsto\rho\circ u^*(f).\end{aligned} \] Choose a map of complexes \(h\colon N(u)^*I\to I_F\) with \(h\circ N(u)^*(\iota)=\iota_F\circ\operatorname{ner}(g)\); it exists as in the proof of Theorem 2.3 (1), \(N(u)^*\iota\) being injective in every degree and a quasi-isomorphism. Let \(\mathrm b\colon N(u)^*i_{0*}i_0^*I\to i_{0*}i_0^*I\) be the map adjoint to the identity of \(i_0^*N(u)^*i_{0*}i_0^*I=i_0^*I\). Then \[ \begin{aligned}&h^{\mathrm{rel}}=\bigl(h,\ i_{0*}i_0^*(h)\circ\mathrm b\bigr)\colon\\&\qquad N(u)^*I^{\mathrm{rel}}\longrightarrow I_F^{\mathrm{rel}}\end{aligned} \] is a map of complexes: both \(\nu_{I_F}\circ h\) and \(i_{0*}i_0^*(h)\circ\mathrm b\circ N(u)^*(\nu_I)\) have the adjoint \(i_0^*h\), since \(i_0^*\nu=\mathrm{id}\) and \(i_0^*\mathrm b=\mathrm{id}\).

Lemma 3.2.

  1. \(h^{\mathrm{rel}}\circ N(u)^*(c)=c_F\circ\operatorname{ner}(\rho)\) as maps \(\operatorname{ner}(u^*J^{\mathrm{rel}})=\) \(N(u)^*\operatorname{ner}(J^{\mathrm{rel}})\to I_F^{\mathrm{rel}}\).
  2. The restriction maps form a map from the triangle of Lemma 1.2 (1) for \((E,M)\) to that for \((u^*E,u^*M)\), whose component on \(R\operatorname{Hom}_T(UE,UM)\) is the identity, through \(Ug\), which represents the identity of \(UM\).

Proof. (1) On the first components, this is the choice of \(h\). On the second, both maps \(\operatorname{ner}(u^*\operatorname{coind}UJ)\to i_{0*}i_0^*I_F\) have the adjoint \(i_0^*(\iota_F)\circ Ug\circ\mathrm{cu}_{UJ}\): for the first, \(i_0^*h\circ i_0^*\iota\circ\mathrm{cu}_{UJ}=i_0^*(\iota_F\circ\operatorname{ner}g)\circ\mathrm{cu}_{UJ}\); for the second, \(i_0^*(\iota_F)\circ\mathrm{cu}^F_{UJ_F}\circ U\kappa\), and \(\mathrm{cu}^F\circ U\kappa=Ug\circ\mathrm{cu}_{UJ}\) by the definition of \(\kappa\). (2) The map \(\rho\circ u^*(-)\) maps the sequence \(0\to\) \(\operatorname{Hom}^\bullet(E,\operatorname{coind}UJ)[-1]\to\) \(\operatorname{Hom}^\bullet(E,J^{\mathrm{rel}})\to\) \(\operatorname{Hom}^\bullet(E,J)\to0\) to the corresponding sequence for \(F\), with the components \(\kappa\circ u^*(-)\) and \(g\circ u^*(-)\); under the adjunctions, \(\kappa\circ u^*(-)\) is the composition with \(Ug\), as the adjoint of \(\kappa\circ u^*f\) is \(Ug\circ\mathrm{cu}\circ Uf\). \(\square\)

Proposition 3.3 (restriction and the canonical identification). Let \(\lambda\), \(M\) and the triangle \(X\to i_{0!}i_0^*\operatorname{ner}\lambda\to\operatorname{ner}\lambda\xrightarrow{\mathrm{at}}X[1]\) be as in Theorem 2.3, and let \(\Theta_F\) be the isomorphism of Theorem 2.3 for \(u^*\lambda\), \(u^*M\) and the image of the triangle under \(N(u)^*\). Then for \(f\in\operatorname{Ext}^{j+1}_{B/S}(\lambda,M)\), \(\Theta_F(\operatorname{res}f)\) is the image of \(\Theta(f)\) under \(N(u)^*\colon\operatorname{Ext}^j(X,\operatorname{ner}M)\to\) \(\operatorname{Ext}^j(N(u)^*X,\operatorname{ner}u^*M)\).

Proof. Let \(\psi\colon X\to I^{\mathrm{rel}}[j]\) correspond to \(f\). By Lemma 3.2 (1), \[ \begin{aligned}&(h^{\mathrm{rel}}\circ N(u)^*\psi)[1]\circ N(u)^*\mathrm{at}\\&\quad=h^{\mathrm{rel}}\circ N(u)^*(c\circ\operatorname{ner}f)\\&\quad=c_F\circ\operatorname{ner}(\rho\circ u^*f),\end{aligned} \] so \(h^{\mathrm{rel}}\circ N(u)^*\psi\) corresponds to \(\operatorname{res}f\), and \(\Theta_F(\operatorname{res}f)\) is its projection to \(I_F\), which is \(h\circ N(u)^*\Theta(f)\). The map \(h\) represents the identity of \(N(u)^*\operatorname{ner}M=\operatorname{ner}u^*M\), as \(h\circ N(u)^*\iota=\iota_F\circ\operatorname{ner}g\). \(\square\)

The same proof gives the statement on the level of complexes: the composites \(\Theta\) and \(\Theta_F\) of the quasi-isomorphisms of Lemma 2.2 and of the inverse of \(R\operatorname{Hom}(\mathrm{at},I^{\mathrm{rel}})\), isomorphisms \(R\operatorname{Hom}_{B/S}(\lambda,M)[1]\cong\) \(R\operatorname{Hom}(X,\operatorname{ner}M)\) in \(D(\mathbf Z)\), form a commutative square with the restriction maps and the maps induced by \(N(u)^*\), since \(R\operatorname{Hom}(-,I^{\mathrm{rel}})\to\) \(R\operatorname{Hom}(N(u)^*(-),I_F^{\mathrm{rel}})\), \(\varphi\mapsto h^{\mathrm{rel}}\circ N(u)^*\varphi\), is a natural transformation of functors on \(D(\mathcal N)\).

4. Relative Ext groups of a homomorphism

Definition 4.1. In the situation of Construction 3.1, for a complex \(\lambda\) of modules of \(B\), the relative derived Hom of \(B\) modulo \(B_F\) is the cocone \[ \begin{aligned}&R\operatorname{Hom}(B/B_F;\lambda,M)\\&\quad=\operatorname{cocone}\bigl(\operatorname{Hom}^\bullet(\lambda,J)\\&\qquad\to\operatorname{Hom}^\bullet(u^*\lambda,J_F)\bigr)\end{aligned} \] of the map \(f\mapsto g\circ u^*f\), which computes the restriction \(R\operatorname{Hom}_B(\lambda,M)\to R\operatorname{Hom}_{B_F}(u^*\lambda,u^*M)\). Its cohomology groups are the relative Ext groups \(\operatorname{Ext}^j(B/B_F;\lambda,M)\), and the relative cohomology is \(R\Gamma(B/B_F,M)=R\operatorname{Hom}(B/B_F;\mathcal O,M)\), with the groups \(H^j(B/B_F,M)\). There is a long exact sequence \[ \begin{aligned}&\cdots\to\operatorname{Ext}^j(B/B_F;\lambda,M)\\&\quad\to\operatorname{Ext}^j_B(\lambda,M)\\&\quad\to\operatorname{Ext}^j_{B_F}(u^*\lambda,u^*M)\\&\quad\to\operatorname{Ext}^{j+1}(B/B_F;\lambda,M)\to\cdots.\end{aligned} \]

Lemma 4.2. The map of cocones induced by Lemma 3.2 (2) is a quasi-isomorphism from the cocone of the restriction map \(R\operatorname{Hom}_{B/S}(\lambda,M)\to\) \(R\operatorname{Hom}_{B_F/S}(u^*\lambda,u^*M)\) onto \(R\operatorname{Hom}(B/B_F;\lambda,M)\).

Proof. By Lemma 3.2 (2), \(\operatorname{res}\) is the middle component of a map of short exact sequences of complexes whose other components are \(f\mapsto g\circ u^*f\) and the composition with \(Ug\) on \(\operatorname{Hom}^\bullet(U\lambda,UJ)[-1]\). Cocones of the components of a map of short exact sequences of complexes form a short exact sequence of complexes. The cocone of the composition with \(Ug\) is acyclic, \(Ug\) being a quasi-isomorphism between bounded below complexes of injective modules; so the map from the cocone of \(\operatorname{res}\) to \(R\operatorname{Hom}(B/B_F;\lambda,M)\) is a quasi-isomorphism. \(\square\)

Theorem 4.3 (the relative Ext groups of the morphism of nerves). Let \(\lambda\in D^b(B)\), \(M\in D^+(B)\) and a triangle \(X\to i_{0!}i_0^*\operatorname{ner}\lambda\to\operatorname{ner}\lambda\to X[1]\) be as in Theorem 2.3, and let \(\operatorname{Ext}^j(\mathcal N/\mathcal N_F;X,\operatorname{ner}M)\) be the relative Ext groups of the arrow topos of the morphism of ringed topoi \(N(u)\colon\mathcal N_F\to\mathcal N\) (Definition 3.1 of The arrow topos and relative Ext groups). Then \[ \begin{aligned}&\operatorname{Ext}^j(\mathcal N/\mathcal N_F;X,\operatorname{ner}M)\\&\quad\cong\operatorname{Ext}^{j+1}(B/B_F;\lambda,M)\end{aligned} \] for every \(j\).

Proof. By Proposition 3.2 of the lesson on the arrow topos, \(R\operatorname{Hom}(\mathcal N/\mathcal N_F;X,\operatorname{ner}M)\) is a cocone of the map \(R\operatorname{Hom}(X,\operatorname{ner}M)\to\) \(R\operatorname{Hom}(N(u)^*X,N(u)^*\operatorname{ner}M)\) given by the functor \(N(u)^*\), which is exact. By Proposition 3.3 and the remark after it, the isomorphisms \(\Theta\) and \(\Theta_F\) carry this map to the restriction map \(R\operatorname{Hom}_{B/S}(\lambda,M)[1]\to\) \(R\operatorname{Hom}_{B_F/S}(u^*\lambda,u^*M)[1]\), through \(N(u)^*\operatorname{ner}M=\operatorname{ner}u^*M\). Two maps isomorphic in the category of arrows of \(D(\mathbf Z)\) have isomorphic cocones [Tag 014A], and Lemma 4.2 identifies the cocone of the restriction map with \(R\operatorname{Hom}(B/B_F;\lambda,M)\). \(\square\)

Corollary 4.4 (deforming the target of a homomorphism). In Theorem 3.1 of Deforming the target of a morphism of diagrams, the groups \(E^j=\operatorname{Ext}^j(N(G)/N(F);L_{N(G)/S},f_G^*I)\) are isomorphic to \(\operatorname{Ext}^{j+1}(B_G/B_F;\ell^{\mathrm{ad}}_G,\varepsilon^*I)\), where \(B_G=B\): the obstruction to deforming \(G\) together with an extension of \(u\) to a given deformation \(F'\) of \(F\) lies in \(\operatorname{Ext}^3(B_G/B_F;\ell^{\mathrm{ad}}_G,\varepsilon^*I)\), the solutions form a torsor under \(\operatorname{Ext}^2\), and the automorphisms of a solution form \(\operatorname{Ext}^1\).

Proof. The groups \(E^j\) are relative Ext groups of the arrow topos of the morphism of Zariski topoi \(N(u)\), a cocone of the map \(R\operatorname{Hom}(L,f_G^*I)\to\) \(R\operatorname{Hom}(LN(u)^*L,f_F^*I)\) given by \(LN(u)^*\) (Proposition 3.2 of the lesson on the arrow topos), with \(L=L_{N(G)/S}\) and \(LN(u)^*f_G^*I=f_F^*I\), the projections \(G^n\to S\) and \(F^n\to S\) being flat. By Lemma 4.2 (2) of the lesson on big fppf sites, with the truncation index \(0\), applying \(L\varepsilon^*\) and composing with \(L\varepsilon^*f_G^*I\to\varepsilon^*f_G^*I=\operatorname{ner}(\varepsilon^*I)\) gives isomorphisms \(R\operatorname{Hom}(L,f_G^*I)\cong\) \(R\operatorname{Hom}(L\varepsilon^*L,\operatorname{ner}\varepsilon^*I)\), and likewise for \(F\); they commute with the maps given by \(LN(u)^*\) and \(N(u)^*\), as \(L\varepsilon^*LN(u)^*\cong N(u)^*L\varepsilon^*\) [Tag 0D6D]. So \(E^j\cong\operatorname{Ext}^j(\mathcal N/\mathcal N_F;L\varepsilon^*L,\operatorname{ner}\varepsilon^*I)\) [Tag 014A]. The triangle of the proof of Corollary 3.5 (2) of the lesson on the adjoint co-Lie complex is of the kind of Theorem 2.3, with \(\lambda=\ell^{\mathrm{ad}}_G\) and \(X=L\varepsilon^*L\), so Theorem 4.3 gives the assertion, and Theorem 3.1 there gives the obstruction, torsor and automorphisms. \(\square\)

For \(F\) the trivial group, \(B_F\) is \(T\), \(u^*\) is \(U\), and \(R\operatorname{Hom}(B/B_F;\lambda,M)\) is \(R\operatorname{Hom}_{B/S}(\lambda,M)\) (Exercise 8.1): the relative groups of Section 1 are those modulo the trivial group.

5. Equivariant deformations of the target

Let \(G\) be a category scheme over \(S\) and \(f\colon X\to Y\) a morphism of \(G\)-schemes, with the notation of Equivariant cotangent complexes: \(B_X=B[G,X]^{\mathrm o}\), \(B_Y=B[G,Y]^{\mathrm o}\), \(\varphi=N(G,f)\colon N(G,X)\to N(G,Y)\), and \(f_B^*\colon\operatorname{Mod}(B_Y)\to\operatorname{Mod}(B_X)\) the functor of Lemma 3.1 there, with \(\operatorname{ner}\circ f_B^*=\varphi_{\mathrm{big}}^*\circ\operatorname{ner}\). The fppf topoi of \(N(G,X)\) and \(N(G,Y)\) are those of the simplicial topoi of the nerves of \([G,X]^{\mathrm o}\) and \([G,Y]^{\mathrm o}\) (Proposition 1.5 (1) there). The underlying module of \(f_B^*M\) is the restriction of that of \(M\) from \(T/Y\) to the slice \(T/X\), an exact functor, and kernels and cokernels of modules of \(B_X\) and of \(B_Y\) are computed on the underlying modules (Proposition 6.2 (1) of the lesson on equivariant sheaves); so \(f_B^*\) is exact. The inverse image \(\varphi_{\mathrm{big}}^*\), the restriction to the slices at every level, is exact as well.

Definition 5.1. For a complex \(\lambda\) of modules of \(B_Y\) and \(M\in D^+(B_Y)\), choose bounded below complexes of injective modules \(J\) representing \(M\) and \(J_X\) representing \(f_B^*M\), and a map of complexes \(r\colon f_B^*J\to J_X\) representing the identity of \(f_B^*M\). The relative derived Hom of \(B_Y\) modulo \(B_X\) is the cocone \[ \begin{aligned}&R\operatorname{Hom}(B_Y/B_X;\lambda,M)\\&\quad=\operatorname{cocone}\bigl(\operatorname{Hom}^\bullet(\lambda,J)\\&\qquad\to\operatorname{Hom}^\bullet(f_B^*\lambda,J_X)\bigr)\end{aligned} \] of the map \(a\mapsto r\circ f_B^*a\), which computes the map \(R\operatorname{Hom}_{B_Y}(\lambda,M)\to\) \(R\operatorname{Hom}_{B_X}(f_B^*\lambda,f_B^*M)\) induced by \(f_B^*\). Its cohomology groups are the relative Ext groups \(\operatorname{Ext}^i(B_Y/B_X;\lambda,M)\); up to isomorphism they depend only on that map [Tag 014A].

Lemma 5.2. Let \(\lambda\) be a bounded above complex of modules of \(B_Y\), \(M\in D^+(B_Y)\), and \(J\), \(J_X\), \(r\) as in Definition 5.1. Choose maps of complexes \(\iota\colon\operatorname{ner}J\to I\) and \(\iota_X\colon\operatorname{ner}J_X\to I_X\) into bounded below complexes of injective modules of the fppf topoi of \(N(G,Y)\) and \(N(G,X)\), injective in every degree and quasi-isomorphisms, and a map of complexes \(s\colon\varphi_{\mathrm{big}}^*I\to I_X\) with \(s\circ\varphi_{\mathrm{big}}^*(\iota)=\iota_X\circ\operatorname{ner}(r)\), which exists as in Construction 3.1. Then:

  1. \(s\circ\varphi_{\mathrm{big}}^*\bigl(\iota\circ\operatorname{ner}(a)\bigr)=\iota_X\circ\) \(\operatorname{ner}(r\circ f_B^*a)\) for every \(a\in\operatorname{Hom}^\bullet(\lambda,J)\);
  2. the maps \(a\mapsto\iota\circ\operatorname{ner}(a)\) and \(b\mapsto\iota_X\circ\operatorname{ner}(b)\) are quasi-isomorphisms \(\operatorname{Hom}^\bullet(\lambda,J)\to\operatorname{Hom}^\bullet(\operatorname{ner}\lambda,I)\) and \(\operatorname{Hom}^\bullet(f_B^*\lambda,J_X)\to\) \(\operatorname{Hom}^\bullet(\operatorname{ner}f_B^*\lambda,I_X)\);
  3. they induce a quasi-isomorphism from \(R\operatorname{Hom}(B_Y/B_X;\lambda,M)\) onto the cocone of the map \(\psi\mapsto s\circ\varphi_{\mathrm{big}}^*\psi\), which computes the map \(R\operatorname{Hom}(\operatorname{ner}\lambda,\operatorname{ner}M)\to\) \(R\operatorname{Hom}(\operatorname{ner}f_B^*\lambda,\operatorname{ner}f_B^*M)\) induced by \(\varphi_{\mathrm{big}}^*\).

Proof. (1) The functors \(\varphi_{\mathrm{big}}^*\circ\operatorname{ner}\) and \(\operatorname{ner}\circ f_B^*\) agree on maps, so the left side is \(s\circ\varphi_{\mathrm{big}}^*(\iota)\circ\operatorname{ner}(f_B^*a)=\iota_X\circ\) \(\operatorname{ner}(r)\circ\operatorname{ner}(f_B^*a)\).

(2) These are the comparison maps (1.1) of the lesson on cohomology of equivariant modules for \(\lambda\) and \(M\), and for \(f_B^*\lambda\) and \(f_B^*M\): the nerve of a map into \(J\), followed by \(\iota\), a quasi-isomorphism into a K-injective complex [Tag 070J]. The simplicial topoi of the nerves of the internal categories \([G,Y]^{\mathrm o}\) and \([G,X]^{\mathrm o}\) of \(T\) are good two-sided pseudo-categories satisfying (I) and (E) (Proposition 5.1 there), so by Theorem 4.3 (2) there both maps are quasi-isomorphisms.

(3) By (1) the two maps of (2) form a commutative square with \(a\mapsto r\circ f_B^*a\) and \(\psi\mapsto s\circ\varphi_{\mathrm{big}}^*\psi\), hence a map of the short exact sequences of the two cocones whose outer components are quasi-isomorphisms; so the map of cocones is a quasi-isomorphism. By its defining identity, \(s\) is a quasi-isomorphism, as \(\varphi_{\mathrm{big}}^*(\iota)\), \(\iota_X\) and \(\operatorname{ner}(r)\) are; it maps into a K-injective complex, and it represents the identity of \(\varphi_{\mathrm{big}}^*\operatorname{ner}M=\operatorname{ner}f_B^*M\), \(\varphi_{\mathrm{big}}^*\) being exact; so \(\psi\mapsto s\circ\varphi_{\mathrm{big}}^*\psi\) computes the map induced by \(\varphi_{\mathrm{big}}^*\). \(\square\)

Theorem 5.3 (equivariant deformations of the target). Let the data 2.1 of Deforming the target of a morphism of diagrams be given, with \(g\) and \(h'\) flat, and let \(E^i\) be the groups of Theorem 2.2 there. Let \(t_{[}L^G_{Y/Z}\) be the equivariant cotangent complex of \(g\) (Theorem 2.4 of the lesson on equivariant cotangent complexes; \(g\) is flat, hence Tor independent of \(G\)), and put \(M=g_B^*K\).

  1. For all integers \(i\) and \(n\leq-i\) there is an isomorphism \(E^i\cong\operatorname{Ext}^i(B_Y/B_X;t_{[n}L^G_{Y/Z},M)\).
  2. For \(n+1\leq-i\), the transition map \(t_{[n}L^G_{Y/Z}\to t_{[n+1}L^G_{Y/Z}\) induces an isomorphism from \(\operatorname{Ext}^i(B_Y/B_X;t_{[n+1}L^G_{Y/Z},M)\) onto \(\operatorname{Ext}^i(B_Y/B_X;t_{[n}L^G_{Y/Z},M)\).
  3. Through these isomorphisms, the obstruction of Theorem 2.2 there is a class in \(\operatorname{Ext}^2(B_Y/B_X;t_{[-2}L^G_{Y/Z},M)\), the isomorphism classes of equivariant solutions form a torsor under \(\operatorname{Ext}^1(B_Y/B_X;t_{[-1}L^G_{Y/Z},M)\), and the automorphism group of an equivariant solution is isomorphic to \(\operatorname{Ext}^0(B_Y/B_X;t_{[0}L^G_{Y/Z},M)\). If \(g\) is a local complete intersection morphism, then \(t_{[n}L^G_{Y/Z}\cong L^G_{Y/Z}\) for \(n\leq-1\) (Proposition 2.5 (2) there), and by (1) \(E^i\cong\operatorname{Ext}^i(B_Y/B_X;L^G_{Y/Z},M)\) for every \(i\).

Proof. Write \(L=L_{N(G,Y)/N(G,Z)}\) and \(\underline K\) for the termwise quasi-coherent module of \(N(G,Z)\) belonging to \(K\). By Section 2 of the lesson on deforming the target and Proposition 3.2 (1) of the lesson on equivariant cotangent complexes, \(\operatorname{ner}M=\varepsilon^*N(G,g)^*\underline K\), and \(\operatorname{ner}f_B^*M=\varphi_{\mathrm{big}}^*\operatorname{ner}M=\varepsilon^*N(G,h)^*\underline K\).

(1) From the Zariski to the fppf topoi. By Proposition 3.2 of The arrow topos and relative Ext groups, \(E^i\) is the cohomology of a cocone of the map \(R\operatorname{Hom}(L,N(G,g)^*\underline K)\to\) \(R\operatorname{Hom}(L\varphi^*L,N(G,h)^*\underline K)\) given by \(L\varphi^*\) on the Zariski topoi, where \(L\varphi^*N(G,g)^*\underline K=N(G,h)^*\underline K\) (Conventions of the lesson on deforming the target). By Lemma 4.2 (2) of the lesson on big fppf sites, with the truncation index \(0\), applying \(L\varepsilon^*\) and composing with \(L\varepsilon^*N(G,g)^*\underline K\to\varepsilon^*N(G,g)^*\underline K\) gives an isomorphism \(R\operatorname{Hom}(L,N(G,g)^*\underline K)\cong\) \(R\operatorname{Hom}(L\varepsilon^*L,\operatorname{ner}M)\), and likewise over \(N(G,X)\); these commute with the maps given by \(L\varphi^*\) and \(\varphi_{\mathrm{big}}^*\), as \(L\varepsilon^*L\varphi^*\cong\varphi_{\mathrm{big}}^*L\varepsilon^*\) [Tag 0D6D]. So \(E^i\cong H^iC\) for a cocone \(C\) of the map \(R\operatorname{Hom}(L\varepsilon^*L,\operatorname{ner}M)\to\) \(R\operatorname{Hom}(\varphi_{\mathrm{big}}^*L\varepsilon^*L,\operatorname{ner}f_B^*M)\) induced by \(\varphi_{\mathrm{big}}^*\) [Tag 014A].

Truncation. Represent \(L\varepsilon^*L\) by a complex \(P\), and choose \(J\), \(J_X\) and \(r\) as in Definition 5.1 for \(M\), and \(I\), \(I_X\) and \(s\) as in Lemma 5.2. Take for \(C\) the cocone of \(\psi\mapsto s\circ\varphi_{\mathrm{big}}^*\psi\) on \(\operatorname{Hom}^\bullet(P,I)\to\operatorname{Hom}^\bullet(\varphi_{\mathrm{big}}^*P,I_X)\), and for \(C_n\) the same cocone with \(\tau_{\geq n}P\) in place of \(P\), which represents \(t_{[n}L\varepsilon^*L\). Composition with the maps of complexes \(P\to\tau_{\geq n}P\) and \(\varphi_{\mathrm{big}}^*P\to\varphi_{\mathrm{big}}^*\tau_{\geq n}P=\tau_{\geq n}\varphi_{\mathrm{big}}^*P\), the functor \(\varphi_{\mathrm{big}}^*\) being exact, commutes with the two maps \(\psi\mapsto s\circ\varphi_{\mathrm{big}}^*\psi\), so it gives a map of cocones \(C_n\to C\). As \(\operatorname{ner}M\) and \(\operatorname{ner}f_B^*M\) are modules, (0.1) of the lesson on equivariant cotangent complexes shows that it induces bijections on \(H^j\) of the two Hom complexes for \(j\leq-n\). For \(n\leq-i\) this holds for \(j=i-1\) and \(j=i\), and the five lemma, applied to the long exact sequences of the cocones, shows that \(H^iC_n\to H^iC\) is bijective.

Nerves. By (2.1) of the lesson on equivariant cotangent complexes, \(\operatorname{ner}\lambda\cong t_{[n}L\varepsilon^*L\) for a bounded complex \(\lambda\) of modules of \(B_Y\) representing \(t_{[n}L^G_{Y/Z}\). So \(C_n\) is isomorphic to the cocone of Lemma 5.2 (3) for \(\lambda\) and \(M\), both being cocones of isomorphic maps [Tag 014A], and by Lemma 5.2 (3) \(H^iC_n\cong\operatorname{Ext}^i(B_Y/B_X;\lambda,M)\). Together, \(E^i\cong H^iC\cong H^iC_n\cong\) \(\operatorname{Ext}^i(B_Y/B_X;t_{[n}L^G_{Y/Z},M)\).

(2) The transition map is the canonical map \(t_{[n}L^G_{Y/Z}\to\tau_{\geq n+1}t_{[n}L^G_{Y/Z}\cong t_{[n+1}L^G_{Y/Z}\) (Theorem 2.4 there and the remark after it). Represent \(t_{[n}L^G_{Y/Z}\) by a bounded complex \(\lambda\) and the transition map by \(\lambda\to\tau_{\geq n+1}\lambda\); as \(f_B^*\) is exact, \(f_B^*\tau_{\geq n+1}\lambda=\tau_{\geq n+1}f_B^*\lambda\). Composition with these maps gives a map from the cocone of Definition 5.1 for \(\tau_{\geq n+1}\lambda\) to that for \(\lambda\). By (0.1) there, applied to the modules \(M\) and \(f_B^*M\), it induces bijections on \(\operatorname{Ext}^j_{B_Y}(-,M)\) and \(\operatorname{Ext}^j_{B_X}(f_B^*(-),f_B^*M)\) for \(j\leq-(n+1)\), so for \(j=i-1\) and \(j=i\), and the five lemma gives the assertion.

(3) The first statement is Theorem 2.2 there, through (1) with \(n=-i\) for \(i=2,1,0\). If \(g\) is a local complete intersection morphism, the transition maps \(t_{[n}L^G_{Y/Z}\to t_{[n+1}L^G_{Y/Z}\) are isomorphisms for \(n<-1\), and \(L^G_{Y/Z}=t_{[-1}L^G_{Y/Z}\) (Proposition 2.5 (2) there). For \(i\geq1\) take \(n=-i\leq-1\) in (1); for \(i\leq0\) take \(n=-1\leq-i\). \(\square\)

6. Two deformations of the target

Let \(u\colon F\to G\) be as in Section 3, with the notation of 3.9 and Proposition 3.10 of the lesson on the adjoint co-Lie complex: \(P=N(G,G)\to N\) and \(P_F=N(F,F)\to N(F)\) are the universal torsors under \(G^{\mathrm o}\) and \(F^{\mathrm o}\), \(P'=N(F)\times_NP\), \(h\colon P_F\to P'\) and \(h_G\colon P'\to P\) are the morphisms of 3.9 there, and \(a\colon N(F^{\mathrm o},P_F)\to N(F)\) is the morphism of Section 1 there for the torsor \(P_F\). This section compares the third maps of three triangles there:

The comparison is a relation between Kodaira–Spencer classes over a fibre product, with a sign. Through the canonical identifications of Section 2, it turns the difference formula for the obstructions attached to two deformations of the target (Proposition 3.5 of the lesson on deformations of flat group schemes) into a formula in relative Ext groups of the classifying topos of \(F\).

For composable morphisms \(A\to B\to C\) of diagrams of schemes of the same type, write the transitivity triangle (Proposition 1.3 of Deformations of diagrams of schemes) as \[ \begin{aligned}&L_{B/C}|_A\xrightarrow{\ v_{ABC}\ }L_{A/C}\xrightarrow{\ w_{ABC}\ }L_{A/B}\\&\quad\xrightarrow{\ K(A/B/C)\ }L_{B/C}|_A[1],\end{aligned} \] where \(E|_A\) is the derived inverse image of \(E\) along the morphism to the diagram on which \(E\) lives, and \(v_{ABC}\), \(w_{ABC}\) are functoriality maps. For a commutative square of diagrams with the top \(A'\to A\), the bottom \(B'\to B\) and the vertical morphisms \(A'\to B'\), \(A\to B\), its functoriality map is the map \(L_{A/B}|_{A'}\to L_{A'/B'}\) of Proposition 1.2 there. Functoriality maps of composite squares are the composites of the functoriality maps (Theorem 2.2 of The cotangent complex).

Lemma 6.1 (Kodaira–Spencer classes over a fibre product). Let \(X\to Y\), \(Y\to Z_1\), \(Y\to Z_2\), \(Z_1\to W\), \(Z_2\to W\) and \(W\to V\) be morphisms of identity type of diagrams of schemes indexed by one small category, such that the square formed by the four middle morphisms commutes and is cartesian at every level, and every level of \(Z_2\to W\) is flat. For \(\{i,i'\}=\{1,2\}\), let \(\xi_i\colon L_{Z_i/W}|_Y\to L_{Y/Z_{i'}}\) be the functoriality map of the square with the top \(Y\to Z_i\) and the bottom \(Z_{i'}\to W\). Then \(\xi_1\) and \(\xi_2\) are isomorphisms, and \[ \begin{aligned}&K(Z_2/W/V)|_X[1]\circ\xi_2^{-1}|_X[1]\\&\qquad\circ K(X/Y/Z_1)\\&\quad=-K(Z_1/W/V)|_X[1]\\&\qquad\circ\xi_1^{-1}|_X[1]\circ K(X/Y/Z_2)\end{aligned} \] as maps \(L_{X/Y}\to L_{W/V}|_X[2]\).

Proof. At every level the square is cartesian and \(Z_1\) and \(Z_2\) are Tor independent over \(W\), \(Z_2\to W\) being flat; so \(\xi_1\) and \(\xi_2\) are isomorphisms at every level (Proposition 1.4 of the lesson on deformations of diagrams), hence isomorphisms. Write \(v_i=v_{YZ_iW}\) and \(w_i=w_{YZ_iW}\). Then \(w_iv_i=0\), two consecutive maps of a distinguished triangle composing to zero [Tag 0146], and \(w_{i'}v_i=\xi_i\), the composite of the squares of \(v_i\) and of \(w_{i'}\) being the square of \(\xi_i\).

Splitting. The map \(s=v_2\xi_2^{-1}\colon L_{Y/Z_1}\to L_{Y/W}\) satisfies \(w_1s=\xi_2\xi_2^{-1}=\mathrm{id}\), so the sum \((v_1,s)\colon L_{Z_1/W}|_Y\oplus L_{Y/Z_1}\to L_{Y/W}\) is an isomorphism [Tag 05QT]. Put \(\sigma=v_1\xi_1^{-1}w_2+v_2\xi_2^{-1}w_1\). Then \(\sigma v_1=v_1\xi_1^{-1}\xi_1+v_2\xi_2^{-1}w_1v_1=v_1\) and \(\sigma s=v_1\xi_1^{-1}w_2v_2\xi_2^{-1}+v_2\xi_2^{-1}w_1s=s\), so \(\sigma=\mathrm{id}\).

Naturality in the base. For \(i=1,2\), the map of triples of rings of \(\operatorname{Top}(X_{\mathrm{Zar}})\) from the inverse images of \(\mathcal O_W\), \(\mathcal O_Y\), \(\mathcal O_X\) to those of \(\mathcal O_{Z_i}\), \(\mathcal O_Y\), \(\mathcal O_X\), the identity on the last two, gives by the naturality of the transitivity triangle a morphism from the triangle of \(X\to Y\to W\) to that of \(X\to Y\to Z_i\), with the components \(w_i|_X\), the functoriality map \(L_{X/W}\to L_{X/Z_i}\), the identity of \(L_{X/Y}\) and \(w_i|_X[1]\) (as in the proof of Proposition 3.5 (2) of the lesson on deformations of flat group schemes). Its third square gives \(K(X/Y/Z_i)=w_i|_X[1]\circ K(X/Y/W)\). In the same way \(K(X/Y/W)=w_{YWV}|_X[1]\circ\) \(K(X/Y/V)\), and \(K(Y/W/V)\circ w_{YWV}=0\) [Tag 0146]; so \(K(Y/W/V)|_X[1]\circ K(X/Y/W)=0\).

Naturality in the source. For \(i=1,2\), let \(\mathcal A\), \(\mathcal B\), \(\mathcal C_i\) be the inverse images of \(\mathcal O_V\), \(\mathcal O_W\), \(\mathcal O_{Z_i}\) on \(\operatorname{Top}(Y_{\mathrm{Zar}})\). The transitivity triangle of \((\mathcal A,\mathcal B,\mathcal C_i)\) is the inverse image along \(Y\to Z_i\) of that of \(Z_i\to W\to V\), inverse images preserving the polynomial constructions (proof of Proposition 1.3 of the lesson on deformations of diagrams); it is a triangle of complexes of flat \(\mathcal C_i\)-modules. The naturality for the map of triples \((\mathcal A,\mathcal B,\mathcal C_i)\to(\mathcal A,\mathcal B,\mathcal O_Y)\) gives \(\mathcal C_i\)-linear maps into the triangle of \(Y\to W\to V\) forming a morphism of triangles in \(D(\mathcal C_i)\); through the adjunction between \(-\otimes_{\mathcal C_i}\mathcal O_Y\) and the restriction of scalars, it is a morphism from the derived inverse image of the triangle of \(Z_i\to W\to V\) to the triangle of \(Y\to W\to V\), with the components the identity of \(L_{W/V}|_Y\), the functoriality map \(L_{Z_i/V}|_Y\to L_{Y/V}\), \(v_i\) and the identity of \(L_{W/V}|_Y[1]\). Its third square gives \(K(Y/W/V)\circ v_i=K(Z_i/W/V)|_Y\).

Conclusion. By the three steps, \[ \begin{aligned}0&=K(Y/W/V)|_X[1]\circ\sigma|_X[1]\\&\qquad\circ K(X/Y/W)\\&=\sum_{i=1,2}K(Z_i/W/V)|_X[1]\\&\qquad\circ\xi_i^{-1}|_X[1]\circ K(X/Y/Z_{i'}),\end{aligned} \] which is the assertion. \(\square\)

Lemma 6.2 (the connecting maps of a homomorphism). In \(D(N(F)_{\mathrm{Zar}})\), \[ \begin{aligned}&LN(u)^*(\mathrm{at})[1]\circ\delta\\&\quad=-K(N(F)/N/S)[1]\circ\mathrm{at}_u\end{aligned} \] as maps \(\ell_u\to LN(u)^*L_{N/S}[2]\).

Proof. Put \(Q_F=N(F^{\mathrm o},P_F)\), \(Y=N(F^{\mathrm o},P')\), \(Z_1=N(F^{\mathrm o},N(F))\), \(Z_2=N(F^{\mathrm o},P)\), \(W=N(F^{\mathrm o},N)\) and \(V=N(F^{\mathrm o})\), with \(F^{\mathrm o}\) acting through \(u\) on \(P'\) and \(P\) and trivially on \(N(F)\) and \(N\): diagrams indexed by \(\Delta^{\mathrm{op}}\times\Delta^{\mathrm{op}}\), \(V\) being constant in the second index, with the morphisms induced by \(h\), \(h_G\), the projections and the structure morphisms. As \(P'=N(F)\times_NP\), the square formed by \(Y\to Z_1\), \(Y\to Z_2\), \(Z_1\to W\) and \(Z_2\to W\) is cartesian at every level, and \(Z_2\to W\) has the flat levels \(\mathrm{id}_{F^r}\times\varphi_n\). Put also \(Q_G=N(G^{\mathrm o},P)\), \(W_G=N(G^{\mathrm o},N)\) and \(V_G=N(G^{\mathrm o})\), with the morphisms \(Z_2\to Q_G\), \(W\to W_G\) and \(V\to V_G\) whose levels are \(u^r\times\mathrm{id}\). The squares they form with \(Z_2\to W\to V\) and \(Q_G\to W_G\to V_G\) are cartesian, and \(Q_G\to W_G\), \(W_G\to V_G\) and \(Q_G\to V_G\) are flat at every level. Let \(a_G\colon Q_G\to N\) be the morphism of Section 1 there for the torsor \(P\), and \(e\colon Q_F\to Q_G\) the composite \(Q_F\to Y\to Z_2\to Q_G\). Then \(N(u)\circ a=a_G\circ e\), both sending \((k;f;x)\) to \(u^n(f)\).

Both sides of the assertion lie in \(D^b_{\mathrm{qc}}(N(F)_{\mathrm{Zar}})\), so by Theorem 1.4 (1) there it suffices to compare their images under \(a^*\). Let \(\mu\colon a^*\ell_u\to L_{Q_F/Y}\) be the isomorphism of Proposition 3.10 (1) there.

  1. By the proof of Theorem 3.11 there, \(a^*(\mathrm{at}_u)=\alpha_1^{-1}[1]\circ\) \(K(Q_F/Y/Z_2)\circ\mu\), where \(\alpha_1\colon a^*L_{N(F)/N}\to L_{Y/Z_2}|_{Q_F}\) is the functoriality map of the square with the top \(Y\to N(F)\) and the bottom \(Z_2\to N\), pulled back to \(Q_F\).
  2. By the proof of Proposition 3.10 (1) there, \(a^*(\delta)=\alpha_2^{-1}[1]\circ K(Q_F/Y/Z_1)\circ\mu\), where \(\alpha_2\colon a^*LN(u)^*\ell_{P/N}\to L_{Y/Z_1}|_{Q_F}\) is the composite of \(a^*\) of the isomorphism \(\ell(h_G)\colon LN(u)^*\ell_{P/N}\to\ell_{P'/N(F)}\) of Lemma 1.8 (2), (4) there, the isomorphism of Theorem 1.5 (1) there for the torsor \(P'\), and the functoriality map of the square with the top \(Y\to N(G^{\mathrm o},P')\) and the bottom \(Z_1\to N(G^{\mathrm o},N(F))\).
  3. By Theorem 1.5 (1), (3) there for the torsor \(P\), with the truncations \(t_{[-1}\) the identity, all objects lying in \(D^{[-1,0]}\) (proof of Theorem 3.3 there), \(a_G^*(\mathrm{at})=\alpha_3^{-1}[1]\circ\) \(K(Q_G/W_G/V_G)\circ\mu_G\), where \(\mu_G\colon a_G^*\ell_{P/N}\to L_{Q_G/W_G}\) is the isomorphism of Theorem 1.5 (1) there and \(\alpha_3\colon a_G^*L_{N/S}\to L_{W_G/V_G}|_{Q_G}\) the functoriality map of the square with the top \(W_G\to N\) and the bottom \(V_G\to S\). And \(a^*LN(u)^*(\mathrm{at})=e^*a_G^*(\mathrm{at})\) [Tag 0D6D].
  4. As in the step naturality in the source of the proof of Lemma 6.1, the maps of triples of rings from the inverse images of the rings of \(V_G\), \(W_G\), \(Q_G\) to those of \(V\), \(W\), \(Z_2\), and from the inverse images of the rings of \(S\), \(N\), \(N(F)\) to those of \(V\), \(W\), \(Z_1\), give \(K(Z_2/W/V)\circ\gamma_1=\gamma_2[1]\circ\) \(K(Q_G/W_G/V_G)|_{Z_2}\) and \(K(Z_1/W/V)\circ\gamma_3=\gamma_4[1]\circ\) \(K(N(F)/N/S)|_{Z_1}\). Here \(\gamma_1\colon L_{Q_G/W_G}|_{Z_2}\to L_{Z_2/W}\), \(\gamma_2\colon L_{W_G/V_G}|_{Z_2}\to L_{W/V}|_{Z_2}\), \(\gamma_3\colon L_{N(F)/N}|_{Z_1}\to L_{Z_1/W}\) and \(\gamma_4\colon L_{N/S}|_{Z_1}\to L_{W/V}|_{Z_1}\) are the functoriality maps of the cartesian squares, isomorphisms by Proposition 1.4 of the lesson on deformations of diagrams: \(Q_G\), \(W_G\) are flat over \(W_G\), \(V_G\), and \(W\), \(Z_1\) are the products with \(V\) over \(S\) of \(N\), \(N(F)\), with \(V\) flat over \(S\).
  5. The identifications agree. By the characterization of \(\ell(h_G)\) in Lemma 1.8 (2) there, \(\alpha_2\) is the composite of \(e^*\mu_G\) with the functoriality map of the composite square with the top \(Y\to Q_G\) and the bottom \(Z_1\to W_G\); this square is also the composite of the squares of \(\gamma_1\) and \(\xi_2\), so \(\alpha_2=\xi_2|_{Q_F}\circ\gamma_1|_{Q_F}\circ e^*\mu_G\). In the same way \(\alpha_1=\xi_1|_{Q_F}\circ\gamma_3|_{Q_F}\), both being functoriality maps of the square with the top \(Y\to N(F)\) and the bottom \(Z_2\to N\), and \(\gamma_2|_{Q_F}\circ e^*\alpha_3=\gamma_4|_{Q_F}\), both being functoriality maps of the square with the top \(W\to N\) and the bottom \(V\to S\), pulled back to \(Q_F\).

By 2, 3, 4 and 5, \[ \begin{aligned}&a^*\bigl(LN(u)^*(\mathrm{at})[1]\circ\delta\bigr)\\&\quad=\gamma_4^{-1}|_{Q_F}[2]\circ K(Z_2/W/V)|_{Q_F}[1]\\&\qquad\circ\xi_2^{-1}|_{Q_F}[1]\circ K(Q_F/Y/Z_1)\circ\mu,\end{aligned} \] and by 1, 4 and 5, \[ \begin{aligned}&a^*\bigl(K(N(F)/N/S)[1]\circ\mathrm{at}_u\bigr)\\&\quad=\gamma_4^{-1}|_{Q_F}[2]\circ K(Z_1/W/V)|_{Q_F}[1]\\&\qquad\circ\xi_1^{-1}|_{Q_F}[1]\circ K(Q_F/Y/Z_2)\circ\mu.\end{aligned} \] Lemma 6.1, with \(X=Q_F\), says that the right sides are opposite. \(\square\)

Theorem 6.3 (connecting maps and the canonical identifications). Let \(M\in D^+(B_F)\). Let \(\Theta_1\colon\operatorname{Ext}^{j+1}_{B_F/S}(u^*\ell^{\mathrm{ad}}_G,M)\to\) \(\operatorname{Ext}^j(N(u)^*L\varepsilon^*L_{N/S},\operatorname{ner}M)\) be the isomorphism of Theorem 2.3, for the group \(F\), for \(u^*\ell^{\mathrm{ad}}_G\), \(M\) and the image under \(N(u)^*\) of the triangle of the proof of Corollary 3.5 (2) of the lesson on the adjoint co-Lie complex (as in Proposition 3.3), and \(\Theta_C\colon\operatorname{Ext}^{j+2}_{B_F/S}(C_u,M)\to\) \(\operatorname{Ext}^{j+1}(L\varepsilon^*L_{N(F)/N},\operatorname{ner}M)\) the isomorphism of Theorem 2.3 for the triangle of the proof of Corollary 3.12 (1) there (Remark 2.4). Let \(\delta_C\colon C_u\to u^*\ell^{\mathrm{ad}}_G[1]\) be the third map of the triangle of Proposition 3.10 (2) there. Then for every \(f\in\operatorname{Ext}^{j+1}_{B_F/S}(u^*\ell^{\mathrm{ad}}_G,M)\) \[ \begin{aligned}&\Theta_C(f[1]\circ\delta_C)\\&\quad=-\Theta_1(f)[1]\circ L\varepsilon^*K(N(F)/N/S).\end{aligned} \]

Proof. The third map of the first triangle is \(\mathrm{at}_1=L\varepsilon^*LN(u)^*(\mathrm{at})\), through \(N(u)^*L\varepsilon^*\cong L\varepsilon^*LN(u)^*\) [Tag 0D6D], that of the second is \(\mathrm{at}_C=L\varepsilon^*(\mathrm{at}_u)\), through \(\operatorname{ner}C_u\cong L\varepsilon^*\ell_u\), and \(\operatorname{ner}(\delta_C)=L\varepsilon^*(\delta)\) (Proposition 3.10 (2) there). By Lemma 6.2, \(\mathrm{at}_1[1]\circ\operatorname{ner}(\delta_C)=\) \(-L\varepsilon^*K(N(F)/N/S)[1]\circ\mathrm{at}_C\). Let \(\psi_1\in\operatorname{Hom}(N(u)^*L\varepsilon^*L_{N/S},I_F^{\mathrm{rel}}[j])\) be the map of Theorem 2.3 for \(f\), with \(\psi_1[1]\circ\mathrm{at}_1=c_F\circ\operatorname{ner}(f)\). Then \[ \begin{aligned}&c_F\circ\operatorname{ner}(f[1]\circ\delta_C)\\&\quad=\psi_1[2]\circ\mathrm{at}_1[1]\circ\operatorname{ner}(\delta_C)\\&\quad=-\psi_1[2]\circ L\varepsilon^*K(N(F)/N/S)[1]\\&\qquad\circ\mathrm{at}_C,\end{aligned} \] so \(-\psi_1[1]\circ L\varepsilon^*K(N(F)/N/S)\) is the map of Theorem 2.3 for \(f[1]\circ\delta_C\), by its uniqueness. Its image in \(\operatorname{Ext}^{j+1}(L\varepsilon^*L_{N(F)/N},\operatorname{ner}M)\) is \(-\Theta_1(f)[1]\circ L\varepsilon^*K(N(F)/N/S)\). \(\square\)

Corollary 6.4 (two deformations of the target in relative Ext groups). In the situation of Deformations of flat group schemes and of homomorphisms, let \(G'\), \(G''\) be deformations of \(G\) over \(S'\), \(d\in\operatorname{Ext}^1(L_{N/S},f_G^*I)\) the element carrying the class of \(G''\) to that of \(G'\) (Theorem 2.1 (2) there), and \(D\in\operatorname{Ext}^2_{B/S}(\ell^{\mathrm{ad}}_G,\varepsilon^*I)\) the class corresponding to \(d\) under the isomorphism of Lemma 4.2 (2) of the lesson on big fppf sites and \(\Theta\) for the triangle of Corollary 3.5 (2) of the lesson on the adjoint co-Lie complex. Under the isomorphisms of Lemma 4.2 (2) and \(\Theta_C\) (Theorem 6.3, with \(M=\varepsilon^*I\)), the difference \(\omega(u,G')-\omega(u,G'')\in\) \(\operatorname{Ext}^2(L_{N(F)/N},f_F^*I)\) of the obstructions of Theorem 3.3 there corresponds to \[ -(\operatorname{res}D)[1]\circ\delta_C\in\operatorname{Ext}^3_{B_F/S}(C_u,\varepsilon^*I), \] where \(\operatorname{res}\) is the restriction along \(u\) of Construction 3.1.

Proof. By Proposition 3.5 (2) there, for the homomorphism \(u\colon F\to G\), \(\omega(u,G')-\omega(u,G'')=\) \((N(u)^*d)[1]\circ K(N(F)/N/S)\). The isomorphisms of Lemma 4.2 (2), applying \(L\varepsilon^*\) and composing with the map to \(\varepsilon^*\) of the coefficients, are natural in the first argument, so they carry composition with \(K(N(F)/N/S)\) to composition with \(L\varepsilon^*K(N(F)/N/S)\); and they commute with the maps given by \(LN(u)^*\) and \(N(u)^*\) [Tag 0D6D], as in the proof of Corollary 4.4. So \(N(u)^*d\) corresponds to \(N(u)^*\Theta(D)\), which is \(\Theta_1(\operatorname{res}D)\) by Proposition 3.3. Hence the difference corresponds to \(\Theta_1(\operatorname{res}D)[1]\circ\) \(L\varepsilon^*K(N(F)/N/S)=\) \(-\Theta_C\bigl((\operatorname{res}D)[1]\circ\delta_C\bigr)\), by Theorem 6.3. \(\square\)

7. Dual coefficients

For modules \(E\), \(L\) of \(B\), write \(\mathcal Hom(E,L)\) for the internal Hom of modules of the ringed topos \(B\), and \(\mathcal Hom^\bullet\) for the internal Hom complex of two complexes. The forgetful functor \(U\) is the restriction to the localization \(B/G_\rho\) (Lemma 3.7 (1) of the lesson on the adjoint co-Lie complex), so \(U\mathcal Hom(E,L)=\mathcal Hom(UE,UL)\) [Tag 0E8H]. Tensor products of modules of \(B\) are computed on the underlying modules [Tag 03EL], so a module of \(B\) is flat if its underlying module is flat, and every object of \(D^-(B)\) is represented by a bounded above complex of flat modules: every module is a quotient of a module \(\operatorname{ind}(Q)\) with \(Q\) free, whose underlying module is flat (proof of Lemma 3.1 (2) of the lesson on equivariant cotangent complexes), and [Tag 05T7] applies. For a homomorphism \(u\colon F\to G\), the restriction \(u^*\) does not change the underlying modules, and it carries tensor products and internal Hom of modules of \(B\) to those of \(B_F\): both are the tensor products and internal Hom of the underlying modules, with the action of \(G\), respectively of \(F\) through \(u\), induced on them.

Lemma 7.1. Let \(P\) be a bounded above complex of flat modules of \(B\) and \(J\) a bounded below complex of injective modules of \(B\), representing \(K\in D^-(B)\) and \(M\in D^+(B)\), and put \(H=\mathcal Hom^\bullet(P,J)\).

  1. \(H\) is a bounded below complex of injective modules representing \(R\mathcal Hom(K,M)\), and \(\Gamma(B,\mathcal Hom^\bullet(P,X))=\operatorname{Hom}^\bullet(P,X)\) for every complex \(X\).
  2. There is an isomorphism of complexes \(\zeta\colon\) \(\mathcal Hom^\bullet(P,\operatorname{coind}UJ)\to\) \(\operatorname{coind}UH\) with \(\zeta\circ\mathcal Hom^\bullet(P,\eta_J)=\eta_H\). Hence \((\mathrm{id},\zeta)\colon\mathcal Hom^\bullet(P,J^{\mathrm{rel}})\to H^{\mathrm{rel}}\) is an isomorphism of complexes.
  3. Let \(u\colon F\to G\) be a homomorphism and \(J_F\), \(g\colon u^*J\to J_F\) as in Construction 3.1. Then \(u^*H=\mathcal Hom^\bullet(u^*P,u^*J)\), the complex \(H_F=\mathcal Hom^\bullet(u^*P,J_F)\) is a bounded below complex of injective modules of \(B_F\) representing \(R\mathcal Hom(u^*K,u^*M)\), and the map \(g_H=\mathcal Hom^\bullet(u^*P,g)\colon u^*H\to H_F\) is a quasi-isomorphism.

Proof. (1) For a module \(E\) of \(B\), \(\operatorname{Hom}(E,\mathcal Hom(P^a,J^b))=\) \(\operatorname{Hom}(E\otimes P^a,J^b)\) [Tag 03EO] is exact in \(E\), \(P^a\) being flat and \(J^b\) injective; so \(\mathcal Hom(P^a,J^b)\) is injective. As \(P\) is bounded above and \(J\) bounded below, every term of \(H\) is a finite product of such modules, and \(H\) is bounded below. The complex \(J\) is K-injective [Tag 070J], so \(H\) represents \(R\mathcal Hom(K,M)\) [Tag 0A94]. Global sections of internal Hom are the Hom groups.

(2) For modules \(E\), \(L\) of \(B\) and a module \(N\) of \(T\), the adjunctions give bijections, natural in \(E\), \(L\) and \(N\), \[ \begin{aligned}&\quad \operatorname{Hom}_B\bigl(E,\mathcal Hom(L,\operatorname{coind}N)\bigr)\\ &\quad =\operatorname{Hom}_B(E\otimes L,\operatorname{coind}N)\\ &\quad \quad =\operatorname{Hom}_T(UE\otimes UL,N)\\ &\quad =\operatorname{Hom}_T\bigl(UE,\mathcal Hom(UL,N)\bigr)\\ &\quad \quad\\ &\quad =\operatorname{Hom}_B\bigl(E,\operatorname{coind}\mathcal Hom(UL,N)\bigr),\end{aligned} \] so by the Yoneda lemma an isomorphism \(\mathcal Hom(L,\operatorname{coind}N)\cong\operatorname{coind}\mathcal Hom(UL,N)\), natural in \(L\) and \(N\). Applied termwise, \(\operatorname{coind}\) commuting with products, and with \(\mathcal Hom(UP^a,UJ^b)=U\mathcal Hom(P^a,J^b)\), it gives \(\zeta\); the differentials correspond by naturality. Under these bijections, a map \(\phi\colon E\to\mathcal Hom(L,\operatorname{coind}N)\) corresponds to the map into \(\operatorname{coind}\mathcal Hom(UL,N)\) whose adjoint \(UE\to\mathcal Hom(UL,N)\) is the tensor adjoint of \(\mathrm{cu}_N\circ U\phi^\flat\), where \(\phi^\flat\colon E\otimes L\to\operatorname{coind}N\) is the tensor adjoint of \(\phi\). For \(\phi=\mathcal Hom^\bullet(P,\eta_J)\), \(\phi^\flat=\eta_J\circ\mathrm{ev}\) with the evaluation \(\mathrm{ev}\colon H\otimes P\to J\), and \(\mathrm{cu}\circ U\eta_J\circ U\mathrm{ev}=U\mathrm{ev}\), whose tensor adjoint is the identification \(UH=\mathcal Hom^\bullet(UP,UJ)\). This is also the adjoint of \(\eta_H\), as \(\mathrm{cu}\circ U\eta_H=\mathrm{id}\). So \(\zeta\circ\mathcal Hom^\bullet(P,\eta_J)=\eta_H\). Finally \(\mathcal Hom^\bullet(P,-)\) commutes with cocones, as \(\operatorname{Hom}^\bullet(P,-)\) does (Conventions), so \(\mathcal Hom^\bullet(P,J^{\mathrm{rel}})\) is the cocone of \(\mathcal Hom^\bullet(P,\eta_J)\), and \((\mathrm{id},\zeta)\) is a map of cocones, an isomorphism.

(3) The first equality holds because \(u^*\) carries internal Hom to internal Hom (see above). The complex \(u^*P\) is a bounded above complex of flat modules representing \(u^*K\), \(u^*\) being exact, so (1) for \(F\) applies to \(H_F\). The underlying map of \(g_H\) is \(\mathcal Hom^\bullet(UP,Ug)\), and \(Ug\colon UJ\to UJ_F\) is a quasi-isomorphism between bounded below complexes of injective modules, hence a homotopy equivalence, which \(\mathcal Hom^\bullet(UP,-)\) preserves; so \(g_H\) is a quasi-isomorphism, quasi-isomorphisms being detected on the underlying complexes. \(\square\)

Theorem 7.2 (dual coefficients). Let \(K\in D^-(B)\) and \(M\in D^+(B)\), and choose \(P\), \(J\) and \(H\) as in Lemma 7.1.

  1. The map \(\Phi=\Gamma\bigl(B,(\mathrm{id},\zeta)\bigr)\) is an isomorphism of complexes from \(R\operatorname{Hom}_{B/S}(K,M)=\operatorname{Hom}^\bullet(P,J^{\mathrm{rel}})\) onto \(\operatorname{Hom}^\bullet(\mathcal O,H^{\mathrm{rel}})=\) \(R\Gamma\bigl(B/S,R\mathcal Hom(K,M)\bigr)\), computed with the resolution \(H\). On cohomology it does not depend on the choices, and it is natural in \(K\).
  2. For a homomorphism \(u\colon F\to G\), \(\Phi\) and the isomorphism \(\Phi_F\) for \(u^*K\), \(u^*M\), \(u^*P\), \(J_F\) commute with the restriction maps of Construction 3.1, formed with \(g\) and with \(g_H\).
  3. With these choices, the map \(f\mapsto g\circ u^*f\) of Definition 4.1 for \(K\) and \(M\) is, through \(\operatorname{Hom}^\bullet(P,J)=\operatorname{Hom}^\bullet(\mathcal O,H)\) and \(\operatorname{Hom}^\bullet(u^*P,J_F)=\operatorname{Hom}^\bullet(\mathcal O,H_F)\), the map \(s\mapsto g_H\circ u^*s\) of Definition 4.1 for \(\mathcal O\) and \(R\mathcal Hom(K,M)\). So \(R\operatorname{Hom}(B/B_F;K,M)=\) \(R\Gamma\bigl(B/B_F,R\mathcal Hom(K,M)\bigr)\).
  4. If \(UK\) is perfect, then \(K\) is perfect and the evaluation map \(K^\vee\otimes^{\mathbf L}M\to R\mathcal Hom(K,M)\) is an isomorphism, natural in \(K\) and \(M\) [Tag 08JJ] and compatible with restriction along homomorphisms. So \(\operatorname{Ext}^j_{B/S}(K,M)\cong\) \(H^j(B/S,K^\vee\otimes^{\mathbf L}M)\) and \(\operatorname{Ext}^j(B/B_F;K,M)\cong\) \(H^j(B/B_F,K^\vee\otimes^{\mathbf L}M)\).

Proof. (1) \(\operatorname{Hom}^\bullet(P,J^{\mathrm{rel}})=\Gamma(B,\mathcal Hom^\bullet(P,J^{\mathrm{rel}}))\) and \(\operatorname{Hom}^\bullet(\mathcal O,H^{\mathrm{rel}})=\Gamma(B,H^{\mathrm{rel}})\), so Lemma 7.1 (2) gives the isomorphism, and \(H\) is a resolution of \(R\mathcal Hom(K,M)\) by Lemma 7.1 (1). The isomorphisms \(\zeta\) and the units are natural in maps of complexes \(P'\to P\) and \(J\to J'\), so \(\Phi\) commutes with the maps that such maps induce. A map \(K'\to K\) in \(D^-(B)\), or another choice of \(P\), is represented by maps of complexes \(P\leftarrow P''\to P'\) with \(P''\) a bounded above complex of flat modules and the first map a quasi-isomorphism [Tag 05T7]; a quasi-isomorphism \(P''\to P\) induces quasi-isomorphisms of both sides, and another choice of \(J\) is connected by a map as in Lemma 1.2 (2). This gives the independence and the naturality on cohomology.

(2) The restriction of \(R\operatorname{Hom}_{B/S}\) is \(f\mapsto\rho\circ u^*f\) with \(\rho=(g,\kappa)\), and that of the relative cohomology is \(s\mapsto\rho_H\circ u^*s\) with \(\rho_H=(g_H,\kappa_H)\), where \(\kappa_H\) is adjoint to \(Ug_H\circ\mathrm{cu}_{UH}\). By Lemma 7.1 (3), \(u^*\mathcal Hom^\bullet(P,J^{\mathrm{rel}})=\mathcal Hom^\bullet(u^*P,u^*J^{\mathrm{rel}})\), and it suffices to show that \((\mathrm{id},\zeta_F)\circ\mathcal Hom^\bullet(u^*P,\rho)=\) \(\rho_H\circ u^*(\mathrm{id},\zeta)\). On the first components both are \(g_H\). On the second, both are maps \(u^*\mathcal Hom^\bullet(P,\operatorname{coind}UJ)\to\operatorname{coind}_FUH_F\); by the description of \(\zeta\) in the proof of Lemma 7.1 (2), the adjoint of each is the map \(\mathcal Hom^\bullet(UP,U\operatorname{coind}UJ)\to\) \(\mathcal Hom^\bullet(UP,UJ_F)\) given by composition with \(Ug\circ\mathrm{cu}_{UJ}=\mathrm{cu}^F\circ U\kappa\).

(3) Both maps send a section of \(H=\mathcal Hom^\bullet(P,J)\) over \(B\), a map \(P\to J\), to \(g\) composed with its restriction along \(u\). Cocones of equal maps are equal (Definition 4.1).

(4) By Lemma 3.7 (2) of the lesson on the adjoint co-Lie complex, \(K\) is perfect. The functor \(u^*\) is exact, carries tensor products to tensor products and flat modules to flat modules, and carries \(R\mathcal Hom(K,M)\) to \(R\mathcal Hom(u^*K,u^*M)\) through \(g_H\) (Lemma 7.1 (3), also for \(M=\mathcal O\)); so it carries the evaluation map of \(K\) and \(M\) to that of \(u^*K\) and \(u^*M\). The relative cohomology groups are functorial in the coefficients: a map of coefficients in \(D^+\) is represented by a map of their resolutions, unique up to homotopy, which induces maps of the cocones of Construction 1.1 and of Definition 4.1, as in the proof of Lemma 1.2 (2); an isomorphism induces isomorphisms of the cohomology groups, by the five lemma. \(\square\)

In particular (1) and (4) give the isomorphism of Proposition 3.8 of the lesson on the adjoint co-Lie complex by a construction that is natural in the first argument and compatible with restriction along homomorphisms.

Corollary 7.3 (dual coefficients in the deformation theorems).

  1. In Corollary 4.4, \(E^j\cong\) \(H^{j+1}\bigl(B_G/B_F,(\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\) for every \(j\): the obstruction to deforming \(G\) together with an extension of \(u\) to a given deformation of \(F\) lies in \(H^3\bigl(B_G/B_F,(\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\), the solutions form a torsor under \(H^2\), and their automorphisms form \(H^1\).
  2. In Corollary 6.4, let \(D'\in H^2\bigl(B/S,(\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\) correspond to \(D\) under Theorem 7.2 (1), (4), and let \(\operatorname{res}D'\) be its restriction along \(u\). Then \(\omega(u,G')-\omega(u,G'')\) corresponds to \(-\delta_C^\vee(\operatorname{res}D')\in\) \(H^3\bigl(B_F/S,C_u^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\), where \(\delta_C^\vee\colon(u^*\ell^{\mathrm{ad}}_G[1])^\vee\to C_u^\vee\) is the dual of \(\delta_C\), acting on \(H^3\bigl(B_F/S,(u^*\ell^{\mathrm{ad}}_G[1])^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)=\) \(H^2\bigl(B_F/S,(u^*\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\).

Proof. (1) Corollary 4.4 and Theorem 7.2 (3), (4), the underlying object \(L\varepsilon^*\ell_G\) of \(\ell^{\mathrm{ad}}_G\) being perfect (Theorem 3.1 of the lesson on big fppf sites). (2) By Corollary 6.4 the difference corresponds to \(-(\operatorname{res}D)[1]\circ\delta_C\). By Theorem 7.2 (2), \(\operatorname{res}D\) corresponds to \(\operatorname{res}D'\), and by the naturality of Theorem 7.2 (1), (4) in the first argument, applied to \(\delta_C\colon C_u\to u^*\ell^{\mathrm{ad}}_G[1]\), composition with \(\delta_C\) corresponds to the map induced by \(R\mathcal Hom(\delta_C,\varepsilon^*I)\), that is, by \(\delta_C^\vee\otimes^{\mathbf L}\varepsilon^*I\). The isomorphisms of Theorem 7.2 commute with the shift of the first argument, all the constructions in Lemma 7.1 being compatible with shifts; this identifies \((\operatorname{res}D)[1]\) with \(\operatorname{res}D'\) in the stated groups. The underlying objects of \(u^*\ell^{\mathrm{ad}}_G\) and \(C_u\) are perfect (Proposition 3.10 (2) of the lesson on the adjoint co-Lie complex). \(\square\)

8. Exercises

8.1. Let \(F\) be the trivial group scheme \(S\), \(u\colon S\to G\) the unit. Show that \(B_F\) is the topos \(T\) with \(u^*=U\), and that one may take \(J_F=UJ\) and \(g=\mathrm{id}\) in Construction 3.1. Deduce that \(R\operatorname{Hom}(B/B_F;\lambda,M)\) is the cocone of \(U\colon\operatorname{Hom}^\bullet(\lambda,J)\to\operatorname{Hom}^\bullet(U\lambda,UJ)\), and that it is quasi-isomorphic to \(R\operatorname{Hom}_{B/S}(\lambda,M)\).

8.2. Let \(u=\mathrm{id}_G\). Show that \(\operatorname{Ext}^j(B/B;\lambda,M)=0\) for all \(j\), and that the groups \(E^j\) of Corollary 4.4 vanish for \(u=\mathrm{id}_G\). Interpret: a deformation of \(F=G\) determines the deformation of the target, uniquely up to unique isomorphism.

8.3. In Lemma 1.2 (2), let \(h\) be a homotopy from \(g\) to \(g_1\). Verify that \(H(x,y)=(h(x),-\operatorname{coind}Uh(y))\) satisfies \(dH+Hd=\) \((g_1,\operatorname{coind}Ug_1)-(g,\operatorname{coind}Ug)\) on \(J^{\mathrm{rel}}\), with the differential of the Conventions, and that the sign of the second component is forced.

8.4. In Definition 5.1, let \(f\) be the identity of \(Y\). Show that \(R\operatorname{Hom}(B_Y/B_Y;\lambda,M)\) is acyclic. Deduce that, in Theorem 2.2 of the lesson on deforming the target with \(X=Y\) and \(f=\mathrm{id}_Y\), the triple \((X',i,h')\) with \(f'=\mathrm{id}_{X'}\) is an equivariant solution, and that every equivariant solution is isomorphic to it by a unique isomorphism.

8.5. In Lemma 6.1, with \(Z_2\to W\) flat at every level as there, let \(Z_1=W\), with the identity \(Z_1\to W\), and identify \(Y=W\times_WZ_2\) with \(Z_2\). Show that \(K(Z_1/W/V)=0\) and that \(\xi_2\) is the identity of \(L_{Z_2/W}\), and deduce from Lemma 6.1 that \(K(Z_2/W/V)|_X[1]\circ K(X/Z_2/W)=0\). Compare with the step naturality in the base of the proof.

9. Complete solutions

8.1. A module of \(B[S,S]^{\mathrm o}\) is a module of \(T\) with an action of the trivial group, that is, with no additional data, so \(B_F=T\) and the restriction along the unit forgets the action: \(u^*=U\). The complex \(UJ\) is a bounded below complex of injective modules (Conventions) and represents \(UM\), so it is a resolution of \(u^*M\), and \(g=\mathrm{id}\) represents the identity. With these choices Definition 4.1 is the cocone of \(f\mapsto Uf\), which by Lemma 1.2 (1) is \(R\operatorname{Hom}_{B/S}(\lambda,M)\) up to the identification \(\operatorname{Hom}^\bullet(\lambda,\operatorname{coind}UJ)=\operatorname{Hom}^\bullet(U\lambda,UJ)\): the cocone of \(\operatorname{Hom}^\bullet(\lambda,\eta_J)\) is carried isomorphically onto the cocone of \(U\) by the adjunction, which is an isomorphism of complexes.

8.2. For \(u=\mathrm{id}\), \(u^*\) is the identity, \(J_F=J\) and \(g=\mathrm{id}\); the map of Definition 4.1 is the identity of \(\operatorname{Hom}^\bullet(\lambda,J)\), whose cocone is acyclic. In Corollary 4.4 the groups \(E^j\) are then isomorphic to \(\operatorname{Ext}^{j+1}(B/B;\ell^{\mathrm{ad}}_G,\varepsilon^*I)=0\); directly, the relative Ext groups of the arrow topos of an identity vanish by the long exact sequence of Proposition 3.2 of the lesson on the arrow topos, whose second map is then the identity. The obstruction vanishes, the solutions form a torsor under \(0\), and the automorphism group is trivial: given \(F'\), the pair \((G',u')=(F',\mathrm{id})\) is the unique solution up to unique isomorphism.

8.3. On \((x,y)\in J^k\oplus(\operatorname{coind}UJ)^{k-1}\), with \(d(x,y)=(dx,\eta(x)-dy)\), \(\eta=\eta_J\), \(\eta'=\eta_{J'}\) and \(\tilde h=\operatorname{coind}Uh\): \(dH(x,y)=(dh(x),\eta'h(x)+d\tilde h(y))\) and \(Hd(x,y)=(hd(x),-\tilde h\eta(x)+\tilde hd(y))\). Naturality of \(\eta\) applied to the module maps \(h^k\) gives \(\tilde h\eta=\eta'h\), so the sum is \(((dh+hd)(x),(d\tilde h+\tilde hd)(y))=\) \(((g_1-g)(x),\operatorname{coind}U(g_1-g)(y))\), as \(\operatorname{coind}U\) is additive and commutes with differentials. With \(+\tilde h\) in the second component, the terms \(\eta'h(x)\) and \(\tilde h\eta(x)\) would add instead of cancelling.

8.4. For \(f=\mathrm{id}_Y\), \(f_B^*\) is the identity; take \(J_X=J\) and \(r=\mathrm{id}\). The map of Definition 5.1 is then the identity of \(\operatorname{Hom}^\bullet(\lambda,J)\). In the long exact cohomology sequence of the termwise split sequence \(0\to\operatorname{Hom}^\bullet(\lambda,J)[-1]\to\operatorname{cocone}\to\) \(\operatorname{Hom}^\bullet(\lambda,J)\to0\), the connecting maps are the maps induced by this identity (Conventions), hence bijective, so the cocone is acyclic. By Theorem 5.3 (1) all groups \(E^i\) vanish; directly, the relative Ext groups of the arrow topos of an identity vanish (Exercise 5.3 of the arrow topos lesson). With \(X=Y\), \(f=\mathrm{id}\) and \(h=g\), the triple \((X',i,h')\), an equivariant deformation of \(g\) over \(Z'\), is an equivariant \(Z'\)-extension of \(Y\) for \(kg\) whose ideal map from \(g_B^*K\) is an isomorphism (Definition 3.1 of the lesson on equivariant extensions), and \(f'=\mathrm{id}_{X'}\) satisfies \(f'\circ i=i\circ f\) and \(h'\circ f'=h'\). So it is an equivariant solution. By Theorem 2.2 there the isomorphism classes of equivariant solutions form a torsor under \(E^1=0\), so there is one class, and the automorphism group \(E^0\) of a solution is trivial, so the isomorphism to \((X',\mathrm{id}_{X'})\) is unique.

8.5. The cotangent complex of an identity morphism is zero, so \(L_{Z_1/W}=L_{W/W}=0\), and \(K(Z_1/W/V)\), a map out of it, is zero; the right side of Lemma 6.1 vanishes. With \(Y=Z_2\), the morphism \(Y\to Z_1\) is \(Z_2\to W\) and \(Y\to Z_2\) is the identity, so the square of \(\xi_2\) has the identities as top and bottom, and its functoriality map \(\xi_2\colon L_{Z_2/W}\to L_{Y/Z_1}=L_{Z_2/W}\) is the identity. The left side of Lemma 6.1 is then \(K(Z_2/W/V)|_X[1]\circ K(X/Z_2/W)\), which therefore vanishes. This is the vanishing \(K(Y/W/V)|_X[1]\circ K(X/Y/W)=0\) of the step naturality in the base, for \(Y=Z_2\): there it follows from the factorization of \(K(X/Y/W)\) through \(K(X/Y/V)\) and from \(K(Y/W/V)\circ w_{YWV}=0\).

Sources

The relative cohomology of a morphism of topoi, its functorial definition and the forms of the deformation theorems for group schemes and for equivariant morphisms with relative Ext groups of classifying topoi, the difference formula for two deformations of the target, and the forms with dual coefficients, are due to Illusie, in his work on the cotangent complex. The cocone constructions and the homological algebra used here are those of the Stacks project.