# Equivariant cotangent complexes

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

A category scheme \(G\) over a scheme \(S\), for instance a group scheme or a groupoid scheme, acts on schemes, and a morphism of schemes with an action of \(G\) has an equivariant cotangent complex. It is built from the nerves. A scheme \(X\) with an action of \(G\) defines a category scheme \([G,X]\) whose objects are the points of \(X\) and whose arrows are the pairs of an arrow of \(G\) and a point of \(X\). Its nerve \(N(G,X)\) is a simplicial scheme, and a morphism \(f\colon X\to Y\) of schemes with an action of \(G\) gives a morphism of simplicial schemes \(N(G,f)\colon N(G,X)\to N(G,Y)\), whose cotangent complex lives on the Zariski total topos of \(N(G,X)\). When \(f\) is Tor independent of \(G\), this complex is determined by its level \(0\) and the action. Its pullback to the big fppf site then descends, truncation by truncation, to an object of the derived category of modules of the classifying category of the opposite category \([G,X]^{\mathrm o}\). This projective system of truncations is the equivariant cotangent complex. Its Ext groups into a module whose underlying module is quasi-coherent are the Ext groups of the cotangent complex of \(N(G,f)\), and the transitivity triangle of the nerves gives the equivariant Kodaira–Spencer class and the long exact sequences of these Ext groups.

**Prerequisites.** [*Décalage, nerves and the homotopy type of a category*](decalage-nerves-and-the-homotopy-type-of-a-category.md) (Sections 2.3 and 2.5); [*Fibred topoi and their total topoi*](fibred-topoi-and-their-total-topoi.md) (Definition 2.1, Examples 1.2 (c), Proposition 5.1); [*Equivariant sheaves on simplicial topoi*](equivariant-sheaves-on-simplicial-topoi.md) (Definitions 1.1 and 2.1, Examples 1.4, Propositions 2.2 and 2.3, Section 6); [*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md) (Corollary 4.4, Propositions 5.1 and 6.1); [*Quasi-coherent and perfect complexes on big fppf sites*](quasi-coherent-and-perfect-complexes-on-big-fppf-sites.md) (Propositions 1.1 and 2.1, Construction 4.1, Lemma 4.2); [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md) (Conventions, Propositions 1.2–1.4). From the programme lesson [*The cotangent complex*](course:AG-HP/the-cotangent-complex#2-independence-of-the-resolution): functoriality maps compose (Theorem 2.2) and local complete intersections (Theorem 7.2). From the AI-integrated Stacks project: representable functors are fppf sheaves [[Stacks, Tag 023Q](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/descent.html#descent-lemma-fpqc-universal-effective-epimorphisms)]; Tor independence [[Tag 08IA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/perfect.html#perfect-definition-tor-independent), [Tag 0660](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-definition-tor-independent)]; tor amplitude [[Tag 08FZ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-definition-tor-amplitude), [Tag 08G1](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-tor-amplitude), [Tag 08H5](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-tor-amplitude-pullback)]; left derived functors through an adapted class [[Tag 05T7](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-subcategory-left-resolution), [Tag 05T9](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-subcategory-left-acyclics), [Tag 05TA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-proposition-enough-acyclics)]; K-flat complexes and derived pullback [[Tag 06YQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-bounded-flat-K-flat), [Tag 06YY](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-derived-base-change)]; maps out of truncated objects and the long exact sequence of a triangle [[Tag 06XS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-negative-exts), [Tag 0149](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-representable-homological)]; Tor independence on affine opens and derived base change [[Tag 0FXV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/perfect.html#perfect-lemma-tor-independent), [Tag 06Y6](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-algebra.html#more-algebra-lemma-derived-base-change)]; affine opens of fibre products [[Tag 01I4](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/schemes.html#schemes-lemma-fibre-product-affine-schemes), [Tag 01JS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/schemes.html#schemes-lemma-affine-covering-fibre-product)].

## Conventions

\(S\) is a scheme, \(T\) the topos of sheaves on a big fppf site \((\mathrm{Sch}/S)_{\mathrm{fppf}}\), and \(\mathcal O\) its structure ring. Representable functors are sheaves [Tag 023Q], and the Yoneda embedding preserves fibre products, so schemes over \(S\) and their fibre products are objects of \(T\). For a scheme \(U\) over \(S\), the slice \(T/U\) is the topos of sheaves on \((\mathrm{Sch}/U)_{\mathrm{fppf}}\), ringed by the inverse image of \(\mathcal O\). A simplicial scheme is a diagram indexed by \(\Delta^{\mathrm{op}}\); for an increasing map \(\alpha\colon[m]\to[n]\) it has a morphism \(N(\alpha)\colon N_n\to N_m\). For a simplicial scheme \(N\), \(\operatorname{Top}(N_{\mathrm{Zar}})\), \(\operatorname{Top}(N_{\mathrm{fppf}})\) and the comparison morphism \(\varepsilon\) between them are those of Construction 4.1 of [*Quasi-coherent and perfect complexes on big fppf sites*](quasi-coherent-and-perfect-complexes-on-big-fppf-sites.md); the fppf topos is the total topos of the simplicial topos \(n\mapsto T/N_n\) (Examples 1.2 (c) of [*Fibred topoi and their total topoi*](fibred-topoi-and-their-total-topoi.md)). An object \(E\) of \(D^-(N_{\mathrm{Zar}})\) or of \(D^-(N_{\mathrm{fppf}})\) is *quasi-cartesian along* \(\alpha\) if its structure map \(LN(\alpha)^*E_m\to E_n\) is an isomorphism.

*Reversal.* For \(\alpha\colon[m]\to[n]\) let \(\alpha^{\mathrm r}\colon[m]\to[n]\), \(\alpha^{\mathrm r}(i)=n-\alpha(m-i)\). This is an increasing map, \((\alpha\beta)^{\mathrm r}=\alpha^{\mathrm r}\beta^{\mathrm r}\), and \(\alpha^{\mathrm r}(m)=n\) if and only if \(\alpha(0)=0\). The *reversal* of a simplicial object \(N\) is \(N^{\mathrm r}\), with the same levels and \(N^{\mathrm r}(\alpha)=N(\alpha^{\mathrm r})\). For a topos fibred over \(\Delta^{\mathrm{op}}\), the categories \(\operatorname{Top}\) of a simplicial topos and of its reversal are identified: an object \(E\) corresponds to the object with the same levels and the structure maps \(E_{\alpha^{\mathrm r}}\) (Definition 2.1 of the lesson on fibred topoi). This identification is compatible with modules, derived categories and the functors \(\varepsilon\).

*Truncations.* As in the lesson on big fppf sites, \(t_{[n}K=\tau_{\geq n}K\). For an object \(K\) of the derived category of an abelian category, an object \(N\) of that category and integers \(i\) and \(n\leq-i\), the map \(K\to t_{[n}K\) induces a bijection
\[
\begin{aligned}&\operatorname{Hom}(t_{[n}K,N[i])\\&\quad \longrightarrow\operatorname{Hom}(K,N[i]).\end{aligned} \tag{0.1}
\]
Indeed, in the long exact sequence of \(\operatorname{Hom}(-,N[i])\) for the triangle \(\tau_{\leq n-1}K\to K\to t_{[n}K\to\tau_{\leq n-1}K[1]\), the groups \(\operatorname{Hom}(\tau_{\leq n-1}K,N[i])\) and \(\operatorname{Hom}(\tau_{\leq n-1}K[1],N[i])\) vanish when \(n\leq-i\), because \(\tau_{\leq n-1}K\in D^{\leq n-1}\) and \(N[i]\in D^{\geq-i}\) [Tag 06XS]. For a morphism of ringed topoi \(\varphi\) and \(K\in D^-\), \(L\varphi^*\) carries \(D^{\leq q}\) into \(D^{\leq q}\), so the triangle \(\tau_{\leq n-1}K\to K\to t_{[n}K\) shows
\[
t_{[n}L\varphi^*K\xrightarrow{\ \sim\ }t_{[n}L\varphi^*t_{[n}K.
\tag{0.2}
\]

## 1. Category schemes acting on schemes

**Definition 1.1.** A *category scheme* over \(S\) is an internal category \(G\) of the category of schemes over \(S\), in the sense of Section 2.3 of [*Décalage, nerves and the homotopy type of a category*](decalage-nerves-and-the-homotopy-type-of-a-category.md): schemes \(G_0\) and \(G_1\) over \(S\), the target \(d_0\) and source \(d_1\colon G_1\to G_0\), the identities \(s_0\colon G_0\to G_1\) and the composition \(G_1\times_{G_0}G_1\to G_1\). It is also an internal category of \(T\), and its nerve \(N(G)\) is a simplicial scheme. A *\(G\)-sheaf* is an equivariant sheaf on the simplicial topos \(n\mapsto T/N_n(G)\) (Definition 1.1 and Examples 1.4 (b) of [*Equivariant sheaves on simplicial topoi*](equivariant-sheaves-on-simplicial-topoi.md)): an object \(p\colon L\to G_0\) of \(T\) with an action
\[
G_1\times_{G_0}L\longrightarrow L,\qquad(g,x)\longmapsto g\cdot x,
\]
the fibre product being formed with the source, such that \(p(g\cdot x)=d_0(g)\), \(1\cdot x=x\) and \(h\cdot(g\cdot x)=(hg)\cdot x\) on points. The \(G\)-sheaves form the category \(B_G\). A *\(G\)-scheme* is a \(G\)-sheaf whose object \(L\) is a scheme, and a morphism of \(G\)-schemes is a morphism of \(G\)-sheaves. The ring \(\mathcal O\) restricted to \(G_0\), with its canonical structure, is a ring of \(B'\) in the sense of Section 6 of that lesson, and \(\operatorname{Mod}(B_G)\) denotes the category of modules of \(B_G\) over it. If \(G\) is a monoid scheme, that is \(G_0=S\), a \(G\)-scheme is a scheme over \(S\) with an action of \(G\) (Examples 1.4 (c) there).

**Construction 1.2.** Let \(L\) be a \(G\)-sheaf. The internal category \([G,L]\) of \(T\) has the objects \(L\), the arrows \(G_1\times_{G_0}L\), the source \((g,x)\mapsto x\), the target \((g,x)\mapsto g\cdot x\), the identities \(x\mapsto(s_0p(x),x)\) and the composition \((h,g\cdot x)(g,x)=(hg,x)\). The axioms follow from those of \(G\) and of the action, checked on points with values in an arbitrary object of \(T\). The maps \((g,x)\mapsto g\) and \(p\) form an internal functor \(\pi_L\colon[G,L]\to G\), and a morphism \(u\colon L\to M\) of \(G\)-sheaves gives the internal functor \([G,u]\), \((g,x)\mapsto(g,u(x))\), over \(G\). For a \(G\)-scheme \(X\), \([G,X]\) is a category scheme over \(S\), its arrows \(G_1\times_{G_0}X\) being a fibre product of schemes. We write \(N(G,X)=N([G,X])\) and \(N(G,f)=N([G,f])\) for a morphism \(f\) of \(G\)-schemes.

**Lemma 1.3.** Let \(L\) be a \(G\)-sheaf. Then \(N_n(G,L)=N_n(G)\times_{G_0}L\), the fibre product being formed with the first vertex \(v_0\colon N_n(G)\to G_0\), and for \(\alpha\colon[m]\to[n]\)
\[
\begin{aligned}&N(G,L)(\alpha)(g_1,\ldots,g_n;x)\\&\quad=\bigl(N(G)(\alpha)(g_1,\ldots,g_n);\\&\qquad g_{\alpha(0)}\cdots g_1\cdot x\bigr).\end{aligned}
\]
In particular the first vertex \(v_0\colon N_n(G,L)\to L\) is the projection, and for a morphism \(u\colon L\to M\) of \(G\)-sheaves the square formed by \(v_0\), \(v_0\), \(N_n(G,u)\) and \(u\) is cartesian. Under the equivalence of Proposition 5.1 of the lesson on fibred topoi, \(N(G,L)\to N(G)\) is the second nerve \(\operatorname{ner}^{\mathrm o}(L)\) of Definition 2.1 of the lesson on equivariant sheaves.

**Proof.** A string of \(n\) composable arrows of \([G,L]\) is \((g_1,x),(g_2,g_1\cdot x),\ldots\), determined by a string \((g_1,\ldots,g_n)\) of \(G\) and its first object \(x\), with \(d_1(g_1)=p(x)\). Its vertex \(k\) is \(g_k\cdots g_1\cdot x\), and \(N(G,L)(\alpha)\) keeps the vertices \(\alpha(0),\ldots,\alpha(m)\), which gives the formula. The square of the second statement is \(N_n(G)\times_{G_0}L\to N_n(G)\times_{G_0}M\) over \(L\to M\), a fibre product. Finally \(\operatorname{ner}^{\mathrm o}(L)_n\) is the pullback of \(L\) along the first vertex of \(N_n(G)\), with the structure map \(a_{0\,\alpha(0)}\) for \(\alpha\), the action of the composite of the arrows from the vertex \(0\) to the vertex \(\alpha(0)\); this is the displayed formula. \(\square\)

**Proposition 1.4.** Let \(L\) be a \(G\)-sheaf. A \([G,L]\)-sheaf \(q\colon M\to L\), with action \(((g,x),m)\mapsto(g,x)\cdot m\), defines the \(G\)-sheaf \(pq\colon M\to G_0\) with \(g\cdot m=(g,q(m))\cdot m\), for which \(q\) is a morphism of \(G\)-sheaves. This is an isomorphism of categories \(B[G,L]\cong B_G/L\), compatible with modules.

**Proof.** The conditions for \(M\) as a \([G,L]\)-sheaf are \(q((g,x)\cdot m)=g\cdot x\), the unit law and \((h,g\cdot x)\cdot((g,x)\cdot m)=(hg,x)\cdot m\). With \(x=q(m)\) they say that \(g\cdot m\) is an action over \(G_0\) for which \(q\) is equivariant. Conversely, for a \(G\)-sheaf \(M\) with a morphism \(q\colon M\to L\) of \(G\)-sheaves, \((g,x)\cdot m=g\cdot m\) for \(q(m)=x\) is a \([G,L]\)-action, since \(q(g\cdot m)=g\cdot x\). The two constructions are inverse, morphisms correspond, and the module structures, defined by the same conditions in both categories, correspond. \(\square\)

**Proposition 1.5 (the opposite category).** Let \(X\) be a \(G\)-scheme and \([G,X]^{\mathrm o}\) the opposite internal category, with source and target exchanged.

1. \(N([G,X]^{\mathrm o})=N(G,X)^{\mathrm r}\). So \(\operatorname{Top}(N([G,X]^{\mathrm o})_{\mathrm{Zar}})\) and \(\operatorname{Top}(N([G,X]^{\mathrm o})_{\mathrm{fppf}})\) are identified with \(\operatorname{Top}(N(G,X)_{\mathrm{Zar}})\) and \(\operatorname{Top}(N(G,X)_{\mathrm{fppf}})\).
2. A module of \(B[G,X]^{\mathrm o}=B([G,X]^{\mathrm o})\) is an \(\mathcal O\)-module \(M\) of \(T/X\) with a map \(m\colon\mathrm{act}^*M\to\mathrm{pr}^*M\) on \(N_1(G,X)=G_1\times_{G_0}X\), where \(\mathrm{act}(g,x)=g\cdot x\) and \(\mathrm{pr}(g,x)=x\), satisfying the two conditions of Definition 1.1 of the lesson on equivariant sheaves. Under (1), its nerve \(\operatorname{ner}(M)\) has the levels \(v_0^*M\), for the first vertex \(v_0\colon N_n(G,X)\to X\).
3. The functor \(\operatorname{ner}\colon\) \(D^b(B[G,X]^{\mathrm o})\to\) \(D^b(\operatorname{Top}(N(G,X)_{\mathrm{fppf}}))\) is fully faithful. Its essential image consists of the objects \(E\) all of whose cohomology modules are quasi-cartesian along every \(\alpha\) with \(\alpha(0)=0\).

**Proof.** (1) A string of arrows of \([G,X]^{\mathrm o}\) is a string of \([G,X]\) read backwards, and the operations on strings are relabeled by \(\alpha\mapsto\alpha^{\mathrm r}\). (2) The source of \([G,X]^{\mathrm o}\) is the target \(\mathrm{act}\) of \([G,X]\), and its target is \(\mathrm{pr}\). The last vertex of a string of \([G,X]^{\mathrm o}\) is the first vertex of the corresponding string of \([G,X]\). (3) By Proposition 5.1 of [*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md), the simplicial topos of the nerve of the internal category \([G,X]^{\mathrm o}\) of \(T\), with the ring \(\mathcal O\), is a good two-sided pseudo-category satisfying (I) and (E); so by its Corollary 4.4 \(\operatorname{ner}\) is fully faithful on \(D^b\), with essential image the objects whose cohomology modules are nerves of modules. By Proposition 2.3 (1) of the lesson on equivariant sheaves, which holds for modules (Section 6 there), these are the modules \(E\) of \(\operatorname{Top}(N([G,X]^{\mathrm o}))\) with invertible \(E_\beta\) for all \(\beta\) with \(\beta(m)=n\); under (1) they are the modules with invertible \(E_{\alpha}\) for \(\alpha=\beta^{\mathrm r}\), that is, with \(\alpha(0)=0\). \(\square\)

**Proposition 1.6 (groupoids).** Suppose that \(G\) is a groupoid scheme, with inversion \(g\mapsto g^{-1}\). For a \(G\)-scheme \(X\), \([G,X]\) is a groupoid, and \((g,x)\mapsto(g^{-1},g\cdot x)\) is an isomorphism of internal categories \([G,X]\to[G,X]^{\mathrm o}\) which is the identity on objects. It induces an isomorphism \(B[G,X]\cong B[G,X]^{\mathrm o}\), so that, by Proposition 1.4, modules of \(B[G,X]^{\mathrm o}\) are the \(G\)-equivariant \(\mathcal O\)-modules over \(X\).

**Proof.** The inverse of the arrow \((g,x)\colon x\to g\cdot x\) is \((g^{-1},g\cdot x)\). The displayed map sends an arrow \(x\to g\cdot x\) of \([G,X]\) to an arrow \(g\cdot x\to x\) of \([G,X]\), which is an arrow \(x\to g\cdot x\) of \([G,X]^{\mathrm o}\); it is compatible with identities and composites, since inversion reverses composites. An isomorphism of internal categories carries equivariant sheaves and modules to equivariant sheaves and modules. \(\square\)

## 2. The equivariant cotangent complex

**Definition 2.1.** A morphism \(f\colon X\to Y\) of \(G\)-schemes and \(G\) are *Tor independent* if, for every \(n\), \(X\) and \(N_n(G,Y)\) are Tor independent over \(Y\) [Tag 08IA], for the first vertex \(v_0\colon N_n(G,Y)\to Y\).

The square formed by \(v_0\), \(v_0\), \(N_n(G,f)\) and \(f\) is cartesian (Lemma 1.3). The condition holds if \(f\) is flat, or if every first vertex \(N_n(G)\to G_0\) is flat, since \(N_n(G,Y)\to Y\) is its base change. The first vertex is flat when the source \(d_1\colon G_1\to G_0\) is flat: \(N_n(G)\to N_{n-1}(G)\), forgetting the first arrow, is a base change of \(d_1\), and the first vertex of \(N_1(G)\) is \(d_1\). This covers flat monoid and group schemes over \(S\).

**Lemma 2.2.** Let \(P\to Q\), \(R\to Q\) and \(R'\to R\) be morphisms of schemes such that \(P\) and \(R\) are Tor independent over \(Q\). Then \(P\times_QR\) and \(R'\) are Tor independent over \(R\) if and only if \(P\) and \(R'\) are Tor independent over \(Q\).

**Proof.** By [Tag 0FXV], both conditions can be checked on affine open coverings. Choose affine opens \(\operatorname{Spec}A\) of \(Q\), \(\operatorname{Spec}B\) of \(R\) mapping into \(\operatorname{Spec}A\), \(\operatorname{Spec}C\) of \(R'\) mapping into \(\operatorname{Spec}B\), and \(\operatorname{Spec}E\) of \(P\) mapping into \(\operatorname{Spec}A\). Such opens cover \(Q\), \(R\), \(R'\) and \(P\) as required for both conditions, the affine opens \(\operatorname{Spec}(E\otimes_AB)\) covering \(P\times_QR\) [Tags 01I4, 01JS]. By [Tag 06Y6] and the Tor independence of \(E\) and \(B\) over \(A\),
\[
\begin{aligned}&E\otimes^{\mathbf L}_AC \cong(E\otimes^{\mathbf L}_AB)\otimes^{\mathbf L}_BC\\&\quad =(E\otimes_AB)\otimes^{\mathbf L}_BC,\end{aligned}
\]
so \(\operatorname{Tor}^A_i(E,C)\cong\operatorname{Tor}^B_i(E\otimes_AB,C)\) for every \(i\) [Tag 0660], and the two conditions agree. \(\square\)

**Proposition 2.3.** Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes, Tor independent of \(G\), and \(L=L_{N(G,X)/N(G,Y)}\) the cotangent complex of the morphism of simplicial schemes \(N(G,f)\) (Definition 1.1 of [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md)).

1. \(L\) is quasi-cartesian along every \(\alpha\colon[m]\to[n]\) with \(\alpha(0)=0\). In particular the functoriality map \(Lv_0^*L_{X/Y}\to L_{N_n(G,X)/N_n(G,Y)}\) is an isomorphism for every \(n\).
2. \(L\varepsilon^*L\in D^{\leq0}(N(G,X)_{\mathrm{fppf}})\), and every cohomology module \(H^q(L\varepsilon^*L)\) is quasi-cartesian along every \(\alpha\) with \(\alpha(0)=0\).

**Proof.** (1) The second assertion is Proposition 1.4 of the lesson on deformations of diagrams, applied to the cartesian, Tor-independent square of Definition 2.1. Let \(\alpha(0)=0\). Then \(v_0\circ N(G,X)(\alpha)=v_0\) as maps \(N_n(G,X)\to X\), and likewise for \(Y\). The structure maps of \(L\) are functoriality maps (Proposition 1.2 (1) of the lesson on deformations of diagrams), and functoriality maps compose (Theorem 2.2 of [*The cotangent complex*](course:AG-HP/the-cotangent-complex#2-independence-of-the-resolution)). So the isomorphism \(Lv_0^*L_{X/Y}\to L_n\) is the composite of \(LN(\alpha)^*\) of the isomorphism \(Lv_0^*L_{X/Y}\to L_m\) and the structure map \(LN(\alpha)^*L_m\to L_n\), which is therefore an isomorphism.

(2) \(L\) is a complex of flat modules zero in positive degrees, so \(L\varepsilon^*L=\varepsilon^*L\) lies in \(D^{\leq0}\). By Lemma 4.2 (1) and the proof of Lemma 4.2 (3) of the lesson on big fppf sites, the structure map of \(L\varepsilon^*L\) along \(\alpha\) is \(L\varepsilon_n^*\) of that of \(L\), hence an isomorphism when \(\alpha(0)=0\); and on the fppf side \(N(\alpha)_{\mathrm{big}}^*\) is exact, so the structure maps of the cohomology modules along \(\alpha\) are isomorphisms as well. \(\square\)

**Theorem 2.4 (the equivariant cotangent complex).** Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes, Tor independent of \(G\), and \(L\) as in Proposition 2.3. For every integer \(n\) there is an object \(t_{[n}L^G_{X/Y}\) of \(D^b(B[G,X]^{\mathrm o})\) with an isomorphism
\[
\begin{aligned}&\operatorname{ner}\bigl(t_{[n}L^G_{X/Y}\bigr)\\&\quad \cong t_{[n}L\varepsilon^*L_{N(G,X)/N(G,Y)},\end{aligned} \tag{2.1}
\]
and for every \(n\) a map \(t_{[n}L^G_{X/Y}\to t_{[n+1}L^G_{X/Y}\) whose nerve is the canonical map of truncations. The objects, the isomorphisms and the maps are unique up to unique isomorphism. The projective system
\[
t_{[}L^G_{X/Y}=\bigl(t_{[n}L^G_{X/Y}\bigr)_{n\in\mathbb Z}
\]
is the *equivariant cotangent complex* of \(f\).

**Proof.** By Proposition 2.3 (2), \(t_{[n}L\varepsilon^*L\) has cohomology only in degrees \([n,0]\), and its cohomology modules are quasi-cartesian along every \(\alpha\) with \(\alpha(0)=0\). By Proposition 1.5 (3) it is \(\operatorname{ner}\) of an object of \(D^b(B[G,X]^{\mathrm o})\), unique up to unique isomorphism compatible with (2.1), and the maps between these objects are the maps between their nerves. \(\square\)

Since \(\operatorname{ner}\) is exact, \(\tau_{\geq n}t_{[n'}L^G_{X/Y}\cong t_{[n}L^G_{X/Y}\) for \(n'\leq n\), compatibly with the transition maps.

**Proposition 2.5.** Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes, Tor independent of \(G\).

1. The underlying module on \(X\) of \(t_{[n}L^G_{X/Y}\) is \(t_{[n}L\varepsilon^*L_{X/Y}\), and \(R\varepsilon_*\) of it is \(t_{[n}L_{X/Y}\).
2. If \(L_{X/Y}\) has tor amplitude in \([a,0]\) [Tag 08FZ], for instance if \(f\) is a local complete intersection morphism, with \(a=-1\) (Theorem 7.2 of *The cotangent complex*, on affine opens through its Proposition 10.1), then the transition maps \(t_{[n}L^G_{X/Y}\to t_{[n+1}L^G_{X/Y}\) are isomorphisms for \(n<a\). We then write \(L^G_{X/Y}=t_{[a}L^G_{X/Y}\in D^b(B[G,X]^{\mathrm o})\).
3. If \(G\) is a groupoid scheme, \(t_{[}L^G_{X/Y}\) is a projective system of \(D^b\) of the \(G\)-equivariant modules over \(X\) (Proposition 1.6).
4. Suppose that \(G\) is a monoid scheme acting trivially on \(X\) and \(Y\), and let \(p^*\) be the functor from \(\mathcal O\)-modules of \(T/X\) to modules of \(B[G,X]^{\mathrm o}\) giving a module its trivial structure. Then \(t_{[n}L^G_{X/Y}\cong t_{[n}p^*L\varepsilon^*L_{X/Y}\) for every \(n\).

**Proof.** (1) The level \(0\) of \(\operatorname{ner}(M)\) is the underlying module of \(M\) (Proposition 1.5 (2)), \(N_0(G,X)=X\), and the level \(0\) of \(L\) is \(L_{X/Y}\) (Proposition 1.2 (1) of the lesson on deformations of diagrams). Restriction to a level commutes with \(\varepsilon^*\) and with truncations. For the second assertion, \(t_{[n}L_{X/Y}\) has quasi-coherent cohomology in degrees \([n,0]\), so it is \(R\varepsilon_*t_{[n}L\varepsilon^*t_{[n}L_{X/Y}\) by Proposition 2.1 of the lesson on big fppf sites, and \(t_{[n}L\varepsilon^*t_{[n}L_{X/Y}=t_{[n}L\varepsilon^*L_{X/Y}\) by (0.2).

(2) By [Tag 08H5] the levels \(Lv_0^*L_{X/Y}\) of \(L\) (Proposition 2.3 (1)) have tor amplitude in \([a,0]\). By [Tag 08G1] they are represented by complexes of flat modules in degrees \([a,0]\), whose \(\varepsilon^*\) lie in degrees \([a,0]\). So \(L\varepsilon^*L\) has cohomology only in degrees \([a,0]\), and \(t_{[n}L\varepsilon^*L\to t_{[n+1}L\varepsilon^*L\) is an isomorphism for \(n<a\); by full faithfulness so is the transition map.

(3) This is Proposition 1.6.

(4) Here \([G,X]\) is \(G\times_SX\) acting trivially, \(N_n(G,X)=N_n(G)\times_SX\), and every face of \(N(G,X)\) is the face of \(N(G)\) times the identity of \(X\); the projections \(q_n\colon N_n(G,X)\to X\) form a morphism \(q\) of simplicial schemes from \(N(G,X)\) to the constant simplicial scheme \(X\). By Lemma 1.3 the squares formed by \(q_n\), \(q_n\), \(N_n(G,f)\) and \(f\) are those of Definition 2.1, so Proposition 1.4 of the lesson on deformations of diagrams gives \(Lq^*L_{X/Y}\cong L\), the structure maps of both sides being functoriality maps. So \(L\varepsilon^*L\cong Lq_{\mathrm{big}}^*L\varepsilon^*L_{X/Y}\). The nerve of \(p^*M\) is \(q_{\mathrm{big}}^*M\), as in the proof of Proposition 6.1 of the lesson on cohomology of equivariant modules, and \(p^*\) and \(q_{\mathrm{big}}^*\) are exact, so \(\operatorname{ner}(t_{[n}p^*L\varepsilon^*L_{X/Y})=t_{[n}L\varepsilon^*L\), and (2.1) gives the assertion. \(\square\)

## 3. Ext groups, pullbacks and the Kodaira–Spencer class

**Lemma 3.1.** Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes and \(\varphi=N(G,f)\).

1. For a module \(M\) of \(B[G,Y]^{\mathrm o}\), with structure \(m\), the module \(f^*M\) on \(X\), with the structure \((\mathrm{id}\times f)^*(m)\) on \(G_1\times_{G_0}X\), is a module \(f_B^*M\) of \(B[G,X]^{\mathrm o}\), and \(\operatorname{ner}(f_B^*M)=\varphi_{\mathrm{big}}^*\operatorname{ner}(M)\).
2. \(f_B^*\) has a left derived functor \(Lf_B^*\colon\) \(D^-(B[G,Y]^{\mathrm o})\to\) \(D^-(B[G,X]^{\mathrm o})\), computed by bounded above complexes of modules whose underlying modules are flat, and \(\operatorname{ner}\circ Lf_B^*\cong L\varphi_{\mathrm{big}}^*\circ\operatorname{ner}\).

**Proof.** (1) Since \(f\) is a morphism of \(G\)-schemes, \(\mathrm{act}\circ(\mathrm{id}\times f)=f\circ\mathrm{act}\) and \(\mathrm{pr}\circ(\mathrm{id}\times f)=f\circ\mathrm{pr}\), so \((\mathrm{id}\times f)^*(m)\) is a map \(\mathrm{act}^*f^*M\to\mathrm{pr}^*f^*M\), and the two conditions are pullbacks of those of \(m\). The levels of \(\operatorname{ner}(f_B^*M)\) are \(v_0^*f^*M=\varphi_n^*v_0^*M\), by the cartesian squares of Lemma 1.3, with the pulled back structure maps.

(2) Let \(\mathcal P\) be the class of modules of \(B[G,Y]^{\mathrm o}\) whose underlying module is flat. Every module is a quotient of \(\operatorname{ind}(P)\) for a free module \(P\) (Proposition 6.2 (3) of the lesson on equivariant sheaves), and \(\operatorname{ind}(P)=t_!s^*P\) has a flat underlying module, by condition (E) (Proposition 5.1 of the lesson on cohomology of equivariant modules). For a short exact sequence \(0\to P'\to P\to P''\to0\) with \(P,P''\in\mathcal P\), \(P'\in\mathcal P\) and \(f_B^*\) of it is exact, as for flat modules. By [Tags 05T7, 05T9, 05TA], \(Lf_B^*\) exists on \(D^-\) and is computed by bounded above complexes in \(\mathcal P\). For such a complex \(Q\), \(\operatorname{ner}(Q)\) is a bounded above complex of modules with flat levels, hence of flat modules of the total topos (tensor products are computed levelwise), so it is K-flat [Tag 06YQ], and \(\operatorname{ner}(f_B^*Q)=\varphi_{\mathrm{big}}^*\operatorname{ner}(Q)\) computes \(L\varphi_{\mathrm{big}}^*\operatorname{ner}(Q)\) [Tag 06YY]. \(\square\)

**Proposition 3.2 (Ext groups).** Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes, Tor independent of \(G\), and \(J\) a module of \(B[G,X]^{\mathrm o}\) whose underlying module is quasi-coherent, \(J=\varepsilon^*J_0\) for a quasi-coherent \(\mathcal O_X\)-module \(J_0\) (Proposition 1.1 of the lesson on big fppf sites).

1. \(\operatorname{ner}(J)=\varepsilon^*\underline J\) for a termwise quasi-coherent module \(\underline J\) of \(\operatorname{Top}(N(G,X)_{\mathrm{Zar}})\), unique up to unique isomorphism, with the levels \(v_0^*J_0\); it is cartesian along every \(\alpha\) with \(\alpha(0)=0\).
2. For all integers \(i\) and \(n\leq-i\), there are isomorphisms, natural in \(J\),
\[
\begin{aligned}&\operatorname{Hom}\bigl(t_{[n}L^G_{X/Y},J[i]\bigr)\\&\quad\cong\operatorname{Ext}^i_{\operatorname{Top}(N(G,X)_{\mathrm{Zar}})}\\&\qquad\bigl(L_{N(G,X)/N(G,Y)},\underline J\bigr),\end{aligned}
\]
compatible with the transition maps. We write \(\operatorname{Ext}^i(t_{[}L^G_{X/Y},J)\) for this group.

**Proof.** (1) The levels \(v_0^*J=\varepsilon^*v_0^*J_0\) of \(\operatorname{ner}(J)\) are quasi-coherent, and \(\varepsilon^*\) is an equivalence from quasi-coherent modules on the Zariski sites onto quasi-coherent modules on the big sites, compatible with pullbacks (Proposition 1.1 of the lesson on big fppf sites and the isomorphisms \(N(\alpha)_{\mathrm{big}}^*\varepsilon^*\cong\varepsilon^*N(\alpha)^*\) of its Construction 4.1). So the levels and structure maps of \(\operatorname{ner}(J)\) come from unique termwise quasi-coherent data \(\underline J\), cartesian along \(\alpha\) with \(\alpha(0)=0\) because \(\operatorname{ner}(J)\) is (Proposition 1.5 (3)).

(2) By Proposition 1.5 (3) and (2.1),
\[
\begin{aligned}&\operatorname{Hom}(t_{[n}L^G_{X/Y},J[i])\\&\quad \cong\operatorname{Hom}(t_{[n}L\varepsilon^*L,\varepsilon^*\underline J[i]),\end{aligned}
\]
which is \(\operatorname{Hom}(L\varepsilon^*L,\varepsilon^*\underline J[i])\) by (0.1), as \(n\leq-i\). Lemma 4.2 (2) of the lesson on big fppf sites, with \(F=\underline J\) and its truncation index \(0\), identifies this with \(\operatorname{Hom}(L,\underline J[i])\), because \(t_{[0}L\varepsilon^*\underline J=\varepsilon^*\underline J\). All these maps are natural and compatible with the truncations. \(\square\)

**Proposition 3.3 (transitivity and the Kodaira–Spencer class).** Let \(X\xrightarrow{f}Y\xrightarrow{g}Z\) be morphisms of \(G\)-schemes, with \(f\) and \(G\), and \(g\) and \(G\), Tor independent, and let \(\varphi=N(G,f)\).

1. \(gf\) and \(G\) are Tor independent, and \(L\varphi^*L_{N(G,Y)/N(G,Z)}\) is quasi-cartesian along every \(\alpha\) with \(\alpha(0)=0\). For every \(n\), \(\operatorname{ner}\bigl(t_{[n}Lf_B^*t_{[n}L^G_{Y/Z}\bigr)\cong\) \(t_{[n}L\varepsilon^*L\varphi^*L_{N(G,Y)/N(G,Z)}\).
2. For every \(n\) there are unique maps in \(D^b(B[G,X]^{\mathrm o})\)
\[
\begin{aligned}&t_{[n}Lf_B^*t_{[n}L^G_{Y/Z} \to t_{[n}L^G_{X/Z}\\&\quad \to t_{[n}L^G_{X/Y}\\&\quad \xrightarrow{\ K_{G,n}\ }t_{[n}\bigl(Lf_B^*t_{[n}L^G_{Y/Z}[1]\bigr)\end{aligned}
\]
whose nerves are \(t_{[n}L\varepsilon^*\) of the transitivity triangle (Proposition 1.3 of the lesson on deformations of diagrams) of \(N(G,X)\to N(G,Y)\to N(G,Z)\). They are compatible with the transition maps. The system \(K_G(X/Y/Z)=(K_{G,n})_n\) is the *equivariant Kodaira–Spencer class* of \(X\to Y\to Z\).
3. For a module \(J\) of \(B[G,X]^{\mathrm o}\) as in Proposition 3.2, the identifications of Proposition 3.2 (2), and the corresponding ones for \(Lf_B^*t_{[}L^G_{Y/Z}\), carry the maps of (2) to the maps of the transitivity triangle of the nerves. So there is a long exact sequence
\[
\begin{aligned}&\cdots \to\operatorname{Ext}^i(t_{[}L^G_{X/Y},J)\\&\quad \to\operatorname{Ext}^i(t_{[}L^G_{X/Z},J)\\&\quad \to\operatorname{Ext}^i(Lf_B^*t_{[}L^G_{Y/Z},J)\\&\quad \xrightarrow{\ K_G^*\ }\operatorname{Ext}^{i+1}(t_{[}L^G_{X/Y},J)\\&\quad \to\cdots,\end{aligned}
\]
where each group is \(\operatorname{Hom}(t_{[n}(-),J[i])\) for any \(n\leq-i-1\).

**Proof.** (1) Apply Lemma 2.2 with \(Q=Z\), \(P=N_n(G,Z)\), \(R=Y\) and \(R'=X\). Here \(P\) and \(R\) are Tor independent over \(Q\) because \(g\) and \(G\) are, and \(P\times_QR=N_n(G,Y)\) by Lemma 1.3, which is Tor independent of \(X\) over \(Y\) because \(f\) and \(G\) are. So \(N_n(G,Z)\) and \(X\) are Tor independent over \(Z\). The structure maps of \(L\varphi^*L_{N(G,Y)/N(G,Z)}\) are \(L\varphi_n^*\) of those of \(L_{N(G,Y)/N(G,Z)}\), which are isomorphisms along \(\alpha\) with \(\alpha(0)=0\) by Proposition 2.3 (1). The last assertion is Lemma 3.1 (2), (2.1) for \(g\), (0.2) for \(\varphi_{\mathrm{big}}\), and the commutation \(L\varphi_{\mathrm{big}}^*L\varepsilon^*\cong L\varepsilon^*L\varphi^*\) of the proof of Lemma 4.2 (3) of the lesson on big fppf sites.

(2) All vertices of \(t_{[n}L\varepsilon^*\) of the triangle are nerves, by (1), Proposition 2.3 (2) and Theorem 2.4. The maps lift uniquely by the full faithfulness of Proposition 1.5 (3).

(3) The identifications of Proposition 3.2 (2) are natural in the first variable, through (2.1), (0.1) and the adjunction of Lemma 4.2 (2) of the lesson on big fppf sites, so they carry the maps of (2) to the maps \(\operatorname{Ext}^i(-,\underline J)\) of the triangle. The long exact sequence of \(\operatorname{Ext}^i(-,\underline J)\) for a distinguished triangle [Tag 0149] gives the sequence. For the term \(\operatorname{Ext}^i(Lf_B^*t_{[}L^G_{Y/Z},J)\) the identification holds by the same argument with (1) in place of (2.1). The connecting map involves \(\operatorname{Ext}^{i+1}\), which is why \(n\leq-i-1\) is required. \(\square\)

The truncations of a distinguished triangle need not form a distinguished triangle, so (2) gives maps, not a distinguished triangle, in \(D^b(B[G,X]^{\mathrm o})\). The long exact sequences of (3) do not depend on this.

## 4. Change of the category scheme

**Proposition 4.1.** Let \(u\colon G'\to G\) be an internal functor of category schemes over \(S\). For a \(G\)-scheme \(X\), let \(X'=G'_0\times_{G_0}X\), with the action \(g'\cdot(c,x)=(d_0(g'),u(g')\cdot x)\) of \(G'\); for a morphism \(f\colon X\to Y\) of \(G\)-schemes, let \(f'=G'_0\times_{G_0}f\).

1. \(N(G',X')=\) \(N(G')\times_{N(G)}N(G,X)\), and \(X'\mapsto\) \(X\) induces an internal functor \(u_X\colon[G',X']\to[G,X]\) and a functor \(u_B^*\) from modules of \(B[G,X]^{\mathrm o}\) to modules of \(B[G',X']^{\mathrm o}\), with a left derived functor \(Lu_B^*\) on \(D^-\) satisfying \(\operatorname{ner}\circ Lu_B^*\cong L\psi_{\mathrm{big}}^*\circ\operatorname{ner}\), where \(\psi=N(u_X)\colon N(G',X')\to N(G,X)\).
2. Suppose that \(f\) and \(G\), and \(f'\) and \(G'\), are Tor independent. For every \(n\) there is a canonical map
\[
\begin{aligned}&t_{[n}Lu_B^*t_{[n}L^G_{X/Y}\\&\quad \longrightarrow t_{[n}L^{G'}_{X'/Y'},\end{aligned} \tag{4.1}
\]
compatible with the transition maps, whose nerve is \(t_{[n}L\varepsilon^*\) of the functoriality map \(L\psi^*L_{N(G,X)/N(G,Y)}\to\) \(L_{N(G',X')/N(G',Y')}\) of the square formed by \(N(u_X)\), \(N(u_Y)\), \(N(G',f')\) and \(N(G,f)\).
3. If moreover \(X\) and \(Y'\) are Tor independent over \(Y\), the maps (4.1) are isomorphisms.

**Proof.** (1) A string of \([G',X']\) is a string \((g'_1,\ldots,g'_n)\) of \(G'\) with a point \((c,x)\) of \(X'\), \(c=d_1(g'_1)\); it maps to the string \((u(g'_1),\ldots,u(g'_n);x)\) of \([G,X]\), and the data are exactly a string of \(G'\) and a string of \([G,X]\) with the same image in \(N(G)\). The functor \(u_B^*\) pulls a module back along \(X'\to X\) with the pulled back structure, and Lemma 3.1 holds for it with the same proof, the cartesian squares of Lemma 1.3 being replaced by \(N_n(G',X')=N_n(G')\times_{N_n(G)}N_n(G,X)\). Condition (E) for \([G,X]^{\mathrm o}\) and Proposition 6.2 (3) of the lesson on equivariant sheaves give the class \(\mathcal P\) of the proof of Lemma 3.1.

(2) The functoriality map of the square of simplicial schemes has levels the functoriality maps of the squares of schemes, compatible with the structure maps, and both sides are quasi-cartesian along \(\alpha\) with \(\alpha(0)=0\) (Propositions 2.3 (1) and 3.3 (1), with \(\psi\) in place of \(\varphi\)). As in Proposition 3.3, \(t_{[n}L\varepsilon^*\) of it is a map between nerves, and it lifts uniquely.

(3) At the level \(n\) the square of schemes is cartesian, \(N_n(G',X')=\) \(N_n(G',Y')\times_{N_n(G,Y)}N_n(G,X)\), by (1). By Proposition 1.4 of the lesson on deformations of diagrams it suffices to show that \(N_n(G,X)\) and \(N_n(G',Y')\) are Tor independent over \(N_n(G,Y)\). First apply Lemma 2.2 with \(Q=Y\), \(P=X\), \(R=Y'\) and \(R'=N_n(G',Y')\): \(X\) and \(Y'\) are Tor independent over \(Y\) by hypothesis, \(X\times_YY'=X'\) (both are \(G'_0\times_{G_0}X\)), and \(X'\) and \(N_n(G',Y')\) are Tor independent over \(Y'\) because \(f'\) and \(G'\) are; so \(X\) and \(N_n(G',Y')\) are Tor independent over \(Y\). Then apply Lemma 2.2 with \(Q=Y\), \(P=X\), \(R=N_n(G,Y)\) and \(R'=N_n(G',Y')\): \(X\) and \(N_n(G,Y)\) are Tor independent over \(Y\) because \(f\) and \(G\) are, and \(X\times_YN_n(G,Y)=N_n(G,X)\) by Lemma 1.3; so \(N_n(G,X)\) and \(N_n(G',Y')\) are Tor independent over \(N_n(G,Y)\). \(\square\)

## 5. Exercises

**5.1.** Let \(S=\operatorname{Spec}R\), \(G=\mathbb G_{m,S}=\operatorname{Spec}R[\lambda,\lambda^{-1}]\) acting on \(X=\mathbb A^1_S=\operatorname{Spec}R[t]\) by \(\lambda\cdot t=\lambda t\), and \(f\colon X\to Y=S\) the structure morphism, with the trivial action on \(S\). Show that \(f\) and \(G\) are Tor independent, that \(L^G_{X/Y}\) (Proposition 2.5 (2)) is the module \(\mathcal O_X\,dt\) in degree \(0\), and that its structure \(\mathrm{act}^*(dt)\mapsto\lambda\,\mathrm{pr}^*(dt)\) is multiplication by the character \(\lambda\).

**5.2.** Let \(G\) be a group scheme over \(S\), with unit \(e\colon S\to G\) and structure morphism \(f\colon G\to S\), acting on \(X=G\) by left translation.

(a) Show that \((g,x)\mapsto(g\cdot x,x)\) is an isomorphism \(G\times_SG\to G\times_SG\), so that \([G,G]\) has exactly one arrow from any point \(x\) to any point \(y\).

(b) Let \(M\) be a module of \(B[G,G]^{\mathrm o}\). Show that its structure is invertible and that \(M\cong f^*e^*M\) as \(\mathcal O\)-modules on \(T/G\).

**5.3.** Let \(G=S\) be the trivial monoid scheme over \(S\), and \(f\colon X\to Y\) a morphism of schemes over \(S\), with the trivial actions. Show that \(B[G,X]^{\mathrm o}\) is the category of \(\mathcal O\)-modules on \(T/X\), that \(f\) and \(G\) are Tor independent, that \(t_{[n}L^G_{X/Y}=t_{[n}L\varepsilon^*L_{X/Y}\), and that \(\operatorname{Ext}^i(t_{[}L^G_{X/Y},\varepsilon^*J_0)=\operatorname{Ext}^i_X(L_{X/Y},J_0)\) for every quasi-coherent \(\mathcal O_X\)-module \(J_0\).

## 6. Complete solutions

**5.1.** The ring \(R[t]\) is free over \(R\), so \(f\) is flat and Tor independent of \(G\) (Definition 2.1). By formula (2.5) of *The cotangent complex* and its Proposition 10.1, \(L_{X/Y}=\mathcal O_X\,dt\) in degree \(0\), of tor amplitude in \([0,0]\). By Proposition 2.5 (2) with \(a=0\), \(L^G_{X/Y}\) is a module in degree \(0\), with underlying module \(\varepsilon^*(\mathcal O_X\,dt)\) (Proposition 2.5 (1)). Its nerve at the level \(1\) is \(\varepsilon^*\) of \(L_{N_1(G,X)/N_1(G,Y)}=\Omega_{R[\lambda^{\pm1},t]/R[\lambda^{\pm1}]}\), free on \(dt\), with \(N_1(G,X)=G\times_SX\) and \(N_1(G,Y)=G\). The structure maps of \(L_{N(G,X)/N(G,Y)}\) along the two vertices of \([1]\) are the functoriality maps of the squares formed by \(\mathrm{pr}\colon(\lambda,t)\mapsto t\) and by \(\mathrm{act}\colon(\lambda,t)\mapsto\lambda t\) over \(G\to S\): they send \(\mathrm{pr}^*(dt)\) to \(dt\) and \(\mathrm{act}^*(dt)\) to \(d(\lambda t)=\lambda\,dt\), since \(d\lambda=0\) relative to \(R[\lambda^{\pm1}]\). In \([G,X]^{\mathrm o}\) the source is \(\mathrm{act}\) and the target \(\mathrm{pr}\) (Proposition 1.5 (2)), and the structure constructed in the proof of Proposition 2.3 (1) of the lesson on equivariant sheaves is the structure map along the source followed by the inverse of the structure map along the target: \(\mathrm{act}^*(dt)\mapsto\lambda\,dt\mapsto\lambda\,\mathrm{pr}^*(dt)\).

**5.2.** (a) On points, the inverse is \((y,x)\mapsto(yx^{-1},x)\). An arrow \((g,x)\) of \([G,G]\) goes from \(x\) to \(g\cdot x\), so arrows from \(x\) to \(y\) correspond to the \(g\) with \(gx=y\), and there is exactly one, \(g=yx^{-1}\).

(b) \(G\) is a groupoid scheme, so the structure is invertible (Proposition 1.6, and Examples 1.4 (b) of the lesson on equivariant sheaves). The structure is an isomorphism \(\mathrm{act}^*M\to\mathrm{pr}^*M\) on \(G\times_SG\) (Proposition 1.5 (2)). Pull it back along \(\sigma\colon G\to G\times_SG\), \(x\mapsto(x,e)\): since \(\mathrm{act}\circ\sigma=\mathrm{id}_G\) and \(\mathrm{pr}\circ\sigma=e\circ f\), it gives an isomorphism \(M\to(ef)^*M=f^*e^*M\).

**5.3.** Here \(G_0=G_1=S\), so \([G,X]\) has the objects \(X\) and only identity arrows, and its nerve is the constant simplicial scheme \(X\). By Examples 1.4 (a) of the lesson on equivariant sheaves, a module of \(B[G,X]^{\mathrm o}\) is an \(\mathcal O\)-module on \(T/X\) with the identity structure. Every \(N_n(G,Y)\to Y\) is the identity, so \(f\) and \(G\) are Tor independent. The underlying-module functor is the identity, so Proposition 2.5 (1) gives \(t_{[n}L^G_{X/Y}=t_{[n}L\varepsilon^*L_{X/Y}\). For \(n\leq-i\), by (0.1) and Corollary 2.2 (1) of the lesson on big fppf sites with the truncation index \(0\),
\[
\begin{aligned}&\operatorname{Hom}(t_{[n}L\varepsilon^*L_{X/Y},\varepsilon^*J_0[i])\\&\quad =\operatorname{Hom}(L\varepsilon^*L_{X/Y},\varepsilon^*J_0[i])\\&\quad =\operatorname{Ext}^i_X(L_{X/Y},J_0).\end{aligned}
\]

## Sources

The equivariant cotangent complex, its construction through the nerves and the classifying topos of the opposite category, and the equivariant Kodaira–Spencer class are due to Illusie, in his work on the cotangent complex.