The adjoint co-Lie complex of a group scheme

Written by Claude Opus 5.5 (Anthropic) on 21 October 2026. Self-checked by the writing AI. Public domain (CC0).

Let \(G\) be a group scheme over \(S\), flat and locally of finite presentation. Conjugation makes \(G\) act on its co-Lie complex \(\ell_G\); on the module of invariant differential forms this is the coadjoint action. This lesson constructs \(\ell_G\) with this action, as an object of the derived category of the classifying topos of \(G\), and uses it to compute the cotangent complex of the nerve \(N(G)\) over \(S\). The adjoint co-Lie complex depends contravariantly on \(G\), and its formation commutes with base change. It is also the equivariant cotangent complex of \(G\) acting on itself by left and right translations, restricted to the unit section, and its twist by a torsor under \(G^{\mathrm o}\) is the co-Lie complex of that torsor. For a homomorphism \(F\to G\), the same method gives a triangle for the cotangent complex of the morphism of nerves, with the cone of the functoriality map of the adjoint co-Lie complexes in place of the adjoint co-Lie complex. The construction is descent along a torsor over a simplicial scheme. The nerve \(P=N(G,G)\) of \(G\) acting on itself by left translations maps to \(N(G)\), and in each degree this map is a torsor under the opposite group \(G^{\mathrm o}\), acting by right translations. The first section extends descent along torsors, and the co-Lie complex and the Atiyah class of a torsor, from schemes to diagrams of schemes, also for group schemes that vary over a diagram. The co-Lie complex of \(P\) over \(N(G)\) is the Zariski form of the adjoint co-Lie complex. The main theorem is a distinguished triangle \[ \begin{aligned}&L_{N(G)/S}\to i_{0!}\ell_G\\&\quad\to\ell_{P/N(G)}\to L_{N(G)/S}[1],\end{aligned} \] in which \(i_{0!}\ell_G\) has a copy of \(\ell_G\) at each vertex of each simplex and the second map is the adjunction map. It identifies the Ext groups of \(L_{N(G)/S}\) with relative Ext groups of the adjoint co-Lie complex. The adjoint co-Lie complex is perfect, and these relative Ext groups are relative cohomology groups of the classifying topos with coefficients in its dual. A spectral sequence computes these from the Ext groups of \(\ell_G\) on the schemes \(G^n\). By the deformation theory of diagrams, the obstructions to deforming the morphism \(N(G)\to S\) over a thickening of \(S\), and the classification of these deformations, lie in such Ext groups of \(L_{N(G)/S}\).

Prerequisites. Torsors, co-Lie complexes and the Atiyah class (its Conventions, Definition 1.1, Lemma 1.2, Proposition 1.3, Theorems 2.5 and 3.3, Corollary 2.6, Lemmas 3.1 and 3.2, Proposition 3.4, Proposition 4.1 and its proof); The co-Lie complex of a group scheme (Definition 1.1, Propositions 1.2 and 1.5, Corollary 2.3); Flat group schemes are local complete intersections (Theorem 3.1); Deformations of diagrams of schemes (Conventions, Propositions 1.2–1.4, Remark 3.5); Equivariant cotangent complexes (Conventions, Lemma 1.3, Proposition 1.5, Theorem 2.4, Proposition 2.5 (2), Lemma 3.1 (1)); Quasi-coherent and perfect complexes on big fppf sites (Corollary 2.2, Theorem 3.1, Construction 4.1, Lemma 4.2); Fibred topoi and their total topoi (Proposition 3.1, Theorem 4.2); Morphisms of fibred topoi and modules on total topoi (Proposition 2.1, Corollary 2.2, Lemma 5.3, Theorem 5.7); Cohomology of total topoi (Lemma 1.1, Remark 4.2); Equivariant sheaves on simplicial topoi (Definitions 1.1 and 2.1, Propositions 2.2, 4.2 and 6.2); Cohomology of equivariant modules (Theorem 4.3, Proposition 5.1); Mixed Hom complexes and Ext groups of equivariant modules (Theorem 3.3 and its proof, Corollary 3.4). From the programme lesson The cotangent complex: Theorem 2.2, Propositions 5.1 and 10.1, and Theorem 6.1. From the AI-integrated Stacks project: initial functors and limits [Stacks, Tag 09WP, Tag 002R]; adjoints and injectives [Tag 015Z, Tag 070J]; derived direct images of modules [Tag 0D6J]; adjunction and derived pullback [Tag 07A6, Tag 0D6D]; descent of isomorphisms [Tag 02L4]; triangles [Tag 0146, Tag 0149, Tag 014A, Tag 014I, Tag 014Q, Tag 05QR, Tag 05QT]; RHom and triangles [Tag 08JC]; duals of perfect complexes [Tag 08DQ, Tag 08JJ]; perfect objects on ringed sites [Tag 08G5]; derived internal Hom, the internal Hom adjunction, global sections and restriction to a localization [Tag 08J9, Tag 08JA, Tag 08JB]; derived tensor products and inverse images [Tag 07A4]; localization of a topos [Tag 04GZ]; the classifying topos is a topos (Corollary 4.2 of Giraud's characterization of topoi); the third isomorphism of triangles and triangles with a zero vertex [Tag 014A, Tag 05QR].

Conventions

\(S\) is a scheme and \(G\) a group scheme over \(S\), flat and locally of finite presentation, with structure morphism \(\pi\colon G\to S\) and unit \(e\colon S\to G\). Its opposite group \(G^{\mathrm o}\) is the scheme \(G\) with the multiplication \(g*h=hg\); it is again flat and locally of finite presentation, and a left action of \(G^{\mathrm o}\) is a right action of \(G\). \(T\) is the big fppf topos of \(S\), and \(\varepsilon\) the comparison morphisms between big fppf and Zariski topoi.

Diagrams of schemes, their Zariski topoi, morphisms of diagrams of a given type and cotangent complexes of morphisms of diagrams are those of Deformations of diagrams of schemes. For a diagram \(X\colon I\to(\mathrm{Sch})\) and an object \(i\) of \(I\), \(i^*\) is the restriction to the level \(i\). \(D_{\mathrm{qc}}(X_{\mathrm{Zar}})\) is the full subcategory of objects with termwise quasi-coherent cohomology, and \(D^b_{\mathrm{qc}}\) its bounded part. A map in the derived category of a Zariski topos of a diagram is an isomorphism if and only if its restrictions to all levels are, cohomology being computed levelwise. For a functor \(u\colon I'\to I\), the diagram \(X\circ u\) maps to \(X\) by the morphism of type \(u\) with identity components; its inverse image is the restriction along \(u\). It is the inverse image of a morphism of ringed topoi, so it is exact and preserves the polynomial constructions; hence it carries the cotangent complex of a morphism of diagrams \(X\to Y\) to that of the restricted morphism, as in the proof of Proposition 1.2 (1) of the lesson on deformations of diagrams.

Simplicial schemes are diagrams indexed by \(\Delta^{\mathrm{op}}\); for \(\alpha\colon[m]\to[n]\) a simplicial scheme \(N\) has \(N(\alpha)\colon N_n\to N_m\), and \(v_k\colon[0]\to[n]\) is the vertex \(k\). Nerves of actions are those of Equivariant cotangent complexes: for a \(G\)-scheme \(X\), \(N_n(G,X)=G^n\times_SX\), the vertex \(k\) of \((g_1,\ldots,g_n;x)\) is \(g_k\cdots g_1x\), and \(N(G,X)(\alpha)(g;x)=\) \(\bigl(N(G)(\alpha)(g);g_{\alpha(0)}\cdots g_1x\bigr)\) (Lemma 1.3 there). Torsors, the comparison of a torsor with the nerve of its action, and the co-Lie complex \(\ell_{X/Y}\) of a torsor over a scheme are those of Torsors, co-Lie complexes and the Atiyah class, with its notation \(\varphi\), \(q\), \(a=q\varphi\) and \(t_{[n}=\tau_{\geq n}\). We write \(B\) for the classifying category \(B[G,S]^{\mathrm o}\). By (0.1) of the lesson on torsors, a module of \(B\) is an \(\mathcal O\)-module \(M\) of \(T\) with a map \(m\colon\pi^*M\to\pi^*M\) on \(G\), whose values \(m_{(g)}\) at points \(g\) of \(G\) satisfy \(m_{(e)}=\mathrm{id}\) and \(m_{(h)}\circ m_{(g)}=m_{(gh)}\): a right action of \(G\) on \(M\). Its underlying module is \(UM=M\).

1. Torsors over diagrams

Definition 1.1. Let \(I\) be a small category, \(H\) a group scheme over \(S\), flat and locally of finite presentation, and \(Y\colon I\to(\mathrm{Sch}/S)\) a diagram of schemes over \(S\). A torsor under \(H\) over \(Y\) is a diagram \(X\colon I\to(\mathrm{Sch}/S)\) with an action of \(H\) on every \(X_i\), such that every \(X_\alpha\) is a morphism of \(H\)-schemes, together with an \(I\)-morphism \(f\colon X\to Y\) of diagrams over \(S\) such that every \(f_i\colon X_i\to Y_i\) is an \(H_{Y_i}\)-torsor (Definition 1.1 of the lesson on torsors).

In this section \(f\colon X\to Y\) is a torsor under \(G\) over \(Y\). The nerves of the actions form a diagram \(N(G,X)\colon\Delta^{\mathrm{op}}\times I\to(\mathrm{Sch})\), with the values \(N_n(G,X_i)\) and, for an arrow \((\alpha,\gamma)\colon([n],i)\to([m],j)\), given by \(\alpha\colon[m]\to[n]\) in \(\Delta\) and \(\gamma\colon i\to j\) in \(I\), the morphism \(N_m(G,X_\gamma)\circ N(G,X_i)(\alpha)\); here \(N(G,X_\gamma)\) is the morphism of nerves induced by the morphism \(X_\gamma\) of \(G\)-schemes, which commutes with the simplicial maps. In the same way we have \(N(G,Y)\), with \(N_n(G,Y_i)=G^n\times_SY_i\), and \(N(G)\), the diagram \(([n],i)\mapsto G^n\), constant in \(i\). The morphisms \(\varphi=N(G,f)\colon N(G,X)\to N(G,Y)\) and the projection \(N(G,Y)\to N(G)\) are \((\Delta^{\mathrm{op}}\times I)\)-morphisms, and \(q\colon N(G,Y)\to Y\), with the components \(q_{(n,i)}\colon G^n\times_SY_i\to Y_i\), and \(a=q\varphi\), with \(a_{(n,i)}=f_i\circ v_0\), are morphisms of type the projection \(p\colon\Delta^{\mathrm{op}}\times I\to I\). The restriction of \(N(G,X)\) to \(\Delta^{\mathrm{op}}\times\{i\}\), its column at \(i\), is the simplicial scheme \(N(G,X_i)\), and the restriction of \(a\) is the morphism \(a^{(i)}\colon N(G,X_i)\to Y_i\) of the lesson on torsors for the torsor \(f_i\).

Lemma 1.2.

1. A morphism of \(G_V\)-torsors over a scheme \(V\) is an isomorphism.
2. For every arrow \(\gamma\colon i\to j\) of \(I\), the square formed by \(X_\gamma\), \(Y_\gamma\), \(f_i\) and \(f_j\) is cartesian.
3. For every arrow \(c\to c'\) of \(\Delta^{\mathrm{op}}\times I\), the square formed by the maps of \(N(G,X)\) and \(N(G,Y)\) along it and by \(\varphi_c\), \(\varphi_{c'}\) is cartesian. Every \(\varphi_c\) and every \(a_c\) is flat, and every \(a_{(0,i)}=f_i\) is flat and surjective.

Proof. (1) Let \(u\colon X_1\to X_2\) be a morphism of \(G_V\)-torsors. By Proposition 1.3 (1) of the lesson on torsors, \(\{X_1\to V\}\) is an fppf covering. After base change along it, \(X_1\times_VX_1\to X_1\) has the diagonal section \(\delta\), and \(u'=u\times_VX_1\) is a morphism of \(G_{X_1}\)-torsors. Since \(u'\) is equivariant, \(u'\circ\psi_\delta=\psi_{u'\delta}\), and both \(\psi\) are isomorphisms by Lemma 1.2 (2) of the lesson on torsors. So \(u'\) is an isomorphism, and so is \(u\) [Tag 02L4].

(2) By Lemma 1.2 (1) of the lesson on torsors, \(X_j\times_{Y_j}Y_i\to Y_i\) is a \(G_{Y_i}\)-torsor, and \((X_\gamma,f_i)\colon X_i\to X_j\times_{Y_j}Y_i\) is a morphism of \(G_{Y_i}\)-torsors, \(X_\gamma\) being equivariant. By (1) it is an isomorphism.

(3) An arrow of \(\Delta^{\mathrm{op}}\times I\) is the composite of an arrow \((\alpha,\mathrm{id}_i)\) and an arrow \((\mathrm{id},\gamma)\), and composites of cartesian squares are cartesian. For \((\alpha,\mathrm{id}_i)\) the square is cartesian by Lemma 3.1 (1) of the lesson on torsors. For \((\mathrm{id}_{[n]},\gamma)\) it is the product with \(G^n\) over \(S\) of the square of (2). The \(\varphi_{(n,i)}=\mathrm{id}_{G^n}\times f_i\) are flat, \(f_i\) being flat (Proposition 1.3 (1) of the lesson on torsors), and \(v_0\colon G^n\times_SX_i\to X_i\) is a base change of the flat \(G^n\to S\). \(\square\)

Lemma 1.3 (direct images along a projection). Let \(K\) and \(I\) be small categories, \(Z\colon K\times I\to(\mathrm{Sch})\) and \(Y\colon I\to(\mathrm{Sch})\) diagrams, and \(a\colon Z\to Y\) a morphism of type the projection \(K\times I\to I\) whose components \(a_{(k,i)}\) are flat. For an object \(i\) of \(I\) let \(Z^{(i)}\colon K\to(\mathrm{Sch})\) be the column \(k\mapsto Z_{(k,i)}\), \(\iota_i^*F=F^{(i)}\) the restriction of an object of \(\operatorname{Top}(Z_{\mathrm{Zar}})\) to it, and \(a^{(i)}\colon Z^{(i)}\to Y_i\) the morphism with the components \(a_{(k,i)}\).

1. \(\iota_i^*\) carries injective abelian sheaves to injective abelian sheaves.
2. For every abelian sheaf \(F\) on \(\operatorname{Top}(Z_{\mathrm{Zar}})\), \(i^*a_*F=a^{(i)}_*F^{(i)}\), naturally in \(F\). For \(F\in D^+(Z_{\mathrm{Zar}})\) there is an isomorphism \(i^*Ra_*F\cong Ra^{(i)}_*F^{(i)}\), natural in \(F\).
3. Under these isomorphisms, for \(E\in D^+(Y_{\mathrm{Zar}})\) the restriction to the level \(i\) of the adjunction map \(E\to Ra_*a^*E\) is the adjunction map \(E_i\to Ra^{(i)}_*a^{(i)*}E_i\), and for \(F\in D^+(Z_{\mathrm{Zar}})\) the restriction to the column \(i\) of the adjunction map \(a^*Ra_*F\to F\) is the adjunction map \(a^{(i)*}Ra^{(i)}_*F^{(i)}\to F^{(i)}\).

Proof. (1) For an abelian sheaf \(L\) on \(\operatorname{Top}(Z^{(i)}_{\mathrm{Zar}})\), put \[ (\iota_{i!}L)_{(k,i')}=\bigoplus_{\gamma\colon i'\to i}Z_{(\mathrm{id}_k,\gamma)}^{-1}L_k, \] where the structure map along an arrow \((\kappa,\gamma')\colon(k,i')\to(k',i'')\) sends the summand of \(\gamma''\colon i''\to i\) to the summand of \(\gamma''\gamma'\) by \(Z_{(\mathrm{id}_k,\gamma''\gamma')}^{-1}(L_\kappa)\), using \(Z_{(\kappa,\gamma''\gamma')}=Z_{(\kappa,\mathrm{id}_i)}\circ Z_{(\mathrm{id}_k,\gamma''\gamma')}\). As in Proposition 3.1 (1) of Fibred topoi and their total topoi, the conditions on composites follow from those of \(L\), and a map \(L\to F^{(i)}\) extends uniquely to \(\iota_{i!}L\to F\), by \(F_{(\mathrm{id}_k,\gamma)}\circ Z_{(\mathrm{id}_k,\gamma)}^{-1}(-)\) on the summand of \(\gamma\); so \(\iota_{i!}\) is left adjoint to \(\iota_i^*\). It is exact, inverse images and sums of abelian sheaves being exact, so \(\iota_i^*\) preserves injectives [Tag 015Z].

(2) By Proposition 2.1 of Morphisms of fibred topoi and modules on total topoi, \((a_*F)_i\) is the limit of the contravariant functor \(((k,i'),\gamma)\mapsto\gamma_*a_{(k,i')*}F_{(k,i')}\) on the category \(\mathcal C\) of pairs of an object \((k,i')\) and an arrow \(\gamma\colon i'\to i\). Let \(\mathcal D\subseteq\mathcal C\) be the full subcategory of the objects \(((k,i),\mathrm{id}_i)\), isomorphic to \(K\). The arrows from an object \(((k,i'),\gamma)\) to objects of \(\mathcal D\) are the pairs \((\kappa,\gamma)\) with \(\kappa\colon k\to k'\); they form a category isomorphic to the category of arrows out of \(k\), which is nonempty and connected, having the initial object \(\mathrm{id}_k\). So \(\mathcal D^{\mathrm{op}}\) is initial in \(\mathcal C^{\mathrm{op}}\) [Tag 09WP], and the limit may be computed on \(\mathcal D\) [Tag 002R]: \((a_*F)_i=\varprojlim_ka_{(k,i)*}F_{(k,i)}\), with the maps adjoint to the \(F_{(\kappa,\mathrm{id}_i)}\). By Corollary 2.2 (a) there, this is \(a^{(i)}_*F^{(i)}\). For \(F\in D^+\), choose a bounded below complex \(J\) of injective abelian sheaves with a quasi-isomorphism \(F\to J\). The functor \(i^*\) is exact and preserves injectives (Lemma 1.1 of Cohomology of total topoi), and so is \(\iota_i^*\), by (1). So \(i^*Ra_*F=i^*a_*J=a^{(i)}_*J^{(i)}\), and \(J^{(i)}\) is a bounded below complex of injectives quasi-isomorphic to \(F^{(i)}\), so this is \(Ra^{(i)}_*F^{(i)}\). Derived direct images of complexes of modules are those of the underlying complexes of abelian sheaves [Tag 0D6J], the topoi being topoi of sheaves on sites (Theorem 4.2 of the lesson on fibred topoi).

(3) The functor \(a^*\) is exact, the \(a_{(k,i)}\) being flat, and \((a^*E)^{(i)}=a^{(i)*}E_i\). In the proof of Proposition 2.1 of the lesson on morphisms of fibred topoi, the component at \(((k,i),\mathrm{id}_i)\) of the level \(i\) of the adjunction map \(E\to a_*a^*E\) is the adjunction map of \(a_{(k,i)}\), and the level \((k,i)\) of the adjunction map \(a^*a_*F\to F\) is \(a_{(k,i)}^*\) of the projection of \((a_*F)_i\) to \(a_{(k,i)*}F_{(k,i)}\) followed by the adjunction map of \(a_{(k,i)}\). These are the descriptions of the adjunction maps of \(a^{(i)}\). With \(a^*E\to J\) and \(F\to J\) injective resolutions as in (2), the same holds for the derived functors. \(\square\)

Theorem 1.4 (descent along a torsor over a diagram). Let \(f\colon X\to Y\) be a torsor under \(G\) over \(Y\).

1. For every \(E\in D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) the adjunction map \(E\to Ra_*a^*E\) is an isomorphism. Hence \(a^*\colon D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\to D(N(G,X)_{\mathrm{Zar}})\) is fully faithful.
2. Let \(K\in D^b(N(G,X)_{\mathrm{Zar}})\) be an object whose columns \(K^{(i)}\) are quasi-cartesian with termwise quasi-coherent cohomology. Then \(Ra_*K\in D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), its level \(i\) is \(Ra^{(i)}_*K^{(i)}\), and the adjunction map \(a^*Ra_*K\to K\) is an isomorphism. The essential image of \(a^*\) on \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) consists of these objects.
3. An object \(E\) of \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) is quasi-cartesian if and only if \(a^*E\) is quasi-cartesian along every arrow \((\mathrm{id}_{[n]},\gamma)\) of \(\Delta^{\mathrm{op}}\times I\).

Proof. By Lemma 1.2 (3), \(a^*\) is exact, so it is its own derived functor, left adjoint to \(Ra_*\) [Tag 07A6].

(1) By Lemma 1.3 (3), the level \(i\) of \(E\to Ra_*a^*E\) is the adjunction map \(E_i\to Ra^{(i)}_*a^{(i)*}E_i\), an isomorphism by Corollary 2.6 (1) of the lesson on torsors, since \(E_i\in D^b_{\mathrm{qc}}((Y_i)_{\mathrm{Zar}})\). For \(E,F\in D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), the map \(u\mapsto a^*u\) followed by the adjunction bijection \(\operatorname{Hom}(a^*E,a^*F)\cong\operatorname{Hom}(E,Ra_*a^*F)\) is composition with the adjunction map of \(F\), a bijection.

(2) By Theorem 2.5 of the lesson on torsors, \(K^{(i)}\cong a^{(i)*}E_i\) for some \(E_i\in D^b_{\mathrm{qc}}((Y_i)_{\mathrm{Zar}})\). By Lemma 1.3 (2) and Corollary 2.6 (1) there, \((Ra_*K)_i\cong Ra^{(i)}_*K^{(i)}\cong E_i\). If \(K\) lies in \(D^{[r,t]}\), then \(a^{(i)*}H^s(E_i)=H^s(K^{(i)})\) vanishes for \(s\notin[r,t]\). Its level \(0\) is \(f_i^*H^s(E_i)\), and \(f_i^*\) is faithful on quasi-coherent modules, \(f_i\) being flat and surjective (Remark 3.5 of the lesson on deformations of diagrams). So \(E_i\in D^{[r,t]}\) for all \(i\), and \(Ra_*K\in D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\). By Lemma 1.3 (3), the column \(i\) of \(a^*Ra_*K\to K\) is the adjunction map of \(a^{(i)}\) at \(K^{(i)}\cong a^{(i)*}E_i\). By the triangle identities its composite with \(a^{(i)*}\) of the adjunction map at \(E_i\), an isomorphism, is the identity; so it is an isomorphism, and so is \(a^*Ra_*K\to K\). Conversely, the columns \(a^{(i)*}E_i\) of \(a^*E\) are quasi-cartesian with termwise quasi-coherent cohomology by Theorem 2.5 of the lesson on torsors.

(3) For \(\gamma\colon i\to j\), \(a_{(n,j)}\circ N(G,X)_{(\mathrm{id},\gamma)}=Y_\gamma\circ a_{(n,i)}\), so the structure map of \(a^*E\) along \((\mathrm{id}_{[n]},\gamma)\) is \(a_{(n,i)}^*\) of the map \(LY_\gamma^*E_j\to E_i\) [Tag 0D6D]. If the latter is an isomorphism, so is the former. Conversely, for \(n=0\), \(a_{(0,i)}=f_i\) is flat and surjective, and the cone of \(LY_\gamma^*E_j\to E_i\) has quasi-coherent cohomology; if \(f_i^*\) of it vanishes, it vanishes. \(\square\)

Theorem 1.5 (the co-Lie complex of a torsor over a diagram). Let \(f\colon X\to Y\) be a torsor under \(G\) over a diagram \(Y\) of \(S\)-schemes.

1. There is an object \(\ell_{X/Y}\) of \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) with an isomorphism \(a^*\ell_{X/Y}\cong L_{N(G,X)/N(G,Y)}\), unique up to unique isomorphism: the co-Lie complex of \(X\) over \(Y\). It is quasi-cartesian, \(\ell_{X/Y}\cong Ra_*L_{N(G,X)/N(G,Y)}\), and its level \(i\) is the co-Lie complex \(\ell_{X_i/Y_i}\) of the lesson on torsors, through isomorphisms under which the structure map along \(\gamma\colon i\to j\) is the base change isomorphism \(LY_\gamma^*\ell_{X_j/Y_j}\cong\ell_{X_i/Y_i}\) of Proposition 3.4 there, \(X_i\) being identified with \(X_j\times_{Y_j}Y_i\) by Lemma 1.2 (2). Its levels are perfect with tor amplitude in \([-1,0]\).
2. For every integer \(n\) there is an object \(\ell_n(X/S)\) of \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) with an isomorphism \(a^*\ell_n(X/S)\cong t_{[n}L_{N(G,X)/N(G)}\), and there are maps \(\ell_n(X/S)\to\ell_{n+1}(X/S)\) whose images under \(a^*\) are the truncation maps, all unique up to unique isomorphism. Moreover \(\ell_n(X/S)\cong Ra_*t_{[n}L_{N(G,X)/N(G)}\), with the levels \(\ell_n(X_i/S)\).
3. For every \(n\leq-1\) there are unique maps forming a distinguished triangle \[ \begin{aligned}&\quad t_{[n}L_{Y/S}\\ &\quad \xrightarrow{\ \beta_n\ }\ell_n(X/S)\\ &\quad \quad \xrightarrow{\ \gamma_n\ }\ell_{X/Y}\\ &\quad \xrightarrow{\ \mathrm{at}_n\ }t_{[n}L_{Y/S}[1]\end{aligned} \tag{1.1} \] whose image under \(a^*\) is the triangle that Lemma 3.2 of the lesson on torsors makes from the transitivity triangle of \(N(G,X)\to N(G,Y)\to N(G)\), through the isomorphisms of (1), (2) and \(a^*t_{[n}L_{Y/S}\cong t_{[n}L\varphi^*L_{N(G,Y)/N(G)}\). Its restriction to every level \(i\) is the triangle (3.1) of the lesson on torsors for \(X_i\to Y_i\to S\). The family \(\mathrm{at}(X/Y/S)=(\mathrm{at}_n)_{n\leq-1}\) is the Atiyah class of \(X\) over \(Y\).

Proof. (1) Let \(L=L_{N(G,X)/N(G,Y)}\). By the Conventions, its column at \(i\) is \(L_{N(G,X_i)/N(G,Y_i)}\), which lies in \(D^{[-1,0]}\), is quasi-cartesian and has termwise quasi-coherent cohomology (Lemma 3.1 (2), (4) of the lesson on torsors). Theorem 1.4 (2) gives \(\ell_{X/Y}=Ra_*L\) with \(a^*\ell_{X/Y}\cong L\), unique up to unique isomorphism by Theorem 1.4 (1), with the levels \(Ra^{(i)}_*L_{N(G,X_i)/N(G,Y_i)}\cong\ell_{X_i/Y_i}\) (Theorem 3.3 (1) there), perfect with tor amplitude in \([-1,0]\). The structure map of \(L\) along \((\mathrm{id}_{[n]},\gamma)\) is the functoriality map of the square formed by the maps of \(N(G,X)\) and \(N(G,Y)\) along this arrow and by \(\varphi_{(n,i)}\), \(\varphi_{(n,j)}\) (Proposition 1.2 (1) of the lesson on deformations of diagrams). This square is cartesian and \(\varphi_{(n,j)}\) is flat (Lemma 1.2 (3)), so the map is an isomorphism (Proposition 1.4 there). Hence \(a^*\ell_{X/Y}\) is quasi-cartesian along these arrows, and \(\ell_{X/Y}\) is quasi-cartesian by Theorem 1.4 (3). The structure map along \(\gamma\) and the base change isomorphism are two isomorphisms \(LY_\gamma^*\ell_{X_j/Y_j}\to\ell_{X_i/Y_i}\) whose images under \(a^{(i)*}\) are the same functoriality map of the nerves; by the uniqueness in Proposition 3.4 of the lesson on torsors they agree.

(2) The columns of \(t_{[n}L_{N(G,X)/N(G)}\) are the \(t_{[n}L_{N(G,X_i)/N(G)}\), restriction being exact. They lie in \(D^{[n,0]}\), are quasi-cartesian and have termwise quasi-coherent cohomology (Lemma 3.1 (4), (5) of the lesson on torsors). Theorem 1.4 gives \(\ell_n(X/S)=Ra_*t_{[n}L_{N(G,X)/N(G)}\) and, by full faithfulness, the transition maps; the levels are identified by Theorem 3.3 (2) there.

(3) The functoriality map \(q^*L_{Y/S}\to L_{N(G,Y)/N(G)}\) of the commutative square formed by \(q\), the projection \(N(G,Y)\to N(G)\) and the structure morphisms to \(S\) has the columns of Lemma 3.1 (3) of the lesson on torsors, which are isomorphisms; so it is an isomorphism, and \(a^*L_{Y/S}=\) \(L\varphi^*q^*L_{Y/S}\cong\) \(L\varphi^*L_{N(G,Y)/N(G)}\). The third vertex \(L\) of the transitivity triangle of \(N(G,X)\to N(G,Y)\to N(G)\) (Proposition 1.3 of the lesson on deformations of diagrams) lies in \(D^{\geq-1}\subseteq D^{\geq n}\). So Lemma 3.2 of the lesson on torsors gives a distinguished triangle with the vertices \(a^*t_{[n}L_{Y/S}\), \(a^*\ell_n(X/S)\) and \(a^*\ell_{X/Y}\). The three objects \(t_{[n}L_{Y/S}\), \(\ell_n(X/S)\) and \(\ell_{X/Y}\) lie in \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), \(L_{Y/S}\) having termwise quasi-coherent cohomology (Proposition 1.2 (2) of the lesson on deformations of diagrams). By Theorem 1.4 (1) the three maps are \(a^*\) of unique maps \(\beta_n\), \(\gamma_n\), \(\mathrm{at}_n\). Complete \(\beta_n\) to a distinguished triangle with third vertex \(C\in D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\). Its image under the exact functor \(a^*\) is distinguished, so there is an isomorphism \(a^*C\cong a^*\ell_{X/Y}\) compatible with the maps [Tag 014A]. By full faithfulness it is \(a^*\) of an isomorphism \(C\cong\ell_{X/Y}\) compatible with the maps, and (1.1) is isomorphic to a distinguished triangle. The restriction to the level \(i\) is exact, and it carries (1.1) to a distinguished triangle whose image under \(a^{(i)*}\) is the triangle of the column \(i\); by the uniqueness in Theorem 3.3 (3) of the lesson on torsors, it is the triangle (3.1) there. \(\square\)

1.6. Group schemes over a diagram. Let \(Z\colon I\to(\mathrm{Sch})\) be a diagram of schemes. A group scheme over \(Z\), flat and locally of finite presentation, is a diagram \(G\colon I\to(\mathrm{Sch})\) with an \(I\)-morphism \(\pi\colon G\to Z\) such that every \(G_i\) is a group scheme over \(Z_i\), flat and locally of finite presentation, and every map \(G_i\to G_j\times_{Z_j}Z_i\) induced by an arrow \(\gamma\colon i\to j\) is a homomorphism of group schemes over \(Z_i\). It is cartesian if these maps are isomorphisms. For a diagram \(Y\) with an \(I\)-morphism \(Y\to Z\), a torsor under \(G\) over \(Y\) is a diagram \(X\) with an action of \(G_i\) on \(X_i\) over \(Z_i\) for every \(i\), such that every \(X_\gamma\) is equivariant along \(G_\gamma\), together with an \(I\)-morphism \(f\colon X\to Y\) over \(Z\) such that every \(f_i\) is a torsor under \(G_i\times_{Z_i}Y_i\). A group scheme \(G\) over the scheme \(S\), with a diagram \(Y\) over \(S\), is the case where \(Z\) and \(G\) are the constant diagrams \(i\mapsto S\) and \(i\mapsto G\), which is cartesian. The nerves are defined as above, with fibre products over the \(Z_i\): \(N(G,X)\) has the values \(G_i^n\times_{Z_i}X_i\) and, along \((\alpha,\gamma)\), the maps \((G_\gamma^n\times X_\gamma)\circ N(G_i,X_i)(\alpha)\), which form a diagram because \(X_\gamma\) is equivariant along \(G_\gamma\); likewise \(N(G,Y)\), with the values \(G_i^n\times_{Z_i}Y_i\), and \(N(G)\), with the values \(G_i^n\). The maps \(\varphi\), \(q\) and \(a=q\varphi\) are defined as above.

Proposition 1.7. Let \(G\) be a group scheme over a diagram \(Z\), flat and locally of finite presentation, and \(f\colon X\to Y\) a torsor under \(G\) over a diagram \(Y\) over \(Z\).

1. Lemma 1.2 (1), the last assertion of Lemma 1.2 (3), Lemma 1.3 and Theorem 1.4 hold. Theorem 1.5 holds with \(Z\) in place of \(S\), that is, with \(N(G)\) the nerve of \(G\) over \(Z\) and \(L_{Y/Z}\) the cotangent complex of the \(I\)-morphism \(Y\to Z\), except that \(\ell_{X/Y}\) need not be quasi-cartesian; its structure map along \(\gamma\colon i\to j\) is the map \(LY_\gamma^*\ell_{X_j/Y_j}\to\ell_{X_i/Y_i}\) whose image under \(a^{(i)*}\) is the functoriality map of the square of nerves along \((\mathrm{id}_{[n]},\gamma)\).
2. If \(G\) is cartesian, Lemma 1.2 and Theorem 1.5 hold as stated.
3. There is an isomorphism \(f^*\ell_{X/Y}\cong L_{X/Y}\) on \(X\). For the trivial torsor \(\pi\colon G\to Z\) under left multiplication, with its unit section \(e\), the co-Lie complex \(\ell_G=Le^*L_{G/Z}\) is \(\ell_{G/Z}\); its levels are the co-Lie complexes \(\ell_{G_i}\), it lies in \(D^{[-1,0]}(Z_{\mathrm{Zar}})\), and \(L_{G/Z}\cong\pi^*\ell_G\).

Proof. (1) At a level \(i\), the group scheme \(G_i\) over \(Z_i\) and the torsor \(f_i\) are in the situation of the lesson on torsors, with \(Z_i\) in place of \(S\), and the columns of the nerves are those of \(f_i\). The proofs of Lemma 1.2 (1), of the squares along the arrows \((\alpha,\mathrm{id}_i)\) in Lemma 1.2 (3) and of the flatness of the \(\varphi_c\) and \(a_c\) are therefore unchanged. Lemma 1.3 concerns any diagram with flat components. The proof of Theorem 1.4 uses only these facts, Lemma 1.3 and the relation \(a_{(n,j)}\circ N(G,X)_{(\mathrm{id},\gamma)}=Y_\gamma\circ a_{(n,i)}\), which holds because \(X_\gamma\) commutes with the vertex \(v_0\) and \(f_jX_\gamma=Y_\gamma f_i\). The proof of Theorem 1.5 is columnwise, except for the quasi-cartesian property and the identification of the structure maps, which use that the squares along \((\mathrm{id}_{[n]},\gamma)\) are cartesian. In general, the structure map of \(L_{N(G,X)/N(G,Y)}\) along \((\mathrm{id}_{[n]},\gamma)\) is the functoriality map of that square (Proposition 1.2 (1) of the lesson on deformations of diagrams), and it is the image under \(a_{(n,i)}^*\) of the structure map of \(\ell_{X/Y}\) along \(\gamma\), since \(a^*\ell_{X/Y}\cong L_{N(G,X)/N(G,Y)}\).

(2) If \(G\) is cartesian, then \(X_j\times_{Y_j}Y_i\to Y_i\) is a torsor under \(G_j\times_{Z_j}Y_i\cong G_i\times_{Z_i}Y_i\) (Lemma 1.2 (1) of the lesson on torsors), and \((X_\gamma,f_i)\colon X_i\to X_j\times_{Y_j}Y_i\) is a morphism of torsors, so it is an isomorphism by Lemma 1.2 (1). This is Lemma 1.2 (2). Since \(G_i^n\times_{Z_i}X_i=G_j^n\times_{Z_j}X_i\), the square along \((\mathrm{id}_{[n]},\gamma)\) is the product with \(G_j^n\) over \(Z_j\) of the square of Lemma 1.2 (2), hence cartesian, and the proofs of Lemma 1.2 (3) and Theorem 1.5 apply as written.

(3) Let \(\iota\colon I\to\Delta^{\mathrm{op}}\times I\) be the functor \(i\mapsto([0],i)\). Restriction along \(\iota\) carries \(N(G,X)\), \(N(G,Y)\) and \(a\) to \(X\), \(Y\) and \(f\), and it carries cotangent complexes to cotangent complexes (Conventions). Restricting the isomorphism \(a^*\ell_{X/Y}\cong L_{N(G,X)/N(G,Y)}\) gives \(f^*\ell_{X/Y}\cong L_{X/Y}\). For \(X=G\) and \(Y=Z\), applying \(Le^*\) gives \(\ell_{G/Z}\cong Le^*\pi^*\ell_{G/Z}\cong Le^*L_{G/Z}=\ell_G\), since \(\pi e=\mathrm{id}\); so \(L_{G/Z}\cong\pi^*\ell_G\). The levels of \(\ell_{G/Z}\) are the \(\ell_{G_i/Z_i}\), which are the \(\ell_{G_i}\) and lie in \(D^{[-1,0]}\) (Proposition 1.2 of the lesson on the co-Lie complex of a group scheme). \(\square\)

For an \(n\)-diagram \(Z\) of schemes (Section 5 of Morphisms of fibred topoi and modules on total topoi), regard the schemes as objects of the big fppf topos \(T_0\) of \(\operatorname{Spec}\mathbb Z\), whose localization at \(Z_c\) is the big fppf topos of \(Z_c\). A group object of the second total topos of \(Z\) whose values at the vertices are represented by group schemes over the \(Z_c\), flat and locally of finite presentation, is by Theorem 5.7 there a group object of \(T_0^{\int I}/Z^\natural\) with these values, that is, a group scheme over the diagram \(Z^\natural\) in the sense of 1.6. So Proposition 1.7 covers families of torsors over \(n\)-diagrams.

Lemma 1.8 (functoriality along a homomorphism). Let \(c\colon S'\to S\) be a morphism of schemes, \(F\) a group scheme over \(S'\) and \(G\) a group scheme over \(S\), both flat and locally of finite presentation, and \(u\colon F\to G\) a morphism over \(c\) such that \(F\to G\times_SS'\) is a homomorphism of group schemes over \(S'\). Let \(f\colon X\to Y\) be a torsor under \(F\) over a diagram \(Y\colon I\to(\mathrm{Sch}/S')\), \(f'\colon X'\to Y'\) a torsor under \(G\) over a diagram \(Y'\colon I\to(\mathrm{Sch}/S)\), \(g\colon Y\to Y'\) an \(I\)-morphism over \(c\), and \(h\colon X\to X'\) an \(I\)-morphism with \(f'h=gf\) and \(h(kx)=u(k)h(x)\). Write \(a\) and \(a'\) for the morphisms of this section for \(f\) and \(f'\).

1. The maps \(u^n\times h_i\) and \(u^n\times g_i\) form morphisms of diagrams \(N(u,h)\colon N(F,X)\to N(G,X')\) and \(N(u,g)\colon N(F,Y)\to N(G,Y')\), with \(N(G,f')\circ N(u,h)=\) \(N(u,g)\circ N(F,f)\) and \(a'\circ N(u,h)=g\circ a\).
2. There is a unique map \(\ell(h)\colon Lg^*\ell_{X'/Y'}\to\ell_{X/Y}\) in \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) whose image under \(a^*\) is the functoriality map \[ \begin{aligned}&LN(u,h)^*L_{N(G,X')/N(G,Y')}\\&\quad\longrightarrow L_{N(F,X)/N(F,Y)}\end{aligned} \] of the square of (1), through the isomorphisms \(a^*\ell_{X/Y}\cong L_{N(F,X)/N(F,Y)}\) and \(a^*Lg^*\ell_{X'/Y'}\cong\) \(LN(u,h)^*a'^*\ell_{X'/Y'}\cong\) \(LN(u,h)^*L_{N(G,X')/N(G,Y')}\) of Theorem 1.5 (1) and [Tag 0D6D].
3. \(\ell(\mathrm{id}_X)=\mathrm{id}\). For a second such datum \((c',u',g',h')\) from \(f'\) to a torsor \(f''\colon X''\to Y''\) under a group scheme \(H\), \(\ell(h'h)=\ell(h)\circ Lg^*\ell(h')\), through \(L(g'g)^*\cong Lg^*Lg'^*\).
4. If \(u\) induces an isomorphism \(F\cong G\times_SS'\) and the squares formed by \(h_i\), \(g_i\), \(f_i\) and \(f'_i\) are cartesian, then \(\ell(h)\) is an isomorphism.

Proof. (1) The map \(u^n\times h_i\colon F^n\times_{S'}X_i\to G^n\times_SX'_i\) commutes with the maps of the nerves along \((\alpha,\mathrm{id}_i)\), because \(u\) is a homomorphism and \(h\) is equivariant along \(u\), and along \((\mathrm{id}_{[n]},\gamma)\), because the \(h_i\) form an \(I\)-morphism; the same holds for \(g\), on which the groups act trivially. The relations hold componentwise, since \(a_{(n,i)}=f_i\circ v_0\) and \(N(u,h)\) commutes with the vertex \(v_0\).

(2) By Lemma 1.2 (3), \(a\) and \(a'\) are flat, so \(a^*\) and \(a'^*\) are exact and \(a^*Lg^*\cong L(ga)^*=\) \(L(a'N(u,h))^*\cong LN(u,h)^*a'^*\) [Tag 0D6D]. Restriction to the level \(i\) is the inverse image of a morphism of ringed topoi (Conventions), so it carries \(Lg^*\ell_{X'/Y'}\) to \(Lg_i^*\ell_{X'_i/Y'_i}\) [Tag 0D6D], which lies in \(D^{[-1,0]}\) with quasi-coherent cohomology, \(\ell_{X'_i/Y'_i}\) being perfect with tor amplitude in \([-1,0]\) (Theorem 1.5 (1)). So \(Lg^*\ell_{X'/Y'}\) lies in \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\), and \(\ell(h)\) exists and is unique because \(a^*\) is fully faithful on \(D^b_{\mathrm{qc}}(Y_{\mathrm{Zar}})\) (Theorem 1.4 (1)).

(3) For \(h=\mathrm{id}_X\) the functoriality map is the identity. The square of nerves for \(h'h\) is the composite of the squares for \(h\) and \(h'\), and functoriality maps of composite squares compose (Theorem 2.2 of The cotangent complex); so both sides have the same image under \(a^*\), and they agree by the uniqueness in (2).

(4) Then \(F^n\times_{S'}X_i\cong\) \((G^n\times_SX'_i)\times_{G^n\times_SY'_i}(F^n\times_{S'}Y_i)\), both sides being \(G^n\times_S(X'_i\times_{Y'_i}Y_i)\); so the square of nerves of (1) is cartesian at every level, and \(N(G,f')\) is flat (Lemma 1.2 (3)). By Proposition 1.4 of the lesson on deformations of diagrams, the functoriality map is an isomorphism at every level, hence an isomorphism (Conventions). As \(a^*\) is fully faithful, \(\ell(h)\) is an isomorphism. \(\square\)

2. The adjoint co-Lie complex

Construction 2.1 (the universal torsor). Let \(N=N(G)\) and \(P=N(G,G)\), the nerve of \(G\) acting on itself by left multiplication, with \(\varphi=N(G,\pi)\colon P\to N\). So \(P_n=G^n\times_SG\), \(\varphi_n\) is the projection to \(G^n=N_n\), and \[ \begin{aligned}&P(\alpha)(g_1,\ldots,g_n;x)\\&\quad=\bigl(N(\alpha)(g);g_{\alpha(0)}\cdots g_1x\bigr).\end{aligned} \] Let \(G^{\mathrm o}\) act on every \(P_n\) by \(h\star(g;x)=(g;xh)\). The maps \(P(\alpha)\) are \(G^{\mathrm o}\)-equivariant, left and right multiplications commuting, and \(\varphi_n\) is invariant. The map \(\varphi_n\) is the base change along \(\pi_n\colon N_n=G^n\to S\) of \(\pi\colon G\to S\), on which \(G^{\mathrm o}\) acts by \(h\star x=xh\): the trivial \(G^{\mathrm o}_S\)-torsor, with the section \(e\). So every \(\varphi_n\) is a \(G^{\mathrm o}_{N_n}\)-torsor (Lemma 1.2 (1) of the lesson on torsors), and \(\varphi\colon P\to N\) is a torsor under \(G^{\mathrm o}\) over the simplicial scheme \(N\), in the sense of Definition 1.1 with \(I=\Delta^{\mathrm{op}}\). By Proposition 1.3 (2) of the lesson on torsors, applied to the \(G_S\)-torsor \(\pi\) with the left action, the vertices \((P(v_0),\ldots,P(v_n))\colon P_n\to G^{n+1}\) form an isomorphism of simplicial schemes from \(P\) onto the Čech nerve of \(\pi\).

Proposition 2.2. Let \(\ell_{P/N}\in D^b_{\mathrm{qc}}(N_{\mathrm{Zar}})\) be the co-Lie complex of the torsor \(P\) over \(N\) (Theorem 1.5 (1), for the group \(G^{\mathrm o}\)).

1. \(\ell_{P/N}\) is quasi-cartesian, and its level \(n\) is isomorphic to \(\pi_n^*\ell_G\), perfect with tor amplitude in \([-1,0]\). Its level \(0\) is the co-Lie complex \(\ell_{G/S}\) of the \(G^{\mathrm o}_S\)-torsor \(\pi\), isomorphic to \(\ell_G\).
2. There is an object \(\ell^{\mathrm{ad}}_G\) of \(D^b(B)\) with an isomorphism \(\operatorname{ner}(\ell^{\mathrm{ad}}_G)\cong L\varepsilon^*\ell_{P/N}\), unique up to unique isomorphism: the adjoint co-Lie complex of \(G\). Its underlying object \(U\ell^{\mathrm{ad}}_G\) is \(L\varepsilon^*\ell_G\).

Proof. (1) By Theorem 1.5 (1), \(\ell_{P/N}\) is quasi-cartesian with the levels \(\ell_{P_n/N_n}\). By Proposition 3.4 of the lesson on torsors, for the base change of \(\pi\) along \(\pi_n\), \(\ell_{P_n/N_n}\cong L\pi_n^*\ell_{G/S}=\pi_n^*\ell_{G/S}\), \(\pi_n\) being flat. By Proposition 1.2 (3) of The co-Lie complex of a group scheme, in its form for the right multiplication, \(\ell_{G/S}\cong Le^*L_{G/S}=\ell_G\), perfect with tor amplitude in \([-1,0]\) (Proposition 1.2 (1) there).

(2) By Lemma 4.2 (1), (3) of the lesson on big fppf sites, \(L\varepsilon^*\ell_{P/N}\) has the levels \(L\varepsilon^*\pi_n^*\ell_G\), which lie in \(D^{[-1,0]}\), and it is quasi-cartesian, so its cohomology modules are cartesian. The simplicial scheme \(N\) is \(N(G,S)\) for the trivial action, and by Proposition 1.5 (3) of the lesson on equivariant cotangent complexes, \(\operatorname{ner}\colon D^b(B)\to D^b(\operatorname{Top}(N_{\mathrm{fppf}}))\) is fully faithful with an essential image containing \(L\varepsilon^*\ell_{P/N}\). The level \(0\) of the nerve of a module is its underlying module (Proposition 1.5 (2) there, the first vertex of \(N_0\) being the identity). \(\square\)

For points of \(G\) with values in an \(S\)-scheme \(V\) and a morphism \(\rho\colon G_V\to G_V\) of \(V\)-schemes fixing \(e_V\), let \(c(\rho)=e_V^*(d\rho)\colon\omega_G\otimes\mathcal O_V\to\omega_G\otimes\mathcal O_V\), as in Section 4 of the lesson on torsors; \(c(\rho\rho')=c(\rho')c(\rho)\). For \(g\in G(V)\) let \(\operatorname{int}(g)(h)=ghg^{-1}\). Then \(\operatorname{int}(gh)=\operatorname{int}(g)\operatorname{int}(h)\) gives \[ \begin{aligned}&c(\operatorname{int}(h))\circ c(\operatorname{int}(g))\\&\quad =c(\operatorname{int}(gh)).\end{aligned} \tag{2.1} \]

Lemma 2.3 (sections). Let \(H\) be a group scheme over \(S\), flat and locally of finite presentation, and \(f\colon X\to Y\) an \(H_Y\)-torsor over a scheme \(Y\). For an open \(V\subseteq Y\) and a section \(s\) of \(f\) over \(V\), let \(\psi_s\colon H_V\to X_V\), \(h\mapsto hs\), and let \[ \begin{aligned}&\tau_s\colon H^0(\ell_{X/Y})|_V \cong s^*\Omega_{X/Y}\\ &\quad =e_V^*\psi_s^*\Omega_{X_V/V}\\ &\quad \xrightarrow{\ e_V^*(d\psi_s)\ }\omega_H\otimes\mathcal O_V,\end{aligned} \] where the first map is \(H^0\) of the isomorphism \(\ell_{X/Y}|_V\cong Ls^*L_{X/Y}\) of Theorem 3.3 (1) of the lesson on torsors, with \(H^0(Ls^*L_{X/Y})=s^*\Omega_{X/Y}\) ((0.1) of the lesson on deformations of diagrams and Theorem 6.1 of The cotangent complex).

1. \(\tau_s\) is an isomorphism, and \(\tau_{hs}=c(\operatorname{int}(h^{-1}))\circ\tau_s\) for \(h\in H(V)\).
2. Let \(u\colon Y'\to Y\) be a morphism, \(X'=X\times_YY'\) with the projection \(u'\colon X'\to X\), and \(s'\) the section of \(X'\) over \(V'=u^{-1}(V)\) with \(u's'=su\). Then \(\tau_{s'}\circ H^0(b)=u^*\tau_s\) on \(V'\), where \(b\colon Lu^*\ell_{X/Y}\cong\ell_{X'/Y'}\) is the base change isomorphism of Proposition 3.4 of the lesson on torsors and \(H^0(Lu^*\ell_{X/Y})=u^*H^0(\ell_{X/Y})\).

Proof. (1) The proof of Proposition 4.1 (2), (3) of the lesson on torsors uses the smoothness of the group only to identify \(\ell_{X/Y}\) and \(L_{X/Y}\) with their \(H^0\). Replace the level \(0\) isomorphism \(\theta\colon f^*\ell_{X/Y}\cong L_{X/Y}\) there by \(H^0(\theta)\colon f^*H^0(\ell_{X/Y})\cong\Omega_{X/Y}\), \(f\) being flat, and the structure maps along the two faces of \(N_1(H,X)\) by their \(H^0\), which are maps of differentials (Theorem 6.1 of The cotangent complex). The argument then applies word for word.

(2) Let \(\theta'\) be the level \(0\) isomorphism for \(X'\) and \(c\colon u'^*L_{X/Y}\to L_{X'/Y'}\) the functoriality map, which on \(H^0\) is the map of differentials \(u'^*\Omega_{X/Y}\to\Omega_{X'/Y'}\). By Proposition 3.4 of the lesson on torsors, the level \(0\) of \(a'^*(b)\) satisfies \(\theta'\circ f'^*(b)=c\circ u'^*(\theta)\). With \(s'^*f'^*=\mathrm{id}\) and \(s'^*u'^*=u^*s^*\), \[ \begin{aligned}\tau_{s'}\circ H^0(b)&=e_{V'}^*(d\psi_{s'})\circ s'^*H^0(c)\\&\qquad\circ u^*s^*H^0(\theta).\end{aligned} \] The map \(\psi_{s'}\) is the base change of \(\psi_s\), so \(u'\psi_{s'}=\psi_su_H\) with \(u_H\colon H_{V'}\to H_V\). The functoriality of differentials gives \(d\psi_{s'}\circ\psi_{s'}^*H^0(c)=du_H\circ u_H^*(d\psi_s)\), where \(e_{V'}^*(du_H)\) is the identification \(u^*(\omega_H\otimes\mathcal O_V)=\omega_H\otimes\mathcal O_{V'}\). Pulling back along \(e_{V'}\), with \(\psi_{s'}e_{V'}=s'\) and \(u_He_{V'}=e_Vu\), gives \(e_{V'}^*(d\psi_{s'})\circ s'^*H^0(c)=u^*e_V^*(d\psi_s)\). Hence \(\tau_{s'}\circ H^0(b)=\) \(u^*\bigl(e_V^*(d\psi_s)\circ s^*H^0(\theta)\bigr)=\) \(u^*\tau_s\). \(\square\)

Proposition 2.4 (the coadjoint action).

1. \(H^0(\ell^{\mathrm{ad}}_G)\) is isomorphic to the module \(\varepsilon^*\omega_G\) of \(B\) with the structure \(m_{(g)}=c(\operatorname{int}(g))\) at every point \(g\) of \(G\).
2. If \(G\) is smooth over \(S\), then \(\ell^{\mathrm{ad}}_G\cong H^0(\ell^{\mathrm{ad}}_G)[0]\).

The structure of (1) satisfies the rule of a module of \(B\) by (2.1). Dually, \(c(\operatorname{int}(g))\) is the transpose of the adjoint action of \(g\) on the Lie algebra.

Proof. (1) Write \(\ell=\ell_{P/N}\) and \(\ell_n\) for its levels. For \(\alpha\colon[m]\to[n]\), \(H^0\) of the structure map \(\ell_\alpha\colon N(\alpha)^*\ell_m\to\ell_n\) is an isomorphism \(N(\alpha)^*H^0(\ell_m)\cong H^0(\ell_n)\) ((0.1) of the lesson on deformations of diagrams), so \(H^0(\ell)\) is a cartesian module, and \(\varepsilon^*H^0(\ell)=\) \(H^0(L\varepsilon^*\ell)\cong\) \(H^0(\operatorname{ner}\ell^{\mathrm{ad}}_G)=\) \(\operatorname{ner}H^0(\ell^{\mathrm{ad}}_G)\), the nerve being exact. For a module \((M,m)\) of \(B\), the nerve has the level \(1\) module \(\pi^*M\), its structure map along the vertex \(v_0\colon[0]\to[1]\) is the identity and along \(v_1\) it is \(m\) (Definition 2.1 of the lesson on equivariant sheaves, with the reversal of Proposition 1.5 (1) of the lesson on equivariant cotangent complexes). If \(w\) is an isomorphism from \(\operatorname{ner}(M)\) to a module \(E\) of \(\operatorname{Top}(N_{\mathrm{fppf}})\), compatibility with these two maps gives \[ \begin{aligned}m&=\pi^*w_0^{-1}\circ\bigl(E_{v_0}^{-1}\circ E_{v_1}\bigr)\\&\qquad\circ\pi^*w_0 .\end{aligned} \] So it suffices to compute \(H^0(\ell_{v_0})^{-1}\circ H^0(\ell_{v_1})\) on \(\pi^*H^0(\ell_0)\).

Use the isomorphisms of Lemma 2.3 for the group \(G^{\mathrm o}\): \(\tau_e\colon H^0(\ell_0)\cong\omega_G\) on \(N_0=S\), and \(\tau_\sigma\colon H^0(\ell_1)\cong\pi^*\omega_G\) on \(N_1=G\) for the section \(\sigma(g)=(g;e)\) of \(\varphi_1\). By Theorem 1.5 (1), \(\ell_{v_k}\) is the base change isomorphism for the cartesian square formed by \(P(v_k)\colon P_1\to P_0\), \(\pi\colon N_1\to N_0\) and \(\varphi\); here \(P(v_0)(g;x)=x\) and \(P(v_1)(g;x)=gx\). The sections of \(\varphi_1\) lying over \(e\) are \(\sigma\) for \(v_0\), since \(P(v_0)\sigma=e\pi\), and \(\sigma_1\colon g\mapsto(g;g^{-1})\) for \(v_1\), since \(P(v_1)\sigma_1=e\pi\). By Lemma 2.3 (2), \[ \begin{aligned}\tau_\sigma\circ H^0(\ell_{v_0})&=\pi^*\tau_e,\\ \tau_{\sigma_1}\circ H^0(\ell_{v_1})&=\pi^*\tau_e .\end{aligned} \] In \(G^{\mathrm o}\), \(\sigma_1=g^{-1}\star\sigma\), and the inner automorphism \(k\mapsto(g^{-1})^{-1}*k*g^{-1}\) of \(G^{\mathrm o}\) is \(k\mapsto g^{-1}kg\), that is, \(\operatorname{int}(g^{-1})\). Lemma 2.3 (1) for \(G^{\mathrm o}\), whose co-Lie module is \(\omega_G\), gives \(\tau_{\sigma_1}=c(\operatorname{int}(g^{-1}))\circ\tau_\sigma\). By (2.1), \(c(\operatorname{int}(g^{-1}))^{-1}=c(\operatorname{int}(g))\). Hence \[ \begin{aligned}&H^0(\ell_{v_0})^{-1}\circ H^0(\ell_{v_1})\\&\quad=\pi^*\tau_e^{-1}\circ c(\operatorname{int}(g))\circ\pi^*\tau_e .\end{aligned} \] Applying \(\varepsilon^*\) and the formula for \(m\), with \(w_0\) followed by \(\varepsilon^*\tau_e\), gives (1).

(2) If \(G\) is smooth, \(\ell_G\cong\omega_G[0]\) (Proposition 1.5 of the lesson on the co-Lie complex), so the levels of \(\ell_{P/N}\) and of \(L\varepsilon^*\ell_{P/N}\) have no cohomology in degree \(-1\). The level \(0\) of \(\operatorname{ner}H^{-1}(\ell^{\mathrm{ad}}_G)\) is the underlying module of \(H^{-1}(\ell^{\mathrm{ad}}_G)\), which therefore vanishes; and \(\ell^{\mathrm{ad}}_G\) lies in \(D^{[-1,0]}\). \(\square\)

Proposition 2.5 (contravariance and base change). Let \(u\colon F\to G\) be a homomorphism of group schemes over \(S\), flat and locally of finite presentation, and \(B_F=B[F,S]^{\mathrm o}\). For a module \((M,m)\) of \(B\), the pair \(u^*(M,m)=(M,u^*m)\), with \((u^*m)_{(k)}=m_{(u(k))}\) at points \(k\) of \(F\), is a module of \(B_F\), the restriction of \(M\) along \(u\). The functor \(u^*\) is exact, \(Uu^*=U\), and \(\operatorname{ner}(u^*M)=N(u)^*\operatorname{ner}(M)\), where \(N(u)\colon N(F)\to N(G)\) is the morphism of nerves, with the levels \(u^n\), and \(N(u)^*\) the inverse image of the fppf total topoi.

1. Let \(h\colon P_F=N(F,F)\to P=N(G,G)\) be the morphism with the levels \(u^n\times u\). There is a unique map \(\lambda_u\colon u^*\ell^{\mathrm{ad}}_G\to\ell^{\mathrm{ad}}_F\) in \(D^b(B_F)\) whose nerve is \(L\varepsilon^*\ell(h)\) (Lemma 1.8), through \(\operatorname{ner}(u^*\ell^{\mathrm{ad}}_G)=\) \(N(u)^*\operatorname{ner}(\ell^{\mathrm{ad}}_G)\cong\) \(N(u)^*L\varepsilon^*\ell_{P/N}\cong\) \(L\varepsilon^*LN(u)^*\ell_{P/N}\). Its underlying map \(U\lambda_u\) is \(L\varepsilon^*\ell(u)\), where \(\ell(u)\colon\ell_G=Le_F^*Lu^*L_{G/S}\to\) \(Le_F^*L_{F/S}=\ell_F\) is \(Le_F^*\) of the functoriality map of \(u\), through the isomorphisms \(\ell_{G/S}\cong\ell_G\) and \(\ell_{F/S}\cong\ell_F\) of Proposition 1.7 (3) for \(G^{\mathrm o}\) and \(F^{\mathrm o}\). On \(H^0\), \(\ell(u)\) is the map \(\omega_G\to\omega_F\) of invariant differential forms induced by \(u\).
2. \(\lambda_{\mathrm{id}_G}=\mathrm{id}\), and \(\lambda_{vu}=\lambda_u\circ u^*\lambda_v\) for a homomorphism \(v\colon G\to H\).
3. Let \(c\colon S'\to S\) be a morphism of schemes, \(G'=G\times_SS'\), \(B'=B[G',S']^{\mathrm o}\), and \(c^*(M,m)=(M|_{S'},m|_{G'})\) the restriction of a module of \(B\) to schemes over \(S'\), a module of \(B'\). The functor \(c^*\) is exact, and \(\operatorname{ner}(c^*M)=N(c)^*\operatorname{ner}(M)\) for the projection \(N(c)\colon N(G')\to N(G)\). Let \(h_c\colon N(G',G')\to N(G,G)\) be the projection. There is a unique isomorphism \(\beta_c\colon c^*\ell^{\mathrm{ad}}_G\cong\ell^{\mathrm{ad}}_{G'}\) whose nerve is \(L\varepsilon^*\ell(h_c)\). For \(u\) as in (1) and \(u'=u\times_SS'\colon F'\to G'\), \(\beta_c^F\circ c^*\lambda_u=\lambda_{u'}\circ u'^*\beta_c\), where \(\beta_c^F\) is the isomorphism for \(F\).

Proof. The rules of a module of \(B_F\) hold for \(u^*m\), \(u\) being a homomorphism: \((u^*m)_{(e)}=m_{(e)}=\mathrm{id}\) and \((u^*m)_{(l)}\circ(u^*m)_{(k)}=\) \(m_{(u(k)u(l))}=(u^*m)_{(kl)}\). Kernels and cokernels are computed on the underlying modules (Proposition 6.2 (1) of the lesson on equivariant sheaves), so \(u^*\) is exact, and \(Uu^*=U\). The nerve of \(M\) has the levels \(\pi_n^*M\) (Proposition 1.5 (2) of the lesson on equivariant cotangent complexes), and its structure maps are inverse images of \(m\) along maps of the simplicial scheme \(N(G)\) into \(N_1(G)=G\) (Definition 2.1 of the lesson on equivariant sheaves). The morphism \(N(u)\) commutes with these maps, so \(N(u)^*\operatorname{ner}(M)\) has the levels \(\pi_n^*M\) and the structure maps built in the same way from \(u^*m\): it is \(\operatorname{ner}(u^*M)\). Both functors being exact, the identification holds on \(D^b\). The same argument applies to \(c^*\), whose underlying functor is the restriction from \(T\) to the big fppf topos \(T/S'\) of \(S'\), an inverse image of a localization.

(1) The levels \(u^n\times u\) form a morphism of simplicial schemes over \(N(u)\), and \(h(k\star x)=u(k)\star h(x)\) for the actions of \(F^{\mathrm o}\) and \(G^{\mathrm o}\) of Construction 2.1, as \(u(xk)=u(x)u(k)\). Lemma 1.8, with \(c=\mathrm{id}_S\), \(I=\Delta^{\mathrm{op}}\), the groups \(F^{\mathrm o}\), \(G^{\mathrm o}\) and the homomorphism \(u\colon F^{\mathrm o}\to G^{\mathrm o}\), gives \(\ell(h)\colon LN(u)^*\ell_{P/N}\to\ell_{P_F/N(F)}\). The comparison morphisms \(\varepsilon\) commute with the morphisms of total topoi induced by \(N(u)\), so \(L\varepsilon^*LN(u)^*\cong N(u)^*L\varepsilon^*\) [Tag 0D6D], and \(L\varepsilon^*\ell(h)\) is a map from \(\operatorname{ner}(u^*\ell^{\mathrm{ad}}_G)\) to \(\operatorname{ner}(\ell^{\mathrm{ad}}_F)\). By Proposition 1.5 (3) of the lesson on equivariant cotangent complexes, \(\operatorname{ner}\) is fully faithful on \(D^b(B_F)\), which gives \(\lambda_u\). The underlying map is the level \(0\) of its nerve, that is, \(L\varepsilon^*\) of the level \(0\) of \(\ell(h)\), as \(N(u)_0=\mathrm{id}_S\). At the level \(0\) the torsors are \(\pi_G\colon G\to S\) and \(\pi_F\colon F\to S\), and \(h_0=u\). Restricting the isomorphisms \(a^*\ell\cong L\) of Theorem 1.5 (1) and the defining property of \(\ell(h)\) to the vertex \(([0],[0])\) of the nerves, as in the proof of Proposition 1.7 (3), shows that \(\pi_F^*\) of the level \(0\) of \(\ell(h)\) is the functoriality map \(Lu^*L_{G/S}\to L_{F/S}\), through \(\pi_F^*\ell_{F/S}\cong L_{F/S}\) and \(\pi_F^*\ell_{G/S}=Lu^*\pi_G^*\ell_{G/S}\cong Lu^*L_{G/S}\). Applying \(Le_F^*\), with \(\pi_Fe_F=\mathrm{id}\) and \(ue_F=e_G\), gives \(\ell(u)\). On \(H^0\), the functoriality map of \(u\) is the map \(u^*\Omega_{G/S}\to\Omega_{F/S}\) of differentials (Theorem 6.1 of The cotangent complex), and \(e_F^*\) of it is \(\omega_G=e_G^*\Omega_{G/S}\to e_F^*\Omega_{F/S}=\omega_F\).

(2) The nerve of \(\lambda_u\circ u^*\lambda_v\) is \(L\varepsilon^*\bigl(\ell(h_u)\circ LN(u)^*\ell(h_v)\bigr)\), since \(\operatorname{ner}(u^*\lambda_v)=N(u)^*\operatorname{ner}(\lambda_v)\). By Lemma 1.8 (3) this is \(L\varepsilon^*\ell(h_vh_u)\), and \(h_vh_u\) is the morphism \(h_{vu}\) for \(vu\); so it is the nerve of \(\lambda_{vu}\), and the maps agree by full faithfulness. For \(u=\mathrm{id}_G\), \(\ell(\mathrm{id})=\mathrm{id}\).

(3) The projection \(h_c\) lies over \(N(c)\), is equivariant along the projection \(G'^{\mathrm o}\to G^{\mathrm o}\), which induces \(G'\cong G\times_SS'\), and the squares formed by \(h_c\), \(N(c)\) and the torsors are cartesian: \(G'^n\times_{S'}G'=(G^n\times_SG)\times_{G^n}G'^n\). By Lemma 1.8 (4), \(\ell(h_c)\colon LN(c)^*\ell_{P/N}\to\ell_{P'/N(G')}\) is an isomorphism, and as in (1) \(L\varepsilon^*\ell(h_c)\) is an isomorphism from \(\operatorname{ner}(c^*\ell^{\mathrm{ad}}_G)\) to \(\operatorname{ner}(\ell^{\mathrm{ad}}_{G'})\), the nerve of a unique map \(\beta_c\), which is an isomorphism, \(\operatorname{ner}\) being fully faithful. The two maps \(c^*u^*\ell^{\mathrm{ad}}_G=u'^*c^*\ell^{\mathrm{ad}}_G\to\ell^{\mathrm{ad}}_{F'}\) have the nerves \(L\varepsilon^*\) of \(\ell(h_c^F)\circ LN(c)^*\ell(h)\) and of \(\ell(h')\circ LN(u')^*\ell(h_c)\), where \(h'\) is the morphism of (1) for \(u'\). By Lemma 1.8 (3) both are \(\ell\) of the composite \(N(F',F')\to N(G,G)\), which is \(h\circ h_c^F=h_c\circ h'\); so the maps agree. \(\square\)

2.6. Left and right translations. Let \(K=G\times_SG^{\mathrm o}\), a group scheme over \(S\), flat and locally of finite presentation, with the multiplication \((g,h)(g',h')=(gg',h'h)\). It acts on \(G\) by \((g,h)\cdot x=gxh\), since \((g,h)\cdot((g',h')\cdot x)=gg'xh'h\). The first projection \(\mathrm{pr}_1\colon K\to G\) is a homomorphism, and so is \(j\colon G\to K\), \(j(g)=(g,g^{-1})\), since \(j(g)j(g')=(gg',g'^{-1}g^{-1})\). The unit section is fixed by \(j(G)\): \(j(g)\cdot e=e\). A module of \(B[K,G]^{\mathrm o}\) is an \(\mathcal O\)-module \(E\) of \(T/G\) with a map \(s\colon\mathrm{act}^*E\to\mathrm{pr}^*E\) on \(K\times_SG\) (Proposition 1.5 (2) of the lesson on equivariant cotangent complexes). At points it consists of maps \(s_{(k,x)}\colon E_{k\cdot x}\to E_x\) with \(s_{(1,x)}=\mathrm{id}\) and \[ s_{(k,x)}\circ s_{(k',k\cdot x)}=s_{(k'k,x)}, \] the two conditions of Definition 1.1 of the lesson on equivariant sheaves, for the strings \(k'k\cdot x\to k\cdot x\to x\) of \([K,G]^{\mathrm o}\). With \(k'=k^{-1}\), and with \((k,x)\) replaced by \((k^{-1},k\cdot x)\) and \(k'=k\), they give \(s_{(k,x)}\circ s_{(k^{-1},k\cdot x)}=\mathrm{id}\) and \(s_{(k^{-1},k\cdot x)}\circ s_{(k,x)}=\mathrm{id}\); so every \(s_{(k,x)}\) is an isomorphism.

Lemma 2.6 (the nerve of \(K\) acting on \(G\)). Let \(\delta\colon\Delta^{\mathrm{op}}\to\Delta^{\mathrm{op}}\times\Delta^{\mathrm{op}}\) be the diagonal, and \(a\colon N(G^{\mathrm o},P)\to N\) the morphism of Section 1 for the \(G^{\mathrm o}\)-torsor \(\varphi\colon P\to N\).

1. The restriction of \(N(G^{\mathrm o},P)\) along \(\delta\) is \(N(K,G)\), through the isomorphisms \[ \begin{aligned}&(G^{\mathrm o})^n\times_SP_n\to K^n\times_SG,\\&\bigl((h_1,\ldots,h_n);(g_1,\ldots,g_n;x)\bigr)\\&\quad\mapsto\bigl((g_1,h_1),\ldots,(g_n,h_n);x\bigr).\end{aligned} \] The restriction of \(N(G^{\mathrm o},N)\) is \(N(K)\), and that of \(N(G^{\mathrm o},\varphi)\) is the projection \(N(K,\pi)\colon N(K,G)\to N(K)\). Let \(\iota\colon N(K,G)\to N(G^{\mathrm o},P)\) be the morphism of type \(\delta\) with identity components. Then \(\psi=a\iota\) has the levels \(\psi_n((g_i,h_i);x)=(g_1,\ldots,g_n)\), that is, \(\psi=N(\mathrm{pr}_1)\circ N(K,\pi)\).
2. \(\psi^*\ell_{P/N}\cong L_{N(K,G)/N(K)}\) in \(D(N(K,G)_{\mathrm{Zar}})\).

Proof. (1) For \(\alpha\colon[m]\to[n]\), the map of \(N(G^{\mathrm o},P)\) along \((\alpha,\alpha)\) is the map along \(\alpha\) of the nerve of the action of \(G^{\mathrm o}\) on \(P_n\), followed by \(P(\alpha)\) on the last factor (Section 1). For the action \(h\star(g;x)=(g;xh)\), Lemma 1.3 of the lesson on equivariant cotangent complexes gives the first map, \[ \begin{aligned}&\bigl((h);(g;x)\bigr)\\&\quad\mapsto\bigl(N(G^{\mathrm o})(\alpha)(h);(g;xh_1\cdots h_{\alpha(0)})\bigr),\end{aligned} \] the product \(h_{\alpha(0)}\cdots h_1\) in \(G^{\mathrm o}\) being \(h_1\cdots h_{\alpha(0)}\) in \(G\). Followed by \(P(\alpha)\) (Construction 2.1), this is \[ \begin{aligned}&\bigl(N(G^{\mathrm o})(\alpha)(h);N(\alpha)(g);\\&\qquad g_{\alpha(0)}\cdots g_1\,x\,h_1\cdots h_{\alpha(0)}\bigr).\end{aligned} \] The map of \(N(K,G)\) along \(\alpha\) sends \(((g_i,h_i);x)\) to \(\bigl(N(K)(\alpha)((g_i,h_i));k_{\alpha(0)}\cdots k_1\cdot x\bigr)\), with \(k_i=(g_i,h_i)\) and \(k_{\alpha(0)}\cdots k_1=(g_{\alpha(0)}\cdots g_1,h_1\cdots h_{\alpha(0)})\) by the multiplication of \(K\); and \(N(K)(\alpha)\) is \(N(\alpha)\times N(G^{\mathrm o})(\alpha)\), the faces of \(N(K)\) multiplying neighbouring entries in each factor. So the isomorphisms of the statement commute with the maps along \(\alpha\). The same computation, \(G^{\mathrm o}\) acting trivially on \(N\), identifies the restriction of \(N(G^{\mathrm o},N)\) with \(N(K)\), and the projections correspond. The components of \(a\) are \(a_{(n,n)}=\varphi_n\circ v_0\), the projection to \(G^n\), the first vertex of \(N_n(G^{\mathrm o},P_n)\) being the projection to \(P_n\).

(2) By the Conventions, \(\iota^*\) is the restriction along \(\delta\), and it carries \(L_{N(G^{\mathrm o},P)/N(G^{\mathrm o},N)}\) to the cotangent complex of the restricted morphism, which is \(L_{N(K,G)/N(K)}\) by (1). By Lemma 1.2 (3) for the \(G^{\mathrm o}\)-torsor \(\varphi\), \(a\) is flat, so \(a^*\) and \(\iota^*\) are exact and \(\psi^*=\iota^*a^*\). Theorem 1.5 (1) for \(\varphi\) gives \(a^*\ell_{P/N}\cong L_{N(G^{\mathrm o},P)/N(G^{\mathrm o},N)}\); apply \(\iota^*\). \(\square\)

Proposition 2.7 (the adjoint co-Lie complex through left and right translations).

1. For a module \((M,m)\) of \(B\), let \(\Phi(M)\) be \(\pi^*M\) on \(T/G\) with \(s_{(k,x)}=m_{(\mathrm{pr}_1(k))}\): the restriction of \(M\) along \(\mathrm{pr}_1\) (Proposition 2.5), followed by the pullback along the morphism of \(K\)-schemes \(\pi\colon G\to S\), \(K\) acting trivially on \(S\) (Lemma 3.1 (1) of the lesson on equivariant cotangent complexes). For a module \((E,s)\) of \(B[K,G]^{\mathrm o}\), let \(\Psi(E)\) be \(e^*E\) with \(m_{(g)}=s_{(j(g),e)}\). Then \(\Phi\) and \(\Psi\) are exact functors, \(\Psi\Phi=\mathrm{id}\), and the maps \(\sigma_x=s_{((1,x),e)}\colon E_x\to E_e\) form an isomorphism \(\sigma\colon E\to\Phi\Psi(E)\), natural in \(E\). So \(\Phi\) is an equivalence of abelian categories with quasi-inverse \(\Psi\), and it induces an equivalence \(D^b(B)\simeq D^b(B[K,G]^{\mathrm o})\). Moreover \(\operatorname{ner}\circ\Phi=\psi_{\mathrm{big}}^*\circ\operatorname{ner}\).
2. Let \(L^K_{G/S}=t_{[-1}L^K_{G/S}\) be the equivariant cotangent complex of \(\pi\colon G\to S\) for the action of \(K\) (Theorem 2.4 and Proposition 2.5 (2) of the lesson on equivariant cotangent complexes). There is a unique isomorphism \(\Phi(\ell^{\mathrm{ad}}_G)\cong L^K_{G/S}\) whose nerve is the composite \[ \begin{aligned}&\operatorname{ner}(\Phi(\ell^{\mathrm{ad}}_G))=\psi_{\mathrm{big}}^*\operatorname{ner}(\ell^{\mathrm{ad}}_G)\\&\quad\cong\psi_{\mathrm{big}}^*L\varepsilon^*\ell_{P/N}\cong L\varepsilon^*\psi^*\ell_{P/N}\\&\quad\cong L\varepsilon^*L_{N(K,G)/N(K)}\end{aligned} \] of Proposition 2.2 (2) and Lemma 2.6 (2). Hence \(\Psi(L^K_{G/S})\cong\ell^{\mathrm{ad}}_G\): the adjoint co-Lie complex is the equivariant cotangent complex of \(G\) under left and right translations, restricted to the unit section with the action of \(G\) through \(j\), that is, by conjugation.

Proof. (1) Kernels and cokernels are computed on the underlying modules (Proposition 6.2 (1) of the lesson on equivariant sheaves); \(\pi^*\) is exact, \(\pi\) being flat, and \(e^*\) is the restriction from \(T/G\) to \(T/S\) along \(e\), also exact. The structure of \(\Phi(M)\) is that of the composite of the two functors named, by Proposition 2.5 and Lemma 3.1 (1) of the lesson on equivariant cotangent complexes, so \(\operatorname{ner}(\Phi(M))=\) \(N(K,\pi)_{\mathrm{big}}^*N(\mathrm{pr}_1)^*\operatorname{ner}(M)=\) \(\psi_{\mathrm{big}}^*\operatorname{ner}(M)\). The map \(m\) of \(\Psi(E)\) is the pullback of \(s\) along \((j,e\pi)\colon G\to K\times_SG\), on which \(\mathrm{act}\) and \(\mathrm{pr}\) are both \(e\pi\), since \(j(g)\cdot e=e\); so it is a map \(\pi^*e^*E\to\pi^*e^*E\) on \(G\). It satisfies \(m_{(1)}=s_{(1,e)}=\mathrm{id}\) and \[ \begin{aligned}&m_{(g)}\circ m_{(g')}=s_{(j(g),e)}\circ s_{(j(g'),e)}\\&\quad=s_{(j(g')j(g),e)}=m_{(g'g)},\end{aligned} \] the rules of a module of \(B\) (Conventions). \(\Psi\Phi(M)=e^*\pi^*M=M\), with \(m_{(\mathrm{pr}_1j(g))}=m_{(g)}\). The map \(\sigma\) is the pullback of \(s\) along \((j',e\pi)\colon G\to K\times_SG\), \(j'(x)=(1,x)\), on which \(\mathrm{act}\) is the identity of \(G\), as \((1,x)\cdot e=x\), and \(\mathrm{pr}\) is \(e\pi\); so it is a map \(E\to\pi^*e^*E\), an isomorphism. It is a morphism of modules of \(B[K,G]^{\mathrm o}\): for \(k=(g,h)\), the structure of \(\Phi\Psi(E)\) at \((k,x)\) is \(s_{(j(g),e)}\), and the cocycle identity gives \[ \begin{aligned}&s_{(j(g),e)}\circ\sigma_{k\cdot x}=s_{((1,gxh)j(g),e)},\\&\sigma_x\circ s_{(k,x)}=s_{(k(1,x),e)},\end{aligned} \] which agree because \((1,gxh)(g,g^{-1})=(g,xh)=(g,h)(1,x)\) in \(K\). For a morphism \(u\colon E\to E'\), the identity \(s'\circ\mathrm{act}^*(u)=\mathrm{pr}^*(u)\circ s\) pulls back along \((j',e\pi)\) to \(\sigma'\circ u=\pi^*e^*(u)\circ\sigma\). Exact functors act termwise on complexes, which gives the equivalence of the derived categories.

(2) The morphism \(\pi\) and \(K\) are Tor independent, \(K\) being flat over \(S\), and \(\pi\) is a local complete intersection (Theorem 3.1 of Flat group schemes are local complete intersections); so by Proposition 2.5 (2) of the lesson on equivariant cotangent complexes \(L^K_{G/S}\) has the nerve \(t_{[-1}L\varepsilon^*L_{N(K,G)/N(K)}\). Here \(t_{[-1}\) is the identity: the level \(n\) of \(L_{N(K,G)/N(K)}\) is the pullback of \(L_{G/S}\) along the projection \(K^n\times_SG\to G\), \(K^n\times_SG\to K^n\) being the base change of \(\pi\) along the flat \(K^n\to S\) (Proposition 1.4 of the lesson on deformations of diagrams), and \(L_{G/S}\) has tor amplitude in \([-1,0]\), so the levels of \(L\varepsilon^*L_{N(K,G)/N(K)}\) lie in \(D^{[-1,0]}\) (Lemma 4.2 (1) of the lesson on big fppf sites). The morphism \(\psi\) is flat at every level, so \(\psi^*\) and \(\psi_{\mathrm{big}}^*\) are exact, and \(\psi_{\mathrm{big}}^*\varepsilon^*=\varepsilon^*\psi^*\) on complexes of flat modules, as in the proof of Lemma 4.2 (3) there; this gives \(\psi_{\mathrm{big}}^*L\varepsilon^*\cong L\varepsilon^*\psi^*\). The displayed composite is therefore an isomorphism between the nerves, and \(\operatorname{ner}\) is fully faithful on \(D^b(B[K,G]^{\mathrm o})\) (Proposition 1.5 (3) there). Applying \(\Psi\), with \(\Psi\Phi=\mathrm{id}\), gives the last statement. \(\square\)

2.8. Twisting by a torsor. Let \(Y\) be a scheme over \(S\) and \(f\colon X\to Y\) a torsor under \(G^{\mathrm o}\) over \(Y\) (Definition 1.1 of the lesson on torsors, for the group \(G^{\mathrm o}\)), with the action \(h\star x\). The inversion \(\mathrm{inv}\colon G^{\mathrm o}\to G\), \(h\mapsto h^{-1}\), is an isomorphism of group schemes. For a module \((M,m)\) of \(B\), let \(M_X\) be the module of \(B[G^{\mathrm o},X]^{\mathrm o}\) obtained by restricting \(M\) along \(\mathrm{inv}\) (Proposition 2.5) and pulling back along the morphism of \(G^{\mathrm o}\)-schemes \(X\to S\), \(G^{\mathrm o}\) acting trivially on \(S\) (Lemma 3.1 (1) of the lesson on equivariant cotangent complexes): the pullback of \(M\) to \(X\), with the structure \(m_{(h^{-1})}\) at \((h,x)\). The twist of \(M\) by \(X\) is the module \[ X\wedge M=f^{G^{\mathrm o}}_*(M_X) \] of \(T/Y\) (Proposition 2.2 of the lesson on torsors). The functor \(M\mapsto X\wedge M\) is exact, as a composite of exact functors, and acts termwise on \(D^b(B)\). For the trivial torsor \(\pi\colon G\to S\) under right translations, the isomorphism \(c_{M_X}\) of Proposition 2.2 (2) of the lesson on torsors, pulled back along the section \(e\), gives \(X\wedge M\cong e^*M_X=UM\).

Let \(\chi=N(\mathrm{inv})\circ N(G^{\mathrm o},X\to S)\colon\) \(N(G^{\mathrm o},X)\to N(G^{\mathrm o})\to N\), with the levels \(\chi_n(h;x)=(h_1^{-1},\ldots,h_n^{-1})\), and let \(a\colon N(G^{\mathrm o},X)\to Y\), \(a_n(h;x)=f(x)\), be the morphism of the lesson on torsors. By Proposition 2.5 and Lemma 3.1 (1) of the lesson on equivariant cotangent complexes, \(\operatorname{ner}(M_X)=\chi_{\mathrm{big}}^*\operatorname{ner}(M)\).

Proposition 2.8 (twisting by a torsor). In 2.8, \(L\varepsilon^*\ell_{X/Y}\cong X\wedge\ell^{\mathrm{ad}}_G\) in \(D^b(T/Y)\).

Proof. The pulled back torsors. Regard \(Y\) and \(X\) as constant simplicial schemes; then \(f\) is a torsor under \(G^{\mathrm o}\) over the constant simplicial scheme \(Y\), whose co-Lie complex is the constant object with the value \(\ell_{X/Y}\) (Theorem 1.5 (1), the structure maps being base change isomorphisms along identities), and \(a\) is a morphism of simplicial schemes to it, since \(f\) is invariant. The base changes \(a^*X=N(G^{\mathrm o},X)\times_YX\) and \(\chi^*P=N(G^{\mathrm o},X)\times_NP\) are torsors under \(G^{\mathrm o}\) over \(N(G^{\mathrm o},X)\). At the level \(n\), a point of \(\chi^*P\) is a pair of a point \((h;x)\) of \(N_n(G^{\mathrm o},X)\) and a point \(y\) of \(G\), the point \((\chi_n(h;x);y)\) of \(P_n\), with the action \(k\star((h;x),y)=((h;x),yk)\). Let \[ \Theta_n\bigl((h;x),y\bigr)=\bigl((h;x),\,y\star x\bigr), \] \(y\) acting through \(G^{\mathrm o}\). It is an isomorphism onto \(N_n(G^{\mathrm o},X)\times_YX\), as \(f\) is a torsor. It is equivariant: \(k\star(y\star x)=(yk)\star x\), the product \(k\) after \(y\) in \(G^{\mathrm o}\) being \(yk\) in \(G\). It commutes with the maps along \(\alpha\colon[m]\to[n]\): put \(c=h_1\cdots h_{\alpha(0)}\) in \(G\), the product \(h_{\alpha(0)}\cdots h_1\) of \(G^{\mathrm o}\). The map of \(N(G^{\mathrm o},X)\) sends \((h;x)\) to \((N(G^{\mathrm o})(\alpha)(h);c\star x)\) (Lemma 1.3 of the lesson on equivariant cotangent complexes); that of \(P\) (Construction 2.1) sends \(y\) to \(h^{-1}_{\alpha(0)}\cdots h^{-1}_1y=c^{-1}y\); that of \(a^*X\) leaves the point of \(X\) unchanged; and \((c^{-1}y)\star(c\star x)=(c\,c^{-1}y)\star x=y\star x\). So \(\Theta\) is an isomorphism of torsors over \(N(G^{\mathrm o},X)\).

Comparison. Lemma 1.8 (4), for the group \(G^{\mathrm o}\), \(u=\mathrm{id}\) and cartesian squares, gives isomorphisms \[ \begin{aligned}&La^*\ell_{X/Y}\cong\ell_{a^*X/N(G^{\mathrm o},X)}\\&\quad\cong\ell_{\chi^*P/N(G^{\mathrm o},X)}\cong L\chi^*\ell_{P/N},\end{aligned} \] for the projection \(a^*X\to X\) over \(a\), for \(\Theta\) over the identity, and for the projection \(\chi^*P\to P\) over \(\chi\). The comparison morphisms \(\varepsilon\) commute with the morphisms of total topoi induced by \(a\) and \(\chi\), whose inverse images on the big fppf topoi are exact, so \(L\varepsilon^*La^*\cong a_{\mathrm{big}}^*L\varepsilon^*\) and \(L\varepsilon^*L\chi^*\cong\chi_{\mathrm{big}}^*L\varepsilon^*\) [Tag 0D6D], and \[ \begin{aligned}&a_{\mathrm{big}}^*L\varepsilon^*\ell_{X/Y}\cong\chi_{\mathrm{big}}^*L\varepsilon^*\ell_{P/N}\\&\quad\cong\chi_{\mathrm{big}}^*\operatorname{ner}(\ell^{\mathrm{ad}}_G)=\operatorname{ner}\bigl((\ell^{\mathrm{ad}}_G)_X\bigr),\end{aligned} \] by Proposition 2.2 (2) and 2.8. By Theorem 2.4 of the lesson on torsors, for the group \(G^{\mathrm o}\), the left side is \(\operatorname{ner}(f^{G^{\mathrm o}*}L\varepsilon^*\ell_{X/Y})\). As \(\operatorname{ner}\) is fully faithful on \(D^b(B[G^{\mathrm o},X]^{\mathrm o})\) (Proposition 1.5 (3) of the lesson on equivariant cotangent complexes), \(f^{G^{\mathrm o}*}L\varepsilon^*\ell_{X/Y}\cong(\ell^{\mathrm{ad}}_G)_X\). Applying the quasi-inverse \(f^{G^{\mathrm o}}_*\) (Proposition 2.2 (3) of the lesson on torsors) gives \(L\varepsilon^*\ell_{X/Y}\cong X\wedge\ell^{\mathrm{ad}}_G\). \(\square\)

So the co-Lie complex of any torsor under \(G^{\mathrm o}\) is the adjoint co-Lie complex twisted by the torsor. For the trivial torsor it gives \(L\varepsilon^*\ell_{G/S}\cong U\ell^{\mathrm{ad}}_G\), the last statement of Proposition 2.2 (2).

3. The cotangent complex of the nerve

Let \(i_0\colon S_{\mathrm{Zar}}\to\operatorname{Top}(N_{\mathrm{Zar}})\) and \(i'_0\colon G_{\mathrm{Zar}}\to\operatorname{Top}(P_{\mathrm{Zar}})\) be the morphisms of the levels \(0\), \(N_0=S\) and \(P_0=G\). Their inverse images \(i_0^*\), \(i'^*_0\) are the restrictions to the level \(0\). They have the left adjoints (Lemma 4.2 (1) of the lesson on big fppf sites) \[ \begin{aligned}(i_{0!}E)_n&=\bigoplus_{k=0}^n\pi_n^*E,\\ (i'_{0!}F)_n&=\bigoplus_{k=0}^nP(v_k)^*F,\end{aligned} \] the summand \(k\) being that of the vertex \(v_k\), with \(N(v_k)=\pi_n\). Every \(P(v_k)\) is flat, being the projection \(G^{n+1}\to G\) under the isomorphism of Construction 2.1, and so is \(\pi_n\). So \(i_{0!}\) and \(i'_{0!}\) are exact, and \(i_0^*\), \(i'^*_0\) carry injective modules to injective modules [Tag 015Z]. On the summand of \(v_k\) at the level \(n\), the adjunction map \(i_{0!}i_0^*K\to K\) is the structure map \(K_{v_k}\colon N(v_k)^*K_0\to K_n\) (Proposition 3.1 (1) of the lesson on fibred topoi), and similarly on \(P\).

Lemma 3.1 (products). Let \(X_0,\ldots,X_m\) be flat \(S\)-schemes and \(X=X_0\times_S\cdots\times_SX_m\), with the projections \(p_k\). The sum of the functoriality maps \(p_k^*L_{X_k/S}\to L_{X/S}\) of the squares formed by \(p_k\) and \(\mathrm{id}_S\) is an isomorphism \(\bigoplus_kp_k^*L_{X_k/S}\cong L_{X/S}\).

Proof. By induction on \(m\), the case \(m=0\) being trivial. Write \(X=X'\times_SX_m\), with \(p'\colon X\to X'=X_0\times_S\cdots\times_SX_{m-1}\). Let \(w\colon L_{X/S}\to L_{X/X'}\) be the second map of the transitivity triangle of \(X\to X'\to S\) (Proposition 1.3 of the lesson on deformations of diagrams), and \(F_m\colon p_m^*L_{X_m/S}\to L_{X/S}\) the functoriality map of \(p_m\). The composite \(w\circ F_m\) is the functoriality map of the cartesian square formed by \(p_m\), \(p'\) and the structure morphisms (Theorem 2.2 of The cotangent complex), an isomorphism by Proposition 1.4 of the lesson on deformations of diagrams, \(X_m\) being flat over \(S\). So \(F_m\circ(w F_m)^{-1}\) is a right inverse of \(w\), and by [Tag 05QT] the sum of \(F_m\) and the first map \(p'^*L_{X'/S}\to L_{X/S}\) of the triangle is an isomorphism \(p'^*L_{X'/S}\oplus p_m^*L_{X_m/S}\cong L_{X/S}\). By induction, \(p'^*L_{X'/S}\) is the sum of the pullbacks of the \(L_{X_k/S}\), \(k<m\), and functoriality maps compose. \(\square\)

Lemma 3.2. The adjunction map \(i'_{0!}i'^*_0L_{P/S}\to L_{P/S}\) is an isomorphism.

Proof. At the level \(n\) its restriction to the summand of \(v_k\) is the structure map of \(L_{P/S}\) along \(v_k\), the functoriality map of the square formed by \(P(v_k)\) and \(\mathrm{id}_S\) (Proposition 1.2 (1) of the lesson on deformations of diagrams). Under the isomorphism \(P_n\cong G^{n+1}\) of Construction 2.1, \(P(v_k)\) is the projection \(p_k\), and functoriality maps are compatible with this isomorphism (Theorem 2.2 of The cotangent complex). So Lemma 3.1, with all \(X_k=G\), shows that the level \(n\) is an isomorphism. \(\square\)

Theorem 3.3 (the cotangent complex of the nerve). There is a distinguished triangle in \(D(N_{\mathrm{Zar}})\) \[ \begin{aligned}&L_{N/S}\longrightarrow i_{0!}i_0^*\ell_{P/N}\\&\quad\xrightarrow{\ \epsilon\ }\ell_{P/N}\xrightarrow{\ \mathrm{at}\ }L_{N/S}[1],\end{aligned} \tag{3.1} \] where \(\epsilon\) is the adjunction map and \(\mathrm{at}\) is the Atiyah class of the torsor \(P\) over \(N\). Here \(i_0^*\ell_{P/N}\cong\ell_G\), so the level \(n\) of the middle term is \(\bigoplus_{k=0}^n\pi_n^*\ell_G\). At every level the triangle is split: the level \(n\) of \(\epsilon\) has a right inverse and the level \(n\) of \(\mathrm{at}\) vanishes.

Proof. By Theorem 3.1 of Flat group schemes are local complete intersections, \(L_{G/S}\) lies in \(D^{[-1,0]}\). By Lemma 3.1, so do \(L_{N_n/S}\) and \(L_{P_n/S}\), with \(N_n=G^n\) and \(P_n\cong G^{n+1}\). The levels of the nerve \(Q=N(G^{\mathrm o},P)\) are \(Q_{(r,n)}=G^r\times_SP_n\), and \(L_{Q_{(r,n)}/G^r}\cong\operatorname{pr}^*L_{P_n/S}\) by Proposition 1.4 of the lesson on deformations of diagrams, \(G^r\) being flat over \(S\). So all these complexes lie in \(D^{[-1,0]}\), on which \(t_{[-1}\) is the identity. Theorem 1.5 (2), (3) for the \(G^{\mathrm o}\)-torsor \(P\) over \(N\), with \(n=-1\), gives \(\lambda=\ell_{-1}(P/S)\) with \(b^*\lambda\cong L_{Q/N(G^{\mathrm o})}\), where \(b\colon Q\to N\) is the morphism called \(a\) in Section 1, and a distinguished triangle \[ \begin{aligned}&L_{N/S}\xrightarrow{\ \beta\ }\lambda\xrightarrow{\ \gamma\ }\ell_{P/N}\\&\qquad\xrightarrow{\ \mathrm{at}\ }L_{N/S}[1].\end{aligned} \]

Step 1: \(i_0^*(\gamma)\) is an isomorphism. By Theorem 1.5 (3), the level \(0\) of this triangle is the triangle (3.1) of the lesson on torsors for \(P_0=G\to N_0=S\to S\), whose first vertex \(t_{[-1}L_{S/S}\) vanishes. Rotating it gives a distinguished triangle with the first map \(i_0^*(\gamma)\) and the third vertex \(0\), so \(i_0^*(\gamma)\) is an isomorphism [Tag 05QR].

Step 2: the adjunction map \(\epsilon_\lambda\colon i_{0!}i_0^*\lambda\to\lambda\) is an isomorphism. Every \(\varphi_n\) is flat and surjective, being a base change of \(\pi\), which has the section \(e\). Both sides have termwise quasi-coherent cohomology, so it suffices to show that \(\varphi^*(\epsilon_\lambda)\) is an isomorphism (Remark 3.5 of the lesson on deformations of diagrams, applied to the cohomology of the cone at each level). Since \(N(v_k)\varphi_n=\varphi_0P(v_k)\), there is an isomorphism \(\varphi^*i_{0!}E\cong i'_{0!}\varphi_0^*E\), natural in \(E\), and under it \(\varphi^*(\epsilon_K)\) is the adjunction map \(\epsilon'_{\varphi^*K}\) of \(\varphi^*K\), the structure maps of \(\varphi^*K\) being \(\varphi^*\) of those of \(K\). The restriction of \(Q\) to the row \(\{0\}\times\Delta^{\mathrm{op}}\) is \(P\), since \(N_0(G^{\mathrm o},P_n)=P_n\), and on it \(b\) restricts to \(\varphi\) and \(N(G^{\mathrm o})\) to the constant diagram \(S\). By the Conventions, restricting \(b^*\lambda\cong L_{Q/N(G^{\mathrm o})}\) to this row gives \(\varphi^*\lambda\cong L_{P/S}\). The adjunction map is natural, so \(\epsilon'_{\varphi^*\lambda}\) is an isomorphism because \(\epsilon'_{L_{P/S}}\) is, by Lemma 3.2.

Step 3. The naturality of the adjunction map gives \(\gamma\circ\epsilon_\lambda=\epsilon\circ i_{0!}i_0^*(\gamma)\). By Steps 1 and 2, \(\mu=i_{0!}i_0^*(\gamma)\circ\epsilon_\lambda^{-1}\colon\lambda\to i_{0!}i_0^*\ell_{P/N}\) is an isomorphism with \(\gamma=\epsilon\circ\mu\). So \((\mathrm{id},\mu,\mathrm{id})\) is an isomorphism from the triangle above onto (3.1), with first map \(\mu\beta\), and (3.1) is distinguished. By Proposition 2.2 (1), \(i_0^*\ell_{P/N}\cong\ell_G\).

At the level \(n\), the summand of \(v_0\) maps to \(\ell_n\) by the structure map \(\ell_{v_0}\), an isomorphism, \(\ell_{P/N}\) being quasi-cartesian. So the restriction \(\epsilon|_n\) of \(\epsilon\) to the level \(n\) has a right inverse \(s\), and the restriction of \(\mathrm{at}\) is \(\mathrm{at}|_n=\mathrm{at}|_n\circ\epsilon|_n\circ s=0\) [Tag 0146]. \(\square\)

So \(L_{N(G)/S}\) sits between a copy of \(\ell_G\) at every vertex and the adjoint co-Lie complex, and at every level it is a complement of one summand \(\ell_G\) (Exercise 4.4).

Corollary 3.4 (Ext groups). For \(M\in D^+(N_{\mathrm{Zar}})\) there is a long exact sequence \[ \begin{aligned}&\quad \cdots \to\operatorname{Ext}^j(\ell_{P/N},M)\\ &\quad \xrightarrow{\ i_0^*\ }\operatorname{Ext}^j_S(\ell_G,i_0^*M)\\ &\quad \quad \to\operatorname{Ext}^j(L_{N/S},M)\\ &\quad \to\operatorname{Ext}^{j+1}(\ell_{P/N},M)\\ &\quad \quad \xrightarrow{\ i_0^*\ }\operatorname{Ext}^{j+1}_S(\ell_G,i_0^*M)\\ &\quad \to\cdots,\end{aligned} \] in which the maps \(i_0^*\) are restriction to the level \(0\) through \(i_0^*\ell_{P/N}\cong\ell_G\). In the derived category of abelian groups, \(R\operatorname{Hom}(L_{N/S},M)[-1]\) is a cocone of the restriction map \(R\operatorname{Hom}(\ell_{P/N},M)\to R\operatorname{Hom}_S(\ell_G,i_0^*M)\): the Ext groups of \(L_{N/S}\) are relative Ext groups of the adjoint co-Lie complex, shifted by one.

Proof. Apply \(\operatorname{Hom}(-,M[j])\) to (3.1) [Tag 0149]. Since \(i_{0!}\) is exact with the exact right adjoint \(i_0^*\), which preserves injectives, \(R\operatorname{Hom}(i_{0!}E,M)=R\operatorname{Hom}_S(E,i_0^*M)\): represent \(M\) by a bounded below complex of injectives \(J\); then \(i_0^*J\) is one, both are K-injective [Tag 070J], and \(\operatorname{Hom}^\bullet(i_{0!}E,J)=\operatorname{Hom}^\bullet(E,i_0^*J)\). Under this identification, composition with \(\epsilon\) is the map induced by \(i_0^*\). For the second statement, \(R\operatorname{Hom}(-,M)\), the derived global sections of the derived sheaf Hom, carries distinguished triangles to distinguished triangles [Tag 08JC]. \(\square\)

Write \(T_n=T/N_n\) for the big fppf topos of \(N_n\), so that \(\operatorname{Top}(N_{\mathrm{fppf}})\) is the total topos of the \(T_n\). For complexes \(E\in D^-(B)\) and \(F\in D^+(B)\), the forgetful functor induces a map \(U\colon R\operatorname{Hom}_B(E,F)\to R\operatorname{Hom}_T(UE,UF)\); write \(R\operatorname{Hom}_{B/S}(E,F)\) for a cocone of it and \(\operatorname{Ext}^j_{B/S}(E,F)\) for its cohomology, the relative Ext groups.

Corollary 3.5 (relative Ext groups on the classifying topos).

1. For \(E\in D^-(B)\) and \(F\in D^+(B)\), \(R\operatorname{Hom}_{B/S}(E,F)\) may be taken to be the total complex of the columns \(n\geq1\) of the double complex (3.2) of the lesson on mixed Hom complexes. There is a convergent spectral sequence \[ \begin{gathered}E_1^{n,k}\\ =\operatorname{Ext}^k_{T_n}(\pi_n^*UE,\pi_n^*UF)\ \ (n\geq1),\\ E_1^{0,k} =0,\quad E_1^{n,k}\\ \Rightarrow\operatorname{Ext}^{n+k}_{B/S}(E,F),\end{gathered} \] whose rows \(E_1^{\bullet,k}\) are the cochain complexes of the cosimplicial groups \([n]\mapsto\operatorname{Ext}^k_{T_n}(\pi_n^*UE,\pi_n^*UF)\) in degrees \(n\geq1\).
2. For \(M\in D^+(B)\) there is an isomorphism \(R\operatorname{Hom}(L\varepsilon^*L_{N/S},\operatorname{ner}M)\cong\) \(R\operatorname{Hom}_{B/S}(\ell^{\mathrm{ad}}_G,M)[1]\) in the derived category of abelian groups.
3. Let \(I\) be a quasi-coherent \(\mathcal O_S\)-module, \(\pi_N^*I\) the cartesian module of \(\operatorname{Top}(N_{\mathrm{Zar}})\) with the levels \(\pi_n^*I\), and \(\varepsilon^*I\) the module of \(B\) with the trivial structure. Then \[ \begin{aligned}&\operatorname{Ext}^j(L_{N/S},\pi_N^*I)\\&\quad \cong\operatorname{Ext}^{j+1}_{B/S}(\ell^{\mathrm{ad}}_G,\varepsilon^*I),\end{aligned} \] and the spectral sequence of (1) for these groups has the terms \[ \begin{aligned}E_1^{n,k}&=\operatorname{Ext}^k_{G^n}(\pi_n^*\ell_G,\pi_n^*I)\\&=H^k\bigl(G^n,\pi_n^*I\otimes^{\mathbf L}(\pi_n^*\ell_G)^\vee\bigr)\end{aligned} \] for \(n\geq1\), the Ext and cohomology groups being those of the Zariski topos of \(G^n\).

Proof. (1) Let \(X\) be the simplicial topos of the nerve of the internal category \([G,S]^{\mathrm o}\) of \(T\), with the levels \(T_n\) (Proposition 1.5 (1) of the lesson on equivariant cotangent complexes); all its vertices \(X_n\to X_0=T\) are the localizations along \(\pi_n\). It is a good two-sided pseudo-category satisfying the conditions (I) and (E) (Proposition 5.1 of the lesson on cohomology of equivariant modules), and \(BX=B\). Represent \(E\) and \(F\) as in the proof of Theorem 3.3 of the lesson on mixed Hom complexes, \(F\) by a bounded below complex \(I\) of injective modules, and let \(C^{\bullet,\bullet}\) be the double complex (3.2) there (written \(G\) there). By that theorem, \(K=\operatorname{Hom}^\bullet_B(E,I)\) is the kernel of the first coface differential on the column \(C^{\bullet,0}\), and its inclusion is a quasi-isomorphism \(K\to\operatorname{Tot}(C)\). The projection \(\operatorname{Tot}(C)\to C^{\bullet,0}\) is a map of complexes, because the coface differential raises \(n\); its kernel is the total complex \(\operatorname{Tot}(C^{\bullet,\geq1})\) of the columns \(n\geq1\), and the sequence is termwise split. The column \(C^{\bullet,0}=\operatorname{Hom}^\bullet(UE,UI)\) computes \(R\operatorname{Hom}_T(UE,UF)\), the modules \(UI^q\) being injective by (I) (Proposition 6.2 (3) of the lesson on equivariant sheaves), and the composite \(K\to C^{\bullet,0}\) is the map induced by \(U\). So the triangle of the termwise split sequence [Tags 014I, 014Q] shows that \(\operatorname{Tot}(C^{\bullet,\geq1})\) is a cocone of \(U\). Filtering it by \(n\), the filtration being finite in each degree as in the proof of Corollary 3.4 of the lesson on mixed Hom complexes, gives the spectral sequence, with the column cohomology \(\operatorname{Ext}^k(X(v_0)^*E,X(v_n)^*F)\) computed there.

(2) Apply \(L\varepsilon^*\), an exact functor, to (3.1). The functor \(L\varepsilon^*\) commutes with \(i_{0!}\), levelwise (Lemma 4.2 (1) of the lesson on big fppf sites) and compatibly with the adjunction maps, the fppf \(i_{0!}\) being defined by the same formula. This gives a distinguished triangle \[ \begin{aligned}&L\varepsilon^*L_{N/S}\to i_{0!}i_0^*\operatorname{ner}\ell^{\mathrm{ad}}_G\\&\quad\xrightarrow{\ \epsilon\ }\operatorname{ner}\ell^{\mathrm{ad}}_G\to L\varepsilon^*L_{N/S}[1].\end{aligned} \] As in the proof of Corollary 3.4, \(R\operatorname{Hom}(i_{0!}i_0^*\operatorname{ner}\ell^{\mathrm{ad}}_G,\operatorname{ner}M)=\) \(R\operatorname{Hom}_T(U\ell^{\mathrm{ad}}_G,UM)\), the level \(0\) of a nerve being the underlying module. By Theorem 4.3 (2) of the lesson on cohomology of equivariant modules, the nerve induces an isomorphism \(R\operatorname{Hom}_B(\ell^{\mathrm{ad}}_G,M)\cong\) \(R\operatorname{Hom}(\operatorname{ner}\ell^{\mathrm{ad}}_G,\operatorname{ner}M)\). Composed with restriction to the level \(0\), it is the map induced by \(U\): on complexes, the nerve of a map of modules of \(B\) has the level \(0\) map itself. So \(R\operatorname{Hom}(L\varepsilon^*L_{N/S},\operatorname{ner}M)[-1]\) and \(R\operatorname{Hom}_{B/S}(\ell^{\mathrm{ad}}_G,M)\) are cocones of the same map [Tag 08JC], hence isomorphic [Tag 014A].

(3) By Lemma 4.2 (2) of the lesson on big fppf sites, with the truncation index \(0\) and \(t_{[0}L\varepsilon^*\pi_N^*I=\varepsilon^*\pi_N^*I\), \(R\operatorname{Hom}(L_{N/S},\pi_N^*I)\cong\) \(R\operatorname{Hom}(L\varepsilon^*L_{N/S},\varepsilon^*\pi_N^*I)\), and \(\varepsilon^*\pi_N^*I=\operatorname{ner}(\varepsilon^*I)\). So (2) gives the isomorphism. In (1), \(\pi_n^*U\ell^{\mathrm{ad}}_G=L\varepsilon^*\pi_n^*\ell_G\) and \(\pi_n^*U\varepsilon^*I=\varepsilon^*\pi_n^*I\), and Corollary 2.2 (1) of the lesson on big fppf sites, over \(G^n\), identifies the Ext groups over \(T_n\) with the Zariski ones. Finally \(\pi_n^*\ell_G\) is perfect, and [Tag 08DQ] gives the cohomological form. \(\square\)

Proposition 3.6 (group schemes over a diagram). Let \(G\) be a group scheme over a diagram \(Z\colon I\to(\mathrm{Sch})\), flat and locally of finite presentation (1.6). Let \(N=N(G)\) and \(P=N(G,G)\), the nerve of \(G\) acting on itself by left multiplication, diagrams indexed by \(\Delta^{\mathrm{op}}\times I\), with \(\varphi\colon P\to N\), \(p\colon\Delta^{\mathrm{op}}\times I\to I\) the projection and \(\pi_N\colon N\to Z\) of type \(p\).

1. The opposite groups \(G^{\mathrm o}_i\) form a group scheme \(G^{\mathrm o}\) over \(Z\), and \(G^{\mathrm o}\) acts on \(P\) by right multiplication on the last factor. With this action, \(\varphi\) is a torsor under the group scheme \(G^{\mathrm o}\circ p\) over the diagram \(N\) over \(Z\circ p\), which is cartesian along the arrows \((\alpha,\mathrm{id}_i)\).
2. The co-Lie complex \(\ell_{P/N}\) of Proposition 1.7 has the levels \(\pi_{(n,i)}^*\ell_{G_i}\), its structure maps along the arrows \((\alpha,\mathrm{id}_i)\) are isomorphisms, and its restriction to the row \(\{[0]\}\times I\) is isomorphic to the co-Lie complex \(\ell_G\) of \(G\) over \(Z\).
3. Let \(i_0\) be the inclusion of the row \(\{[0]\}\times I\), on which \(N\) restricts to \(Z\). Theorem 3.3 holds: the triangle (3.1), formed with these \(N\), \(P\) and \(i_0\), is distinguished, the second map being the adjunction map and the third the Atiyah class of the torsor \(P\) over \(N\), and it is split at every level.

Proof. (1) Left and right multiplications commute, and every \(G_\gamma\) is a homomorphism, so the maps of \(P\) are equivariant along the maps \(G^{\mathrm o}_\gamma\), and \(\varphi\) is invariant. At \(([n],i)\), \(\varphi\) is the torsor of Construction 2.1 for the group \(G_i\) over \(Z_i\). Along \((\alpha,\mathrm{id}_i)\) the group does not change.

(2) The levels are those of Proposition 2.2 (1) for \(G_i\) over \(Z_i\). Along \((\alpha,\mathrm{id}_i)\) the squares of the nerve of the torsor \(P\) are cartesian, as in Lemma 1.2 (3), the group being constant along these arrows; so, as in the proof of Theorem 1.5 (1), the structure maps of \(\ell_{P/N}\) along them are isomorphisms. The restriction to the row is the co-Lie complex of the restricted torsor \(\pi\colon G\to Z\) under \(G^{\mathrm o}\), by the uniqueness in Theorem 1.5 (1), since restriction along the row commutes with \(a^*\) and carries the cotangent complexes of the nerves to those of the restricted nerves (Conventions). By Proposition 1.7 (3), for the right multiplication, it is \(Le^*L_{G/Z}=\ell_G\).

(3) As in Proposition 3.1 (1) of the lesson on fibred topoi and Lemma 1.3 (1), the restriction \(i_0^*\) to the row has the exact left adjoint \(i_{0!}\) with \((i_{0!}E)_{([n],i)}=\bigoplus_{k=0}^n\pi_{(n,i)}^*E_i\), the summand \(k\) being that of the vertex \(v_k\). Along an arrow \((\alpha,\gamma)\colon([n],i)\to([m],j)\), with \(\alpha\colon[m]\to[n]\), the summand \(k\) maps to the summand \(\alpha(k)\) by the pullback of \(E_\gamma\). On the summand of \(v_k\) the adjunction map \(i_{0!}i_0^*K\to K\) is the structure map of \(K\) along \((v_k,\mathrm{id}_i)\), and similarly on \(P\). Lemmas 3.1 and 3.2 hold at every \(([n],i)\), over \(Z_i\). The morphism \(N\to Z\circ p\) and the morphism \(\pi_N\) of type \(p\) give the same ring of \(\operatorname{Top}(N_{\mathrm{Zar}})\), so \(L_{N/Z\circ p}=L_{N/Z}\). With these replacements the proof of Theorem 3.3 applies. Theorem 1.5 (2), (3) for the torsor \(P\) over \(N\) under \(G^{\mathrm o}\circ p\) are Proposition 1.7 (1). In Step 1, the row of the triangle of Theorem 1.5 (3) has the first vertex \(t_{[-1}L_{Z/Z}=0\). In Step 2, every \(\varphi_{(n,i)}\) is flat and surjective, and the restriction of the nerve \(Q\) of the torsor \(P\) to its row of index \(0\) is \(P\), on which the base \(N(G^{\mathrm o}\circ p)\) restricts to \(Z\circ p\). The splitting uses the structure maps of \(\ell_{P/N}\) along the arrows \((v_k,\mathrm{id}_i)\), isomorphisms by (2). \(\square\)

Lemma 3.7 (perfect objects of the classifying topos). The category \(B\) is a topos, being \(BX\) for the good pseudo-category \(X\) of the proof of Corollary 3.5 (Corollary 4.2 of the lesson on Giraud's characterization of topoi), and it is ringed by \(\mathcal O\) with the trivial action. Let \(G_\rho\) be the sheaf \(G\) of \(T\) with the action of \(G\) by right multiplication, an object of \(B\).

1. The functor \(F\mapsto F\times G_\rho\), with the projection to \(G_\rho\) and \(G\) acting on the second factor, is an equivalence of ringed topoi from \(T\) to the localization \(B/G_\rho\). Under it, restriction to \(B/G_\rho\) is the forgetful functor \(U\), and \(G_\rho\to*\) is an epimorphism of \(B\).
2. An object \(K\) of \(D(B)\) is perfect if \(UK\) is perfect, and \(U(K^\vee)\cong(UK)^\vee\), where \(K^\vee=R\mathcal Hom(K,\mathcal O)\).
3. For perfect \(K\), \(M\in D(B)\) and \(F\in D^+(T)\) there are isomorphisms, natural in \(M\) and \(F\), \[ \begin{aligned}&R\operatorname{Hom}_B(K,M)\\&\quad\cong R\Gamma(B,K^\vee\otimes^{\mathbf L}M),\\&K^\vee\otimes^{\mathbf L}R\operatorname{coind}F\\&\quad\cong R\operatorname{coind}\bigl(U(K^\vee)\otimes^{\mathbf L}F\bigr),\end{aligned} \] the second induced by the counit \(UR\operatorname{coind}F\to F\). Here \(R\operatorname{coind}F\) is \(\operatorname{coind}\) of a bounded below complex \(J\) of injective modules representing \(F\). The functor \(\operatorname{coind}\) is right adjoint to the exact functor \(U\) and carries injective modules to injective modules (Propositions 4.2 and 6.2 of the lesson on equivariant sheaves), and bounded below complexes of injective modules are K-injective [Tag 070J]; so \(\operatorname{Hom}_{D(B)}(W,R\operatorname{coind}F)=\) \(\operatorname{Hom}_{D(T)}(UW,F)\) for every \(W\in D(B)\), both sides being homotopy classes of maps of complexes.

Proof. (1) For an object \(V\) of \(B\) with an equivariant map \(p\colon V\to G_\rho\), let \(\Phi(V)=V\times_{G_\rho,e}S\) be its fibre over the unit section, a sheaf of \(T\). The map \(\Phi(V)\times G_\rho\to V\), \((x,g)\mapsto xg\), is equivariant and lies over \(G_\rho\), since \(p(xg)=p(x)g=g\); it is an isomorphism, with inverse \(y\mapsto(y\,p(y)^{-1},p(y))\). Also \(\Phi(F\times G_\rho)=F\). These identifications respect the rings, so \(B/G_\rho\) is the ringed topos \((T,\mathcal O)\) [Tag 04GZ]. The restriction of an object \(V\) of \(B\) to \(B/G_\rho\) is the projection \(V\times G_\rho\to G_\rho\), whose fibre over the unit is \(V\) with its action forgotten, that is, \(UV\). The functor \(U\) is faithful, a morphism of \(B\) being a morphism of \(T\) that satisfies a condition, and it carries \(G_\rho\to*\) to \(\pi\colon G\to S\), an epimorphism split by \(e\); so \(G_\rho\to*\) is an epimorphism.

(2) By (1), restriction to \(B/G_\rho\) is \(U\), and for every object \(V\) of \(B\) the projection \(V\times G_\rho\to V\), a base change of the epimorphism \(G_\rho\to*\), is an epimorphism, hence a covering. If \(UK\) is perfect, the restriction of \(K\) to \(B/G_\rho\) is perfect, so every \(V\) has a covering, refining \(\{V\times G_\rho\to V\}\), on whose members \(K\) is strictly perfect: \(K\) is perfect [Tag 08G5]. Restriction to a localization commutes with derived internal Hom [Tag 08JB], so \(U(K^\vee)=(UK)^\vee\).

(3) For the first isomorphism, \(R\operatorname{Hom}_B(K,M)\) is \(R\Gamma(B,R\mathcal Hom(K,M))\), computed with a K-injective complex representing \(M\) [Tag 08JA], and \(R\mathcal Hom(K,M)\cong K^\vee\otimes^{\mathbf L}M\) [Tag 08JJ]. For the second, \(U\), the inverse image of the localization at \(G_\rho\), commutes with derived tensor products [Tag 07A4]. For a perfect object \(P\) and objects \(W\), \(Y\) of the derived category of a ringed topos, \(\operatorname{Hom}(W,P^\vee\otimes^{\mathbf L}Y)=\) \(\operatorname{Hom}(W,R\mathcal Hom(P,Y))=\) \(\operatorname{Hom}(W\otimes^{\mathbf L}P,Y)\) [Tags 08JJ, 08J9, 08JA]. So for every \(W\in D(B)\), the tensor products being derived, \[ \begin{aligned}&\operatorname{Hom}(W,K^\vee\otimes R\operatorname{coind}F)\\&\quad=\operatorname{Hom}(W\otimes K,R\operatorname{coind}F)\\&\quad=\operatorname{Hom}(UW\otimes UK,F)\\&\quad=\operatorname{Hom}(UW,(UK)^\vee\otimes F)\\&\quad=\operatorname{Hom}\bigl(W,R\operatorname{coind}(U(K^\vee)\otimes F)\bigr),\end{aligned} \] naturally in \(W\), using (2) in the last step. By the Yoneda lemma this gives an isomorphism; with \(W=K^\vee\otimes R\operatorname{coind}F\), the identity is carried to the map adjoint to \(\mathrm{id}\otimes\text{counit}\colon\) \(U(K^\vee)\otimes UR\operatorname{coind}F\to\) \(U(K^\vee)\otimes F\). \(\square\)

Proposition 3.8 (the dual form). For \(F\in D^+(B)\) put \(R\Gamma(B/S,F)=R\operatorname{Hom}_{B/S}(\mathcal O,F)\), the relative cohomology of \(B\) over \(S\), with the groups \(H^j(B/S,F)\). Let \(K\in D^b(B)\) with \(UK\) perfect, and \(M\in D^+(B)\). Then \(K^\vee\otimes^{\mathbf L}M\in D^+(B)\), and \[ \begin{aligned}&R\operatorname{Hom}_{B/S}(K,M)\\&\quad\cong R\Gamma(B/S,K^\vee\otimes^{\mathbf L}M).\end{aligned} \] In particular, as \(U\ell^{\mathrm{ad}}_G=L\varepsilon^*\ell_G\) (Proposition 2.2 (2)) is perfect (Theorem 3.1 of the lesson on big fppf sites), \[ \begin{aligned}&R\operatorname{Hom}(L\varepsilon^*L_{N/S},\operatorname{ner}M)\\&\quad\cong R\Gamma\bigl(B/S,(\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}M\bigr)[1],\end{aligned} \] and \(\operatorname{Ext}^j(L_{N/S},\pi_N^*I)\cong\) \(H^{j+1}\bigl(B/S,(\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\) in the situation of Corollary 3.5 (3).

Proof. By Lemma 3.7 (2), \(K\) is perfect, and \(K^\vee\otimes^{\mathbf L}M\cong R\mathcal Hom(K,M)\) [Tag 08JJ]. If \(K\) is represented by a complex in degrees \(\leq b\) and \(M\) by a complex \(J\) of injective modules in degrees \(\geq m\), which is K-injective [Tag 070J], then \(R\mathcal Hom(K,M)\) is represented by \(\mathcal Hom^\bullet(K,J)\), which vanishes in degrees \(<m-b\); so \(K^\vee\otimes^{\mathbf L}M\in D^+(B)\).

Let \(E\in D^-(B)\), \(M\in D^+(B)\) and \(\eta_M\colon M\to R\operatorname{coind}UM\) the unit. Under the adjunction isomorphism \(R\operatorname{Hom}_B(E,R\operatorname{coind}UM)\cong\) \(R\operatorname{Hom}_T(UE,UM)\), the map induced by \(U\) is \(R\operatorname{Hom}_B(E,\eta_M)\): with \(M\) represented by a bounded below complex \(J\) of injective modules, \(UJ\) is one by (I) (Proposition 6.2 (3) of the lesson on equivariant sheaves), and on the complexes \(\operatorname{Hom}^\bullet(E,J)\to\) \(\operatorname{Hom}^\bullet(E,\operatorname{coind}UJ)=\) \(\operatorname{Hom}^\bullet(UE,UJ)\) it is \(f\mapsto Uf\), by the identity \(\text{counit}\circ U\eta_J=\mathrm{id}\). Choose a distinguished triangle \(M^{\mathrm{rel}}\to M\xrightarrow{\eta_M}R\operatorname{coind}UM\to M^{\mathrm{rel}}[1]\). Applying \(R\operatorname{Hom}_B(E,-)\) gives a distinguished triangle [Tag 08JC], so \(R\operatorname{Hom}_B(E,M^{\mathrm{rel}})\) is a cocone of the map induced by \(U\), as \(R\operatorname{Hom}_{B/S}(E,M)\) is by definition; two cocones of one map are isomorphic [Tag 014A]. Hence \[ \begin{aligned}&R\operatorname{Hom}_{B/S}(K,M)\\&\quad\cong R\operatorname{Hom}_B(K,M^{\mathrm{rel}})\\&\quad\cong R\Gamma(B,K^\vee\otimes^{\mathbf L}M^{\mathrm{rel}})\end{aligned} \] by Lemma 3.7 (3). Tensoring the triangle with \(K^\vee\) gives a distinguished triangle whose third term \(K^\vee\otimes^{\mathbf L}R\operatorname{coind}UM\) is \(R\operatorname{coind}U(K^\vee\otimes^{\mathbf L}M)\) by Lemma 3.7 (3) with \(F=UM\). Under this identification the map from \(K^\vee\otimes^{\mathbf L}M\) is the unit \(\eta_{K^\vee\otimes M}\), since both maps have the adjoint \(\mathrm{id}\): the identification is adjoint to \(\mathrm{id}\otimes\text{counit}\), and \(\text{counit}\circ U\eta_M=\mathrm{id}\). So \(K^\vee\otimes^{\mathbf L}M^{\mathrm{rel}}\) is a cocone of \(\eta_{K^\vee\otimes M}\), and the first step, applied to \(E=\mathcal O\) and to \(K^\vee\otimes^{\mathbf L}M\in D^+(B)\) in place of \(M\), gives \(R\Gamma(B,K^\vee\otimes^{\mathbf L}M^{\mathrm{rel}})\cong\) \(R\operatorname{Hom}_{B/S}(\mathcal O,K^\vee\otimes^{\mathbf L}M)\). The last assertions follow from Corollary 3.5 (2), (3). \(\square\)

3.9. Homomorphisms. Let \(u\colon F\to G\) be a homomorphism of group schemes over \(S\), flat and locally of finite presentation, and \(N(u)\colon N(F)\to N\) the morphism of nerves. Put \(P'=N(F)\times_NP\), the base change of the \(G^{\mathrm o}\)-torsor \(\varphi\colon P\to N\) along \(N(u)\). Its levels are \(P'_n=F^n\times_SG\), with the maps \(P'(\alpha)(f;x)=\) \(\bigl(N(\alpha)f;u(f_{\alpha(0)})\cdots u(f_1)x\bigr)\): \(P'\) is the nerve of the action of \(F\) on \(G\) by left multiplication through \(u\). Let \(h\colon P_F=N(F,F)\to P'\) be the morphism with the levels \(\mathrm{id}_{F^n}\times u\), and \(h_G\colon P'\to P\) the projection, with the levels \(u^n\times\mathrm{id}_G\); then \(h_Gh\) is the morphism of Proposition 2.5 (1). Let \(F^{\mathrm o}\) act on \(P_F\) as in Construction 2.1, on \(P'\) and on \(P\) by right multiplication through \(u\), and trivially on \(N(F)\) and \(N\); \(h\), \(h_G\) and the projections are equivariant. For \(F^{\mathrm o}\) acting on \(P_F\), \(P'\) or \(P\), the squares formed by the maps of the nerves of the actions along an arrow of \(\Delta^{\mathrm{op}}\) in the direction of the action and by the maps induced by \(h\) or \(h_Gh\) are cartesian: for the action maps, \((k,x)\mapsto\bigl((k,h(x)),xk\bigr)\) is an isomorphism from \(F\times P_{F,n}\) onto the fibre product of \(F\times P'_n\to P'_n\), \((k,y)\mapsto yu(k)\), with \(h\), with inverse \(((k,y),z)\mapsto(k,zk^{-1})\), and likewise for \(h_Gh\); for the projections this is clear. The maps of these nerves are flat, the actions being free. Write \(a\colon N(F^{\mathrm o},P_F)\to N(F)\) for the morphism of Section 1 for the \(F^{\mathrm o}\)-torsor \(\varphi_F\colon P_F\to N(F)\).

Lemma 3.9 (products over a homomorphism).

1. Let \(X_k\to Y_k\), \(k=0,\ldots,m\), be morphisms of \(S\)-schemes with all \(X_k\) and \(Y_k\) flat over \(S\), and \(X=\prod_kX_k\to Y=\prod_kY_k\) their product over \(S\), with the projections \(p_k\). The sum of the functoriality maps \(p_k^*L_{X_k/Y_k}\to L_{X/Y}\) is an isomorphism \(\bigoplus_kp_k^*L_{X_k/Y_k}\cong L_{X/Y}\).
2. For the morphism \(h_Gh\colon P_F\to P\), the adjunction map \(i'_{0!}i'^*_0L_{P_F/P}\to L_{P_F/P}\) is an isomorphism, and \(i'^*_0L_{P_F/P}=L_{F/G}\).

Proof. (1) By induction on \(m\), as in Lemma 3.1. Write \(X=X'\times_SX_m\), \(Y=Y'\times_SY_m\), and factor \(X\to Y\) as \(X'\times X_m\to X'\times Y_m\to Y'\times Y_m\). In the transitivity triangle of this factorization, the first vertex is \(p'^*L_{X'/Y'}\) and the third \(p_m^*L_{X_m/Y_m}\), by Tor-independent base change (Proposition 1.4 of the lesson on deformations of diagrams), \(Y_m\) and \(X'\) being flat over \(S\). The functoriality map \(F_m\colon p_m^*L_{X_m/Y_m}\to L_{X/Y}\), followed by the second map of the triangle, is the functoriality map of the cartesian square formed by \(p_m\) and \(X'\times Y_m\to Y_m\), which is that isomorphism; so the second map has a right inverse, and the sum of \(F_m\) and the first map is an isomorphism [Tag 05QT]. The first map is the functoriality map of the square formed by \(p'\colon X\to X'\) and \(Y\to Y'\), functoriality maps composing (Theorem 2.2 of The cotangent complex), and the induction hypothesis applies to \(X'\to Y'\).

(2) Under the isomorphisms of Construction 2.1 with the Čech nerves, \(P_{F,n}\cong F^{n+1}\), \(P_n\cong G^{n+1}\) and \(h_Gh\) is \(u^{n+1}\), the vertices of \((u(f);u(x))\) being the images of the vertices of \((f;x)\). The restriction of the adjunction map to the summand of \(v_k\) at the level \(n\) is the functoriality map of the square formed by the projections \(p_k\), and (1) with \(X_k=F\), \(Y_k=G\) shows that the level \(n\) is an isomorphism, as in Lemma 3.2. At the level \(0\), \(P_{F,0}=F\) and \(P_0=G\). \(\square\)

Proposition 3.10 (the co-Lie complex of a homomorphism).

1. There is an object \(\ell_u\) of \(D^b_{\mathrm{qc}}(N(F)_{\mathrm{Zar}})\) with an isomorphism \(a^*\ell_u\cong L_{N(F^{\mathrm o},P_F)/N(F^{\mathrm o},P')}\), unique up to unique isomorphism, and a distinguished triangle \[ \begin{aligned}&LN(u)^*\ell_{P/N}\\&\quad\xrightarrow{\ \ell(h_Gh)\ }\ell_{P_F/N(F)}\\&\quad\to\ell_u\\&\quad\to LN(u)^*\ell_{P/N}[1],\end{aligned}\tag{3.2} \] whose image under \(a^*\) is the transitivity triangle of \(N(F^{\mathrm o},P_F)\to N(F^{\mathrm o},P')\to\) \(N(F^{\mathrm o},N(F))\), the first map being that of Lemma 1.8. The object \(\ell_u\) is quasi-cartesian, and its level \(0\) is \(\ell_{F/G}=Le_F^*L_{F/G}\), a cone of the map \(\ell(u)\colon\ell_G\to\ell_F\) of Proposition 2.5 (1); its levels are perfect.
2. There is an object \(C_u\) of \(D^b(B_F)\) with an isomorphism \(\operatorname{ner}(C_u)\cong L\varepsilon^*\ell_u\), unique up to unique isomorphism, and a distinguished triangle \[ \begin{aligned}&u^*\ell^{\mathrm{ad}}_G\xrightarrow{\ \lambda_u\ }\ell^{\mathrm{ad}}_F\\&\quad\to C_u\to u^*\ell^{\mathrm{ad}}_G[1]\end{aligned} \] whose nerve is \(L\varepsilon^*\) of (3.2). Its underlying object \(UC_u=L\varepsilon^*\ell_{F/G}\) is perfect.

Proof. (1) The transitivity triangle of \(N(F^{\mathrm o},P_F)\to N(F^{\mathrm o},P')\to\) \(N(F^{\mathrm o},N(F))\) (Proposition 1.3 of the lesson on deformations of diagrams) has the middle vertex \(L_{N(F^{\mathrm o},P_F)/N(F^{\mathrm o},N(F))}\cong a^*\ell_{P_F/N(F)}\) (Theorem 1.5 (1)). For the first vertex, the morphism \(N(F^{\mathrm o},P')\to N(G^{\mathrm o},P')\) with the levels \(u^r\times\mathrm{id}\) and the morphism \(N(F^{\mathrm o},N(F))\to N(G^{\mathrm o},N(F))\) form cartesian squares with the projections, the second projection being flat, so \(L_{N(F^{\mathrm o},P')/N(F^{\mathrm o},N(F))}\) is the inverse image of \(L_{N(G^{\mathrm o},P')/N(G^{\mathrm o},N(F))}\) (Proposition 1.4 of the lesson on deformations of diagrams), that is, of \(\ell_{P'/N(F)}\) (Theorem 1.5 (1) for the \(G^{\mathrm o}\)-torsor \(P'\)); and \(\ell_{P'/N(F)}\cong LN(u)^*\ell_{P/N}\) by Lemma 1.8 (4). So the first vertex, pulled back along \(N(F^{\mathrm o},h)\), is \(a^*LN(u)^*\ell_{P/N}\). The third vertex \(L=L_{N(F^{\mathrm o},P_F)/N(F^{\mathrm o},P')}\) has the columns \(L_{N(F^{\mathrm o},P_{F,n})/N(F^{\mathrm o},P'_n)}\), whose levels are pullbacks of \(L_{P_{F,n}/P'_n}=\operatorname{pr}^*L_{F/G}\), in \(D^{[-2,0]}\) with quasi-coherent cohomology, \(L_{F/G}\) being the cone of the map of perfect complexes \(Lu^*L_{G/S}\to L_{F/S}\) in \(D^{[-1,0]}\) (Theorem 3.1 of Flat group schemes are local complete intersections); the columns are quasi-cartesian, the squares of 3.9 being cartesian and flat (Proposition 1.4 there). By Theorem 1.4 (2), \(L\cong a^*\ell_u\) for \(\ell_u=Ra_*L\), unique up to unique isomorphism by Theorem 1.4 (1). The three maps are \(a^*\) of unique maps between objects of \(D^b_{\mathrm{qc}}(N(F)_{\mathrm{Zar}})\), and, as in the proof of Theorem 1.5 (3), the resulting triangle is distinguished. Its first map is \(\ell(h_Gh)\): the functoriality map of the square of nerves for \(h_Gh\) of Lemma 1.8 is the composite of the functoriality maps of the squares for \(h\), for \(N(F^{\mathrm o},P')\to N(G^{\mathrm o},P')\) and for \(h_G\), and both have the same image under \(a^*\), which is fully faithful. The structure maps of the first two vertices are isomorphisms (Proposition 2.2 (1), and \(LN(u)^*\) preserves this property), so those of \(\ell_u\) are isomorphisms [Tag 014A]. At the level \(0\), Proposition 2.5 (1) identifies the first map with \(\ell(u)\), and restricting \(a^*\ell_u\cong L\) to the vertex \(([0],[0])\) gives \(\pi_F^*\ell_{u,0}\cong L_{F/G}\), so \(\ell_{u,0}\cong Le_F^*\pi_F^*\ell_{u,0}\cong Le_F^*L_{F/G}\), a perfect complex; the levels \(\pi_n^*\ell_{F/G}\) are perfect.

(2) Let \(C_u\) be a cone of \(\lambda_u\) in \(D^b(B_F)\). The nerve is exact, so \(\operatorname{ner}(C_u)\) and \(L\varepsilon^*\ell_u\) are cones of \(\operatorname{ner}(\lambda_u)=L\varepsilon^*\ell(h_Gh)\) (Proposition 2.5 (1)), and there is an isomorphism between them compatible with the maps [Tag 014A]. By Proposition 1.5 (3) of the lesson on equivariant cotangent complexes, \(C_u\) is unique up to unique isomorphism with this property, and the maps of the triangle are the unique maps with the given nerves. Its underlying object is the level \(0\) of its nerve, \(L\varepsilon^*\ell_{F/G}\), perfect by Theorem 3.1 of the lesson on big fppf sites. \(\square\)

Theorem 3.11 (the cotangent complex of the nerve of a homomorphism). There is a distinguished triangle in \(D(N(F)_{\mathrm{Zar}})\) \[ \begin{aligned}&L_{N(F)/N}\longrightarrow i_{0!}i_0^*\ell_u\\&\quad\xrightarrow{\ \epsilon\ }\ell_u\to L_{N(F)/N}[1],\end{aligned} \] where \(\epsilon\) is the adjunction map and \(i_0^*\ell_u\cong\ell_{F/G}\), so that the level \(n\) of the middle term is \(\bigoplus_{k=0}^n\pi_n^*\ell_{F/G}\).

Proof. Consider the transitivity triangle of \(N(F^{\mathrm o},P_F)\to N(F^{\mathrm o},P')\to N(F^{\mathrm o},P)\). Its first vertex, the inverse image of \(L_{N(F^{\mathrm o},P')/N(F^{\mathrm o},P)}\), is \(a^*L_{N(F)/N}\): the squares formed by \(N(F^{\mathrm o},P')\to N(F^{\mathrm o},P)\) and \(N(u)\colon N(F)\to N\) are cartesian, \(P'\) being the base change of \(P\), and \(N(F^{\mathrm o},P)\to N\) is flat (Proposition 1.4 of the lesson on deformations of diagrams). The columns of its middle vertex \(L_{N(F^{\mathrm o},P_F)/N(F^{\mathrm o},P)}\) are quasi-cartesian with quasi-coherent cohomology in \(D^{[-2,0]}\), as in the proof of Proposition 3.10 (1), so it is \(a^*\lambda'\) for an object \(\lambda'\) of \(D^b_{\mathrm{qc}}(N(F)_{\mathrm{Zar}})\) (Theorem 1.4 (2)). Its third vertex is \(a^*\ell_u\). As in the proof of Proposition 3.10 (1), there is a distinguished triangle \[ \begin{aligned}&L_{N(F)/N}\xrightarrow{\ \beta\ }\lambda'\xrightarrow{\ \gamma\ }\ell_u\\&\qquad\to L_{N(F)/N}[1].\end{aligned} \] Step 1. At the level \(0\), \(N(F)_0=N_0=S\), so the first vertex vanishes and \(i_0^*(\gamma)\) is an isomorphism [Tag 05QR].

Step 2. The adjunction map \(\epsilon_{\lambda'}\colon i_{0!}i_0^*\lambda'\to\lambda'\) is an isomorphism. As in Step 2 of the proof of Theorem 3.3, it suffices to show that \(\varphi_F^*(\epsilon_{\lambda'})\) is an isomorphism, and it is the adjunction map of \(\varphi_F^*\lambda'\). Restricting \(a^*\lambda'\cong L_{N(F^{\mathrm o},P_F)/N(F^{\mathrm o},P)}\) to the row of index \(0\) of the nerves of the actions gives \(\varphi_F^*\lambda'\cong L_{P_F/P}\), whose adjunction map is an isomorphism by Lemma 3.9 (2).

Step 3. As in Step 3 of the proof of Theorem 3.3, \(\mu=i_{0!}i_0^*(\gamma)\circ\epsilon_{\lambda'}^{-1}\colon\lambda'\to i_{0!}i_0^*\ell_u\) is an isomorphism with \(\gamma=\epsilon\circ\mu\), and the triangle of the statement is distinguished. By Proposition 3.10 (1), \(i_0^*\ell_u\cong\ell_{F/G}\). \(\square\)

Corollary 3.12 (relative Ext groups of a homomorphism). Let \(M\in D^+(B_F)\).

1. There are isomorphisms in the derived category of abelian groups \[ \begin{aligned}&R\operatorname{Hom}(L\varepsilon^*L_{N(F)/N},\operatorname{ner}M)\\&\quad\cong R\operatorname{Hom}_{B_F/S}(C_u,M)[1]\\&\quad\cong R\Gamma(B_F/S,K\otimes^{\mathbf L}M)[1],\end{aligned} \] where \(K=C_u^\vee\) is the first vertex of a distinguished triangle \(K\to(\ell^{\mathrm{ad}}_F)^\vee\xrightarrow{\lambda_u^\vee}(u^*\ell^{\mathrm{ad}}_G)^\vee\to K[1]\), \(\lambda_u^\vee\) being the dual of the functoriality map.
2. For a quasi-coherent \(\mathcal O_S\)-module \(I\), \(\operatorname{Ext}^j(L_{N(F)/N},\pi_{N(F)}^*I)\cong\) \(\operatorname{Ext}^{j+1}_{B_F/S}(C_u,\varepsilon^*I)\cong\) \(H^{j+1}(B_F/S,K\otimes^{\mathbf L}\varepsilon^*I)\), and the spectral sequence of Corollary 3.5 (1) for these groups has \(E_1^{n,k}=\operatorname{Ext}^k_{F^n}(\pi_n^*\ell_{F/G},\pi_n^*I)\) for \(n\geq1\).

Proof. (1) Apply \(L\varepsilon^*\) to Theorem 3.11: \(L\varepsilon^*\ell_u\cong\operatorname{ner}(C_u)\), and \(L\varepsilon^*\) commutes with \(i_{0!}\) and the adjunction maps (Lemma 4.2 (1) of the lesson on big fppf sites). The proof of Corollary 3.5 (2), with \(C_u\) in place of \(\ell^{\mathrm{ad}}_G\), gives the first isomorphism, and Proposition 3.8, \(UC_u\) being perfect, the second. The functor \(R\mathcal Hom(-,\mathcal O)\) carries the distinguished triangle of Proposition 3.10 (2) to a distinguished triangle [Tag 08JC], which is the triangle of the statement.

(2) As in the proof of Corollary 3.5 (3): Lemma 4.2 (2) of the lesson on big fppf sites, (1) with \(M=\varepsilon^*I\), and Corollary 2.2 (1) there over \(F^n\), the levels of \(\operatorname{ner}(C_u)\) being \(L\varepsilon^*\pi_n^*\ell_{F/G}\). \(\square\)

4. Exercises

4.1. Let \(G\) be commutative. Show that the structure of \(H^0(\ell^{\mathrm{ad}}_G)\) is trivial, and that if \(G\) is moreover smooth, \(\ell^{\mathrm{ad}}_G\) is isomorphic to \(\varepsilon^*\omega_G\) in degree \(0\) with the trivial structure.

4.2. Let \(G=\operatorname{Spec}\mathcal O_S[a,a^{-1},b]\) with \((a,b)(a',b')=(aa',ab'+b)\), the group of affine transformations \(x\mapsto ax+b\), with unit \((1,0)\). Show that \(\omega_G\) is free on the classes of \(da\) and \(db\), and that for a point \(g=(c,d)\), \[ \begin{aligned}&c(\operatorname{int}(g))(\alpha\,da+\beta\,db)\\&\quad=(\alpha-d\beta)\,da+c\beta\,db .\end{aligned} \] Check (2.1) directly. For \(S=\operatorname{Spec}k\), \(k\) a field, show that the sections of \(H^0(\ell^{\mathrm{ad}}_G)\) fixed by the structure, that is, \(\operatorname{Hom}_B(\mathcal O,H^0(\ell^{\mathrm{ad}}_G))\), form the line \(k\,da\).

4.3. Show that \(G\) is étale over \(S\) if and only if \(L_{N(G)/S}=0\), and deduce from Theorem 3.3 that then \(\ell_{P/N}=0\), all Ext groups of Corollary 3.4 vanishing.

4.4. (a) Show that at every level \(n\), (3.1) gives an isomorphism \(L_{G^n/S}\cong\bigoplus_{k=1}^n\pi_n^*\ell_G\). (b) Let \(G\) be smooth. Show that the level \(1\) of (3.1) is a short exact sequence \[ \begin{aligned}0&\to\Omega_{G/S}\to\pi^*\omega_G\oplus\pi^*\omega_G\\&\to\pi^*\omega_G\to0,\end{aligned} \] whose second map, in the isomorphisms \(\tau_e\) on the two summands and \(\tau_\sigma\) on the target of the proof of Proposition 2.4, is \((u,v)\mapsto u+c(\operatorname{int}(g))(v)\). Deduce an isomorphism \(\Omega_{G/S}\cong\pi^*\omega_G\).

5. Complete solutions

4.1. For a commutative group \(\operatorname{int}(g)=\mathrm{id}\), so \(c(\operatorname{int}(g))=\mathrm{id}\), and Proposition 2.4 (1) gives the trivial structure. If \(G\) is also smooth, Proposition 2.4 (2) gives \(\ell^{\mathrm{ad}}_G\cong H^0(\ell^{\mathrm{ad}}_G)[0]\).

4.2. \(G\) is the open subscheme \(a\neq0\) of the affine plane, so \(\Omega_{G/S}\) is free on \(da,db\) and \(\omega_G=e^*\Omega_{G/S}\) is free on their classes. The inverse of \(g=(c,d)\) is \((c^{-1},-c^{-1}d)\), and \[ \begin{aligned}&\operatorname{int}(g)(a,b)\\&\quad=(c,d)(a,b)(c^{-1},-c^{-1}d)\\&\quad=(a,\,cb+d-ad).\end{aligned} \] Over \(V\), \(c\) and \(d\) are constants, so \(\operatorname{int}(g)^*da=da\) and \(\operatorname{int}(g)^*db=c\,db-d\,da\); at the unit this is the stated formula. For \(h=(c',d')\), \(gh=(cc',cd'+d)\), and \[ \begin{aligned}&c(\operatorname{int}(h))c(\operatorname{int}(g))(\alpha,\beta)\\&\quad=(\alpha-d\beta-d'c\beta,\ c'c\beta),\end{aligned} \] which is \(c(\operatorname{int}(gh))(\alpha,\beta)\), as \(cd'+d\) is the second coordinate of \(gh\). A map \(\mathcal O\to M\) of modules of \(B\) is a section \(v\) of \(M\) with \(m(\pi^*v)=\pi^*v\) on \(G\). For \(v=\alpha\,da+\beta\,db\) with \(\alpha,\beta\in k\), the universal point \(g=(c,d)\in G(G)\) gives \(\alpha-d\beta=\alpha\) and \(c\beta=\beta\) in \(k[c,c^{-1},d]\), so \(\beta=0\). Conversely \(da\) is fixed. By Proposition 2.4 (1) the fixed sections form \(k\,da\).

4.3. The level \(1\) of \(L_{N(G)/S}\) is \(L_{G/S}\), and \(L_{G^n/S}\cong\bigoplus\operatorname{pr}_k^*L_{G/S}\) by Lemma 3.1. So \(L_{N(G)/S}=0\) if and only if \(L_{G/S}=0\). If \(L_{G/S}=0\), then \(\ell_G=Le^*L_{G/S}=0\), and \(G\) is étale by Corollary 2.3 of the lesson on the co-Lie complex. If \(G\) is étale, \(L_{G/S}=0\) (Propositions 5.1 and 10.1 of The cotangent complex), so \(L_{N(G)/S}=0\) and \(\ell_G=Le^*L_{G/S}=0\). In (3.1) the first vertex vanishes, so \(\epsilon\) is an isomorphism [Tag 05QR], and its source \(i_{0!}i_0^*\ell_{P/N}\cong i_{0!}\ell_G\) vanishes; so \(\ell_{P/N}=0\). Then the terms \(\operatorname{Ext}^j(\ell_{P/N},M)\) and \(\operatorname{Ext}^j_S(\ell_G,i_0^*M)\) of Corollary 3.4 vanish, and so do the \(\operatorname{Ext}^j(L_{N/S},M)\), as they must.

4.4. (a) Restrict (3.1) to the level \(n\): with \(L_n=L_{G^n/S}\), \(A=(i_{0!}i_0^*\ell_{P/N})_n=\bigoplus_{k=0}^n\pi_n^*\ell_0\) and \(\beta\colon L_n\to A\) the first map, it is a distinguished triangle whose second map \(\epsilon|_n\) has the right inverse \(s=\iota_0\circ\ell_{v_0}^{-1}\), \(\iota_0\) being the inclusion of the summand of \(v_0\) (Theorem 3.3). By [Tag 05QT], \((\beta,s)\colon L_n\oplus\ell_n\to A\) is an isomorphism, hence so is \((\beta,\iota_0)\colon L_n\oplus\pi_n^*\ell_0\to A\). Also \((\iota_0,j)\colon\pi_n^*\ell_0\oplus C\to A\) is an isomorphism, \(C\) being the sum of the summands \(k\geq1\) and \(j\) its inclusion. Let \(p_L,p_D\) and \(p'_D,p_C\) be the projections of the two decompositions, so that \(\beta p_L+\iota_0p_D=\mathrm{id}=\iota_0p'_D+jp_C\), \(p_L\beta=\mathrm{id}\), \(p_Cj=\mathrm{id}\) and \(p_L\iota_0=0=p_C\iota_0\). Then \(p_Lj\colon C\to L_n\) and \(p_C\beta\colon L_n\to C\) are inverse: \(p_Ljp_C\beta=p_L(\mathrm{id}-\iota_0p'_D)\beta=\mathrm{id}\) and \(p_C\beta p_Lj=p_C(\mathrm{id}-\iota_0p_D)j=\mathrm{id}\). So \(L_{G^n/S}\cong C\cong\bigoplus_{k=1}^n\pi_n^*\ell_G\).

(b) If \(G\) is smooth, \(L_{G/S}=\Omega_{G/S}\) and \(\ell_G=\omega_G\) are modules in degree \(0\), and so is every level of \(\ell_{P/N}\). The level \(1\) of (3.1) is a distinguished triangle of modules in degree \(0\), and its long exact cohomology sequence [Tag 0149] is the short exact sequence stated. The level \(1\) of \(\epsilon\) is \(\ell_{v_0}\) on the first summand and \(\ell_{v_1}\) on the second. The proof of Proposition 2.4 computes them: \(\tau_\sigma\circ\ell_{v_0}=\pi^*\tau_e\) and \(\tau_\sigma\circ\ell_{v_1}=c(\operatorname{int}(g))\circ\pi^*\tau_e\). So the map is \((u,v)\mapsto u+c(\operatorname{int}(g))(v)\), and its kernel consists of the pairs \((-c(\operatorname{int}(g))(v),v)\). Projecting to the second summand gives \(\Omega_{G/S}\cong\pi^*\omega_G\).

Sources

The adjoint co-Lie complex, the distinguished triangle for the cotangent complex of the nerve of a group scheme, and its expression through relative cohomology of the classifying topos are due to Illusie, in his work on the cotangent complex, as is the extension of the co-Lie complex to torsors over diagrams of schemes.