# Lie complexes of module schemes

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

Let \(A\) be a ring scheme over \(S\) and \(G\) an \(A\)-module scheme, flat and locally of finite presentation. The Lie complex of \(G\), a perfect complex with cohomology in degrees \(0\) and \(1\), carries an action of \(A\) through the endomorphisms of \(G\) given by multiplication by local sections of \(A\). This lesson shows that the action exists on the complex itself, not only on its cohomology: there is an object \({}_A\ell^\vee_G\) of the derived category of \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\), unique up to unique isomorphism, whose restriction of scalars to \(\mathcal O\) is the Lie complex. The derived tensor product appears because \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\to A\otimes_{\mathbb Z}\mathcal O\) is not a quasi-isomorphism when \(\operatorname{Tor}_1^{\mathbb Z}(A,\mathcal O)\neq0\) (Exercise 5.3).

The construction uses the spectral nerve \(N(A,G)\), the nerve of the action of the spectrum of \(A\) on the spectrum of \(G\). It is a diagram of products of copies of \(A\) and \(G\), and a commutative group scheme over the diagram \(N(A,S)\) of products of copies of \(A\). Its Lie complex is a second nerve, so it comes from the classifying category of the spectral action; by the computation of the Lie complex of the spectrum ([*Lie complexes of commutative group schemes*](lie-complexes-of-commutative-group-schemes.md)), its underlying object is of constant type, and the comparison theorem of [*Ext groups of modules as cohomology of spectral nerves*](ext-groups-of-modules-as-cohomology-of-spectral-nerves.md) turns it into \({}_A\ell^\vee_G\). The main theorem identifies the Ext groups of the cotangent complex of the spectral nerve with Ext groups of \(A\)-modules:
\[
\begin{aligned}&\operatorname{RHom}\bigl(Lu^*L\varepsilon^*L_{N(A,G)/N(A,S)},M_{N(A,F)}\bigr)\\&\quad\cong\operatorname{RHom}_A\bigl(F,{}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M\bigr)\end{aligned}
\]
for a homomorphism \(u\colon F\to G\) of \(A\)-module schemes. These are the groups in which deformations of module schemes are classified.

**Prerequisites.** [*Lie complexes of commutative group schemes*](lie-complexes-of-commutative-group-schemes.md) (Definition 1.1, Propositions 1.3, 1.4, 1.6 and 3.5, Lemmas 1.5 and 3.2); [*Ext groups of modules as cohomology of spectral nerves*](ext-groups-of-modules-as-cohomology-of-spectral-nerves.md) (Conventions, Lemmas 1.1 and 2.1, Definition 3.2, Theorem 3.3, Propositions 3.1, 3.4, 3.5, 4.1, 4.2 and 4.4, Theorem 4.3, Remark 4.5); [*The stable ring of a ring and the Mac Lane resolution*](the-stable-ring-of-a-ring-and-the-mac-lane-resolution.md) (Construction 2.1, Proposition 2.4); [*Graded monoids and the nerves of their actions*](graded-monoids-and-the-nerves-of-their-actions.md) (Conventions, Proposition 3.3); [*Graded rings and the spectra of modules*](graded-rings-and-the-spectra-of-modules.md) (2.2, Definition 3.2); [*Diagonal rings and adjunction formulas*](diagonal-rings-and-adjunction-formulas.md) (Construction 2.1, Propositions 2.3 and 3.4); [*Differential graded modules in a topos*](differential-graded-modules-in-a-topos.md) (Propositions 2.5 and 2.7, Exercise 4.2); [*Equivariant sheaves on simplicial topoi*](equivariant-sheaves-on-simplicial-topoi.md) (Definition 2.1, Proposition 2.3, Section 6); [*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md) (Lemmas 1.2 and 2.2, Propositions 3.1 and 3.2, Theorem 4.3, Corollary 4.4); [*Nerves of induced and coinduced modules*](nerves-of-induced-and-coinduced-modules.md) (Theorem 3.4); [*Derived left adjoints on the second total topos and local inductive limits*](derived-left-adjoints-on-the-second-total-topos-and-local-inductive-limits.md) (Theorem 3.1, Remark 3.2, Corollary 4.2, Proposition 4.4); [*A duality formula for diagrams in a topos*](a-duality-formula-for-diagrams-in-a-topos.md) (Lemmas 1.2 and 3.1, Theorem 2.3); [*Quasi-coherent and perfect complexes on big fppf sites*](quasi-coherent-and-perfect-complexes-on-big-fppf-sites.md) (Construction 4.1, Lemma 4.2); [*The co-Lie complex of a group scheme*](the-co-lie-complex-of-a-group-scheme.md) (Propositions 1.4 and 2.2); [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md) (Propositions 1.3 and 1.4). From the AI-integrated Stacks project: extension by zero [[Stacks, Tag 03DJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-extension-by-zero-exact), [Tag 03EV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-j-shriek-flat)]; triangles [[Tag 014A](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-third-isomorphism-triangle)]; duals of perfect complexes on ringed sites [[Tag 08JJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-dual-perfect-complex)].

## Conventions

\(S\) is a scheme, \(T\) the topos of sheaves on \((\mathrm{Sch}/S)_{\mathrm{fppf}}\) with its structure ring \(\mathcal O\), and \(\varepsilon\) the comparison morphisms; diagrams of \(S\)-schemes, their topoi, group schemes over diagrams, co-Lie complexes \(\ell\), Lie complexes \(\ell^!\) and the notation \(M_Y\) are those of [*Lie complexes of commutative group schemes*](lie-complexes-of-commutative-group-schemes.md). \(A\) is a *ring scheme* over \(S\): a ring of \(T\), associative and unital, represented by an \(S\)-scheme. An *\(A\)-module scheme* is an \(A\)-module of \(T\) represented by an \(S\)-scheme that is flat and locally of finite presentation over \(S\); a homomorphism of \(A\)-module schemes is an \(A\)-linear map.

The notation of [*Ext groups of modules as cohomology of spectral nerves*](ext-groups-of-modules-as-cohomology-of-spectral-nerves.md) is used for the ring \(A\) of \(T\) and the commutative ring \(\mathcal O\): the rings \(Q=Q_A\), \(Q_{\mathcal O}\), \(B_A^{\mathcal O}\) and \(R_{\mathcal O}\), which represents \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\); the maps \(\pi_{\mathcal O}\colon B^{\mathcal O}_A\to R_{\mathcal O}\) and \(\bar\varepsilon\colon R_{\mathcal O}\to A\otimes\mathcal O\); the underlying complex \(M|_A\) of \(A\)-modules of \(M\in D(R_{\mathcal O})\); the functors \(\Phi\), \(\Phi_{\mathcal O}\), \(\Psi\colon D^b(R_{\mathcal O})\to D^b(Q_{\mathcal O})\); objects of constant type; the simplicial topoi \(\mathcal X(G',X)\), \(\mathcal X_A\), \(\mathcal X_G\), the category \(BA\) and the morphism \(p\colon\mathcal X_G\to\mathcal X_A\). For an \(A\)-module \(G\) let
\[
\begin{aligned}&N(A,G)\\&\quad =\operatorname{Ner}\bigl(C_\bullet(A(1)^\times),C_\bullet(G(1)^\times)\bigr),\end{aligned}
\]
a diagram of \(T\) indexed by the total category \(\mathfrak L\) of the bisimplicial shape \((a,q)\mapsto C_q(S_1)^{a+1}\), so that \(\operatorname{Top}^{\mathrm o}(\mathcal X_G)=T^{\mathfrak L}/N(A,G)\) (Remark 4.5 there), and \(N(A,S)=N(A,0)\), the diagram of \(\mathcal X_A\). The direction \(a\) is the direction of the nerve of the action. For an increasing map \(\alpha\colon[m]\to[n]\) and an object \(k\) of the level \(n\), the arrow of \(\mathfrak L\) over \(\alpha\) whose second component is an identity is written \((\alpha,\mathrm{id})\); the maps of a module \(E\) of \(\operatorname{Top}^{\mathrm o}(\mathcal X_A)\) along these arrows are the maps \(E_\alpha\colon E_n\to\mathcal X_A(\alpha)^*E_m\) of the simplicial topos \(\mathcal X_A\). The modules of \(BA\) with coefficients in \(\mathcal O\) are the \(Q_{\mathcal O}\)-modules (Proposition 4.1 (2) there); \(D(BA)\) denotes \(D(Q_{\mathcal O})\), \(\operatorname{ner}^{\mathrm o}\colon D(BA)\to D(\operatorname{Top}^{\mathrm o}(\mathcal X_A))\) the second nerve, and \(U_{\mathcal O}\) the functor that forgets the action of \(Q\) and keeps the \(\mathcal O\)-modules of type \(C_\bullet(S_1)\). The object \(o\) of \(\int C_\bullet(S_1)\) is that of Lemma 3.2 of the lesson on commutative group schemes, at which \(C_\bullet(G(1))\) has the value \(G\).

## 1. The spectral nerve of a module scheme

**Lemma 1.1.** Let \(G\) be an \(A\)-module scheme.

1. \(N(A,G)\) is a diagram of \(S\)-schemes, and with the projection \(q\colon N(A,G)\to N(A,S)\) it is a commutative group scheme over \(N(A,S)\): its value at every object \(\lambda\) of \(\mathfrak L\) is \(N(A,S)_\lambda\times_SG^{P(\lambda)}\) for a finite set \(P(\lambda)\), with the group law of \(G^{P(\lambda)}\), and its maps are homomorphisms over those of \(N(A,S)\). A homomorphism \(u\colon F\to G\) gives a homomorphism \(N(A,u)\colon N(A,F)\to N(A,G)\) over \(N(A,S)\).
2. The restriction of \(q\) to the level \(a=0\), the full subcategory \(\int C_\bullet(S_1)\) of \(\mathfrak L\), is the group scheme \(C_\bullet(G(1))\) over \(S_K\), \(K=\int C_\bullet(S_1)\).
3. For every arrow \((\alpha,\mathrm{id})\) with \(\alpha(0)=0\), the square formed by the maps of \(N(A,G)\) and \(N(A,S)\) along it and by \(q\) is cartesian.

**Proof.** By the conventions of the lesson on graded monoids and Remark 4.5 of the lesson on Ext groups, the value of \(N(A,G)\) at an object \((q,(k_1,\ldots,k_a,k))\) is the product of the \(C_q(A(1)^\times)(k_j)\), which are products \(A^{P_j}\), and of \(C_q(G(1)^\times)(k)=G^{P}\); for \(G=0\) the last factor is \(S\). These are \(S\)-schemes, which gives (1) for the values, with \(P(\lambda)=P\), and (2), since at \(a=0\) there are no factors \(A^{P_j}\). The maps of \(N(A,G)\) along the arrows of a fixed level \(a\) act on each factor by the maps of \(C_\bullet(A(1)^\times)\) and of \(C_\bullet(G(1)^\times)\), on the factor \(G^P\) by homomorphisms given by integer matrices (Lemma 3.2 of the lesson on commutative group schemes). Along \((\alpha,\mathrm{id})\) they are the maps of the nerve of the action (Proposition 3.3 of the lesson on graded monoids and its proof): on a point \((g_1,\ldots,g_n;x)\) the factors of \(A\) are multiplied according to \(\alpha\), and the factor of \(G\) becomes \(g_{\alpha(0)}\cdots g_1x\), formed with the action of \(C_\bullet(A(1)^\times)\) on \(C_\bullet(G(1)^\times)\). This action is the spectrum of the action \(A\times G\to G\) (Definition 3.2 of the lesson on graded rings), additive in the second variable. So for fixed values of the factors of \(A\), every map is a homomorphism in the factor of \(G\), that is, a homomorphism over the corresponding map of \(N(A,S)\). A homomorphism \(u\) acts by powers of \(u\) on the factors of \(G\) and commutes with all these maps, being additive and \(A\)-linear. (3) If \(\alpha(0)=0\), the factor of \(G\) is \(x\), carried identically, with its index \(k\) unchanged, and the square is the product of the square of \(N(A,S)\) with the identity of \(G^P\). \(\square\)

**Lemma 1.2 (the action of \(A\) on the Lie complex).** Let \(G\) be an \(A\)-module scheme. For an \(S\)-scheme \(V\) and a point \(a\in A(V)\), multiplication by \(a\) is an endomorphism \(a_G\) of the group scheme \(G_V\) over \(V\). The maps induced by the \(\ell^!(a_G)\) on \(H^i(\ell^!_G)|_V\) make every \(H^i(\ell^!_G)\) a module over \(A\otimes\mathcal O\).

**Proof.** The Lie complex commutes with base change to \(V\) (Proposition 1.4 of [*The co-Lie complex of a group scheme*](the-co-lie-complex-of-a-group-scheme.md), and Remark 3.2 of the lesson on big fppf sites for \(L\varepsilon^*\), \(\ell_G\) being perfect), and the maps \(\ell^!(a_G)\) are compatible with base change along morphisms \(V'\to V\), the functoriality maps being compatible with base change. The \(S\)-schemes generate \(T\), so these maps define a map of sheaves from \(A\) to the sheaf of \(\mathcal O\)-linear endomorphisms of \(H^i(\ell^!_G)\). It is a ring homomorphism: \((a+b)_G=a_G+b_G\) gives additivity by Lemma 1.5 (3) of the lesson on commutative group schemes, \((ab)_G=a_G\circ b_G\) and \(1_G=\mathrm{id}\) give multiplicativity and unitality, \(\ell^!\) being a covariant functor. The action commutes with \(\mathcal O\), being \(\mathcal O\)-linear. \(\square\)

## 2. Second nerves on the spectral pseudo-categories

**Lemma 2.1.** Let \(\mathcal X=\mathcal X(G',X)\) be the simplicial topos of Section 4 of the lesson on Ext groups, for an \(S_1\)-graded monoid \(G'\) and an \(S_1\)-graded \(G'\)-object \(X\) of \(T\), with the ring \(\mathcal O\). Then \(\operatorname{ner}^{\mathrm o}\colon D^b(B\mathcal X)\to D^b(\operatorname{Top}^{\mathrm o}(\mathcal X))\) is fully faithful, and its essential image consists of the objects \(E\) all of whose cohomology modules have invertible maps along the arrows \((\alpha,\mathrm{id})\) with \(\alpha(0)=0\).

**Proof.** By Proposition 4.1 (3) of the lesson on Ext groups, \(\mathcal X\) is a good two-sided pseudo-category satisfying (I). Corollary 4.4 of [*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md) for the second nerve rests on Theorem 4.3 (1) there, whose proof applies Lemma 1.2 there with the vanishing of Proposition 3.2 there. The proof of Proposition 3.2 uses goodness and (I) (through Theorem 3.4 of [*Nerves of induced and coinduced modules*](nerves-of-induced-and-coinduced-modules.md) and Lemma 2.2 of the lesson on equivariant modules), and condition (E) only through Theorem 3.1 (3) and Proposition 4.4 of [*Derived left adjoints on the second total topos and local inductive limits*](derived-left-adjoints-on-the-second-total-topos-and-local-inductive-limits.md), applied to the augmentation \(\lambda\colon\operatorname{Dec}^1(\mathcal X)\to\mathcal X_0\). By Proposition 3.3 (3) of the lesson on graded monoids, \(\operatorname{Dec}^1(\mathcal X)\) with its augmentation is equivalent to the fibred topos of an augmented simplicial object of the topos \(\mathcal X_0\): its transition morphisms and the components of \(\lambda\) are localizations. The left adjoints of their module inverse images are extensions by zero, which are exact and carry flat modules to flat modules [Stacks, Tags 03DJ, 03EV]. So the conditions (L), (H\(^{\mathrm o}\)) and (F) of that lesson hold for \(\lambda\), and by its Remark 3.2 the formulas of its Theorem 3.1 (2) and Corollary 4.2 (2) hold for every bounded above complex. The proof of its Proposition 4.4 used the levelwise flat representative of \(M\) only to apply Corollary 4.2 (2); it therefore applies to every bounded above complex representing \(M\), and condition (F) for \(\mathcal X\) itself is not needed. Hence Proposition 3.2, Theorem 4.3 (1) and Corollary 4.4 of the lesson on equivariant modules hold for \(\mathcal X\) and the second nerve. By Proposition 2.3 (2) of the lesson on equivariant sheaves, which holds for modules (Section 6 there), a module of \(\operatorname{Top}^{\mathrm o}(\mathcal X)\) is a second nerve exactly when its maps \(E_\alpha\) with \(\alpha(0)=0\) are invertible. \(\square\)

## 3. The Lie complex with its action

For a complex \(Y\) of objects of \(\mathcal M_1(\mathcal O)\) (the \(\mathcal O\)-modules of type \(C_\bullet(S_1)\)), the value of \(Y^{\mathrm{red}}(-1)\) at the object of \(\int C_\bullet(S_0)\) corresponding to \(o\) is \(Y_o\), there being no simplicial coordinates at \(o\) (2.2 of the lesson on graded rings). Let
\[
\begin{aligned}&\nu_Y \colon Y_o \longrightarrow\Phi(Y^{\mathrm{red}}(-1))\\&\quad =\operatorname{Tot}N\Delta^2(Y^{\mathrm{red}}(-1))\end{aligned}
\]
be the inclusion of \(Y_o\) as the summand of the \(0\)-simplex \([-1]\) in the degree \(0\) of \(\Delta^2\) (Construction 2.1 of the lesson on diagonal rings), followed by the inclusion of the row \(0\) of the total complex, which is a subcomplex. It is \(\mathcal O\)-linear and natural in \(Y\).

**Lemma 3.1.** If \(UY\) is of constant type, \(\nu_Y\) is a quasi-isomorphism.

**Proof.** Evaluation at \(o\), reduction, \((-1)\) and \(\Phi\) preserve quasi-isomorphisms (Proposition 2.3 (2) of the lesson on diagonal rings), so \(\nu\) is a natural transformation between functors on \(D(\mathcal M_1)\), and it suffices to treat \(UY=(cH)(1)\) for a complex \(H\) of abelian sheaves. Then \(Y^{\mathrm{red}}(-1)\cong cH\) (Proposition 1.1 (3) of the lesson on diagonal rings), and \(\Delta^2cH=H\otimes\mathbb Z(N(\bar\Delta'))\), with the faces and degeneracies of the nerve of \(\bar\Delta'\) (proof of Proposition 3.4 (2) there). The map \(\nu\) is the inclusion of \(H\otimes[-1]\), the vertex \([-1]\). The augmentation \(N\mathbb Z(N(\bar\Delta'))\to\mathbb Z\) is a chain homotopy equivalence, the nerve of \(\bar\Delta'\) being contractible, and the vertex \([-1]\) is a section of it. So the inclusion of the vertex is a homotopy equivalence, and so is \(\nu\) after tensoring with \(H\) and totalizing. \(\square\)

**Proposition 3.2 (the Lie complex with coefficients in \(A\otimes^{\mathbf L}\mathcal O\)).** Let \(G\) be an \(A\)-module scheme.

1. There is an object \({}_A\ell^\vee_G\) of \(D^{[0,1]}(R_{\mathcal O})\) with an isomorphism \(\ell^!_{N(A,G)}\cong\operatorname{ner}^{\mathrm o}\Psi({}_A\ell^\vee_G)\) in \(D(\operatorname{Top}^{\mathrm o}(\mathcal X_A))\), and the pair is unique up to unique isomorphism. A homomorphism \(u\colon F\to G\) induces a unique map \({}_A\ell^\vee(u)\colon{}_A\ell^\vee_F\to{}_A\ell^\vee_G\) compatible with \(\ell^!(N(A,u))\), and \(G\mapsto{}_A\ell^\vee_G\) is a functor.
2. The restriction of scalars of \({}_A\ell^\vee_G\) to \(\mathcal O\) is isomorphic to \(\ell^!_G\), naturally in \(G\). The underlying complex of abelian sheaves of \(({}_A\ell^\vee_G)|_A\) is that of \(\ell^!_G\).
3. The cohomology modules of \({}_A\ell^\vee_G\), which are modules over \(H^0(R_{\mathcal O})=A\otimes\mathcal O\), are the \(H^i(\ell^!_G)\) with the action of Lemma 1.2.

**Proof.** (1) By Lemma 1.1 (3) and Proposition 1.3 (2) of the lesson on commutative group schemes, the maps of \(\ell^!_{N(A,G)}\) along the arrows \((\alpha,\mathrm{id})\) with \(\alpha(0)=0\) are invertible, and so are those of its cohomology modules. By Lemma 2.1 there is \(L\in D^b(BA)\) with \(\operatorname{ner}^{\mathrm o}(L)\cong\ell^!_{N(A,G)}\), unique up to unique isomorphism. The level \(0\) of the second nerve of \(L\) is \(U_{\mathcal O}L\) (Definition 2.1 of the lesson on equivariant sheaves), so \(U_{\mathcal O}L\) is the restriction of \(\ell^!_{N(A,G)}\) to the level \(0\), which is \(\ell^!_{C_\bullet(G(1))}\) by Lemma 1.1 (2) and Proposition 1.3 (3) of the lesson on commutative group schemes. By Proposition 3.5 there, \(UL\cong(cU\ell^!_G)(1)\), with \(U\ell^!_G\in D^{[0,1]}\). By Proposition 3.4 of the lesson on Ext groups, \(L\cong\Psi(M)\) for an object \(M\) of \(D^{[0,1]}(R_{\mathcal O})\), unique up to unique isomorphism; put \({}_A\ell^\vee_G=M\). A homomorphism \(u\) gives \(\ell^!(N(A,u))\), which comes from unique maps in \(D^b(BA)\) and in \(D^{[0,1]}(R_{\mathcal O})\), by the full faithfulness of \(\operatorname{ner}^{\mathrm o}\) (Lemma 2.1) and of \(\Psi\) on \(D^{[0,1]}\) (Proposition 3.4 there). Uniqueness makes the construction functorial.

(2) Let \(M={}_A\ell^\vee_G\) and \(L=\Psi(M)\). By Definition 3.2 of the lesson on Ext groups, \(\Phi_{\mathcal O}(L)\cong\pi_{\mathcal O}^*M\) as \(B_A^{\mathcal O}\)-modules. Restrict scalars along \(\mathcal O\to B_A^{\mathcal O}\), \(b\mapsto1\otimes b\). On \(\pi_{\mathcal O}^*M\) this is the restriction of \(M\) to \(\mathcal O\), \(\pi_{\mathcal O}\) being a map of \(\mathcal O\)-algebras. On \(\Phi_{\mathcal O}(L)=\Phi(U_{\mathcal O}L^{\mathrm{red}}(-1))\) it is the \(\mathcal O\)-module structure of the components of \(L\), by the construction of the action of \(B_A^{\mathcal O}\) (Lemma 1.1 (2) there). By Lemma 3.1, \(\nu\colon(U_{\mathcal O}L)_o\to\Phi(U_{\mathcal O}L^{\mathrm{red}}(-1))\) is an \(\mathcal O\)-linear quasi-isomorphism, and \((U_{\mathcal O}L)_o\) is the value of \(\ell^!_{N(A,G)}\) at the object \(o\) of the level \(0\), which is \(\ell^!_G\) (Proposition 1.3 (3) of the lesson on commutative group schemes). So \(M|_{\mathcal O}\cong\ell^!_G\), naturally in \(G\), all the maps being natural. By Theorem 3.3 (1) and (3) of the lesson on Ext groups, \(U\Psi(M)\) is isomorphic both to \((cH)(1)\), with \(H\) the underlying complex of abelian sheaves of \(M\), and to \((cH')(1)\), with \(H'\) that of \(M|_A\); since \(H\mapsto(cH)(1)\) is fully faithful (Proposition 3.6 (2) of the lesson on diagonal rings), \(H'\cong H\), which is the underlying complex of \(\ell^!_G\).

(3) By Theorem 3.3 (1) of the lesson on Ext groups, \(\Psi\) is exact and carries \(D^{\leq a}\) and \(D^{\geq a}\) into themselves, so it commutes with truncations and \(H^i(\Psi(M))\cong\Psi(H^i(M))\). The module \(H^i(M)\), placed in degree \(0\), is \(\bar\varepsilon^*H_i\) for the \(A\otimes\mathcal O\)-module \(H_i=H^i(M)\) (Exercise 4.2 of the lesson on differential graded modules), and \(\Psi(\bar\varepsilon^*H_i)\cong C_\bullet(H_i(1))\) (Theorem 3.3 (2) there). So \(H^i(\ell^!_{N(A,G)})\cong\operatorname{ner}^{\mathrm o}C_\bullet(H_i(1))\). At the object \(o\) of the level \(0\) this gives an \(\mathcal O\)-linear isomorphism \(H_i\cong H^i(\ell^!_G)\). At the level \(1\), take the object \((o,o)\), at which \(N(A,G)\) is \(A\times_SG\) over \(N(A,S)_{(o,o)}=A\), and the face \(\delta^0\colon[0]\to[1]\), which keeps the vertex \(1\). Along \((\delta^0,\mathrm{id})\), the second nerve of \(C_\bullet(H_i(1))\) has the action of the tautological section of \(A\) on \(H_i|_A\) (Definition 2.1 of the lesson on equivariant sheaves, the action being \(g(1,1)^\times\) at the index \((o,o)\), that is, the action of \(A\) on \(H_i\)). The group scheme \(N(A,G)\) has along this arrow the map \(A\times_SG\to G\), \((a,g)\mapsto ag\), over \(A\to S\); by Proposition 1.3 (1) and (2) there, the map of \(H^i(\ell^!_{N(A,G)})\) along it is induced by \(\ell^!\) of the endomorphism \((a,g)\mapsto(a,ag)\) of \(A\times_SG\) over \(A\), multiplication by the tautological section, which acts on \(H^i(\ell^!_G)|_A\) as in Lemma 1.2. So the isomorphism \(H_i\cong H^i(\ell^!_G)\) is \(A\)-linear for the tautological section, hence for every section, all sections being pulled back from it. \(\square\)

The isomorphism class of \({}_A\ell^\vee_G\) is more than its cohomology together with its underlying complex of \(\mathcal O\)-modules: Exercise 5.3 of the lesson on Ext groups shows that objects of \(D(R_{\mathcal O})\) with the same underlying \(Q\)-module can differ. When \(\operatorname{Tor}_1^{\mathbb Z}(A,\mathcal O)\neq0\), \(R_{\mathcal O}\) has cohomology in degree \(-1\) and is not quasi-isomorphic to \(A\otimes\mathcal O\) (Exercise 5.3).

**Proposition 3.3 (tensor products).** For \(M\in D^b(\mathcal O)\) there is an isomorphism, natural in \(M\),
\[
\begin{aligned}&\ell^!_{N(A,G)}\otimes^{\mathbf L}M_{N(A,S)}\\&\quad\cong\operatorname{ner}^{\mathrm o}\Psi\bigl({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M\bigr).\end{aligned}
\]

**Proof.** Represent \(M\) by a bounded above complex \(P\) of flat \(\mathcal O\)-modules of \(T\). The levels of the second nerve are inverse images, which commute with tensor products, so \(\operatorname{ner}^{\mathrm o}(Y)\otimes P_{N(A,S)}=\operatorname{ner}^{\mathrm o}(Y\otimes_{\mathcal O}P)\) for every complex \(Y\) of \(Q_{\mathcal O}\)-modules, the tensor product on the right being formed componentwise. Both sides compute derived tensor products (Proposition 3.5 of the lesson on Ext groups, and Proposition 1.6 of the lesson on commutative group schemes for the left side). So the left side is \(\operatorname{ner}^{\mathrm o}(\Psi({}_A\ell^\vee_G)\otimes^{\mathbf L}_{\mathcal O}M)\). The complex \({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M\) is bounded, its underlying complex being \(\ell^!_G\otimes^{\mathbf L}M\) by Proposition 3.2 (2), with \(\ell^!_G\) perfect; so Proposition 3.5 of the lesson on Ext groups gives \(\Psi({}_A\ell^\vee_G)\otimes^{\mathbf L}_{\mathcal O}M\cong\Psi({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M)\). \(\square\)

**Proposition 3.4 (change of ring).** Let \(\rho\colon A'\to A\) be a homomorphism of ring schemes, \(A\)-module schemes being \(A'\)-module schemes through \(\rho\), and \(\rho_R\colon R'_{\mathcal O}\to R_{\mathcal O}\) the induced map of the rings representing \(A'\otimes^{\mathbf L}\mathcal O\) and \(A\otimes^{\mathbf L}\mathcal O\). Then \(\rho_R^*({}_A\ell^\vee_G)\cong{}_{A'}\ell^\vee_G\), naturally in \(G\). In particular, for the constant ring scheme \(\mathbb Z\), \(R_{\mathcal O}\) is quasi-isomorphic to \(\mathcal O\), and \({}_{\mathbb Z}\ell^\vee_G\) is \(\ell^!_G\).

**Proof.** The spectrum \(A(1)^\times\) is functorial in \(A\) (Definition 3.2 of the lesson on graded rings), and the rings \(Q_A\), \(B_A\), \(P_A=t_1(B_A)\), \(R_{\mathcal O}=P_A\otimes\mathcal O\) and the maps \(\varepsilon\), \(\pi\), \(w\), \(\pi_{\mathcal O}\) are obtained from it by functors (Construction 2.1 of the lesson on the stable ring). So \(\rho\) induces compatible ring maps \(Q_{A'}\to Q_A\), \(B^{\mathcal O}_{A'}\to B^{\mathcal O}_A\) and \(\rho_R\), and restriction of scalars along them commutes with \(\Phi_{\mathcal O}\), which is computed on components, and with \(\pi_{\mathcal O}^*\). For \(M\in D^b(R_{\mathcal O})\), the restriction of \(\Psi(M)\) to \(Q_{A'}\) is therefore of constant type with \(\Phi_{\mathcal O}\) isomorphic to \(\pi'^*_{\mathcal O}\rho_R^*M\), and by the uniqueness in Definition 3.2 of the lesson on Ext groups it is \(\Psi'(\rho_R^*M)\). On the other side, \(\rho\) induces morphisms of diagrams \(r\colon N(A',S)\to N(A,S)\) and \(N(A',G)\to N(A,G)\) of identity type, and the squares they form with the projections are cartesian, both diagrams of group schemes being products with the same \(G^{P(\lambda)}\). The functoriality map of a cartesian square of group schemes over diagrams, \(G\) being flat, is an isomorphism at every level (Proposition 1.4 of the lesson on the co-Lie complex), so \(\ell_{N(A',G)}\cong Lr^*\ell_{N(A,G)}\), and \(\ell^!_{N(A',G)}\cong r^{\mathrm o*}\ell^!_{N(A,G)}\) (Lemma 3.1 of the lesson on the duality formula). The second nerve of the restriction to \(Q_{A'}\) of a module \(L\) of \(BA\) is \(r^{\mathrm o*}\operatorname{ner}^{\mathrm o}(L)\), its levels being the inverse images along the vertices \(0\), which commute with \(r\). So \(\operatorname{ner}^{\mathrm o}\Psi'(\rho_R^*{}_A\ell^\vee_G)\cong r^{\mathrm o*}\ell^!_{N(A,G)}\cong\ell^!_{N(A',G)}\), and Proposition 3.2 (1) for \(A'\) gives the isomorphism. For \(A'=\mathbb Z\), \(P_{\mathbb Z}\to\mathbb Z\) is a quasi-isomorphism of bounded complexes of flat sheaves, so \(R_{\mathcal O}=P_{\mathbb Z}\otimes\mathcal O\to\mathcal O\) is one, restriction of scalars along it is an equivalence (Proposition 2.5 of the lesson on differential graded modules), and Proposition 3.2 (2) identifies \({}_{\mathbb Z}\ell^\vee_G\) with \(\ell^!_G\). \(\square\)

**Proposition 3.5 (exact sequences).** Let \(0\to E\xrightarrow{\iota}F\xrightarrow{\rho}G\to0\) be a sequence of \(A\)-module schemes that is exact as a sequence of group schemes (Section 2 of the lesson on the co-Lie complex). There is a distinguished triangle in \(D(B_A^{\mathcal O})\)
\[
\begin{aligned}&\pi_{\mathcal O}^*\,{}_A\ell^\vee_E\to\pi_{\mathcal O}^*\,{}_A\ell^\vee_F\to\pi_{\mathcal O}^*\,{}_A\ell^\vee_G\\&\quad\to\pi_{\mathcal O}^*\,{}_A\ell^\vee_E[1]\end{aligned}
\]
whose first two maps are induced by \({}_A\ell^\vee(\iota)\) and \({}_A\ell^\vee(\rho)\). Under the isomorphisms of Proposition 3.2 (2), its restriction of scalars to \(\mathcal O\) is the image of the triangle (2.1) of the lesson on the co-Lie complex under \(R\underline{\operatorname{Hom}}(L\varepsilon^*(-),\mathcal O)\).

**Proof.** At every level the sequence \(N(A,E)\to N(A,F)\to N(A,G)\) is the base change to \(N(A,S)_\lambda\) of a product of copies of the given exact sequence, hence exact. The proof of Proposition 2.2 of the lesson on the co-Lie complex applies over the diagram \(N(A,S)\), with the transitivity triangle of \(N(A,F)\to N(A,G)\to N(A,S)\) (Proposition 1.3 of the lesson on deformations of diagrams) and the cartesian square formed by \(N(A,\iota)\) and the unit of \(N(A,G)\), at every level (Proposition 1.4 there). It gives a distinguished triangle \(\ell_{N(A,G)}\to\) \(\ell_{N(A,F)}\to\) \(\ell_{N(A,E)}\to\) \(\ell_{N(A,G)}[1]\) with the first two maps \(\ell(N(A,\rho))\) and \(\ell(N(A,\iota))\). Applying \(R\underline{\operatorname{Hom}}^{\mathrm o}(L\varepsilon^*(-),\mathcal O)\) gives a distinguished triangle \(\ell^!_{N(A,E)}\to\) \(\ell^!_{N(A,F)}\to\) \(\ell^!_{N(A,G)}\to\) \(\ell^!_{N(A,E)}[1]\). By Proposition 3.2 (1), its vertices and first two maps are the images under \(\operatorname{ner}^{\mathrm o}\Psi\) of those of \({}_A\ell^\vee\), and by full faithfulness its third map is the image of a unique map \(\Psi({}_A\ell^\vee_G)\to\Psi({}_A\ell^\vee_E)[1]\). Complete \(\Psi({}_A\ell^\vee(\iota))\) to a distinguished triangle in \(D^b(BA)\); its image under the exact functor \(\operatorname{ner}^{\mathrm o}\) is distinguished, so it is isomorphic to the triangle above by a morphism of triangles that is the identity on the first two vertices [Stacks, Tag 014A], and by full faithfulness this isomorphism comes from \(D^b(BA)\). So the triangle of the \(\Psi({}_A\ell^\vee)\) is distinguished in \(D^b(BA)\). Its image under the exact functor \(\Phi_{\mathcal O}\), with \(\Phi_{\mathcal O}\Psi\cong\pi_{\mathcal O}^*\), is the triangle of the statement. Restricting scalars to \(\mathcal O\) and using the natural quasi-isomorphisms \(\nu\) of Lemma 3.1, it becomes the triangle of the values at the object \(o\) of the level \(0\), which is the dual triangle for \(E\to F\to G\) over \(S\), by Proposition 1.3 (3) of the lesson on commutative group schemes. \(\square\)

**Proposition 3.6 (the co-Lie complex with coefficients).** The ring \(\mathcal O\) is central in \(R_{\mathcal O}\). Put
\[
{}_A\ell_G=R\mathcal Hom_{\mathcal O}({}_A\ell^\vee_G,\mathcal O)\in D(R_{\mathcal O}^{\mathrm{op}}),
\]
by Proposition 2.7 of the lesson on differential graded modules. Its restriction of scalars to \(\mathcal O\) is isomorphic to \(L\varepsilon^*\ell_G\), it lies in \(D^{[-1,0]}\), and the canonical map \({}_A\ell^\vee_G\to R\mathcal Hom_{\mathcal O}({}_A\ell_G,\mathcal O)\) is an isomorphism in \(D(R_{\mathcal O})\).

**Proof.** Composed with restriction of scalars to \(\mathcal O\), \(R\mathcal Hom_{\mathcal O}\) is the derived internal Hom of \(\mathcal O\)-modules (Proposition 2.7 there). So the restriction of \({}_A\ell_G\) is \(R\mathcal Hom(\ell^!_G,\mathcal O)\cong L\varepsilon^*\ell_G\), by Proposition 3.2 (2) and the biduality of perfect complexes [Stacks, Tag 08JJ]; it lies in \(D^{[-1,0]}\). The canonical map to the double dual is \(R_{\mathcal O}\)-linear, being defined by evaluation, with which the action commutes; applying Proposition 2.7 to the ring \(R_{\mathcal O}^{\mathrm{op}}\), its target lies in \(D(R_{\mathcal O})\). On underlying complexes of \(\mathcal O\)-modules it is the biduality map of the perfect complex \(\ell^!_G\), an isomorphism [Stacks, Tag 08JJ]. \(\square\)

## 4. Ext groups

**Theorem 4.1 (the Ext formula).** Let \(u\colon F\to G\) be a homomorphism of \(A\)-module schemes, \(p\colon N(A,F)\to N(A,S)\) the projection, and \(M\in D^b(\mathcal O)\). There is an isomorphism, natural in \(M\),
\[
\begin{aligned}&\operatorname{RHom}\bigl(Lu^*L\varepsilon^*L_{N(A,G)/N(A,S)},M_{N(A,F)}\bigr)\\&\quad\cong\operatorname{RHom}_A\bigl(F,({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M)|_A\bigr),\end{aligned}
\]
where the left side is formed in \(D(\operatorname{Top}(N(A,F)_{\mathrm{fppf}}))\) and \(u\) stands for \(N(A,u)\).

**Proof.** Let \(q\colon N(A,G)\to N(A,S)\) be the projection, so that \(qu=p\), and \(E=L\varepsilon^*\ell_{N(A,G)}\). By the shear (Proposition 1.4 of the lesson on commutative group schemes) \(L_{N(A,G)/N(A,S)}\cong q^*\ell_{N(A,G)}\). Derived inverse images compose and commute with \(L\varepsilon^*\) [Stacks, Tag 0D6D], and \(p\) is flat at every level, so \(Lu^*L\varepsilon^*L_{N(A,G)/N(A,S)}\cong\) \(L\varepsilon^*Lp^*\ell_{N(A,G)}=\) \(p^*E\). Then
\[
\begin{aligned}&\operatorname{RHom}(p^*E,M_{N(A,F)})\\&\cong\Gamma_F\,R\underline{\operatorname{Hom}}^{\mathrm o}(p^*E,M_{N(A,F)})\\&\cong\Gamma_F\,p^{\mathrm o*}R\underline{\operatorname{Hom}}^{\mathrm o}(E,M_{N(A,S)})\\&\cong\Gamma_F\,p^{\mathrm o*}(\ell^!_{N(A,G)}\otimes^{\mathbf L}M_{N(A,S)})\\&\cong\Gamma_F\,p^*\operatorname{ner}^{\mathrm o}\Psi(M')\\&\cong\operatorname{RHom}_A(F,M'|_A),\end{aligned}
\]
with \(\Gamma_F=R\Gamma(\operatorname{Top}^{\mathrm o}(\mathcal X_F),-)\) and \(M'={}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M\in D^b(R_{\mathcal O})\). The first isomorphism is Theorem 2.3 of the lesson on the duality formula, for the diagram \(N(A,F)\) of \(T\), \(\operatorname{Top}^{\mathrm o}(\mathcal X_F)\) being its second total topos. The second is Lemma 3.1 there, for the morphism \(p\) of identity type, with \(p^{\mathrm o*}M_{N(A,S)}=M_{N(A,F)}\). The third is Proposition 1.6 of the lesson on commutative group schemes. The fourth is Proposition 3.3, with \(p^{\mathrm o*}\operatorname{ner}^{\mathrm o}=\operatorname{ner}^{\mathrm o}p^*\) (Proposition 4.2 (3) of the lesson on Ext groups). The last is Theorem 4.3 there, for the \(A\)-module \(F\) and \(Y=\Psi(M')|_Q\), cohomology of complexes of \(\mathcal O\)-modules being computed on the underlying abelian sheaves. \(\square\)

**Theorem 4.2 (the relative cotangent complex).** Let \(u\colon F\to G\) be a homomorphism of \(A\)-module schemes and \(K\) the first vertex of a distinguished triangle
\[
K\to{}_A\ell^\vee_F\xrightarrow{\ {}_A\ell^\vee(u)\ }{}_A\ell^\vee_G\to K[1]
\]
in \(D(R_{\mathcal O})\). For \(M\in D^b(\mathcal O)\) there is an isomorphism, natural in \(M\),
\[
\begin{aligned}&\operatorname{RHom}\bigl(L\varepsilon^*L_{N(A,F)/N(A,G)},M_{N(A,F)}\bigr)\\&\quad\cong\operatorname{RHom}_A\bigl(F,(K\otimes^{\mathbf L}_{\mathcal O}M)|_A\bigr).\end{aligned}
\]
If \(u\) is faithfully flat with kernel \(E\), then \(K\cong{}_A\ell^\vee_E\).

**Proof.** By the transitivity triangle of \(N(A,F)\to N(A,G)\to N(A,S)\) (Proposition 1.3 of the lesson on deformations of diagrams) and the naturality of the shear (Proposition 1.4 (2) of the lesson on commutative group schemes), there is a distinguished triangle \(p^*\ell_{N(A,G)}\xrightarrow{p^*\ell(u)}\) \(p^*\ell_{N(A,F)}\to\) \(L_{N(A,F)/N(A,G)}\to\) \(p^*\ell_{N(A,G)}[1]\). So \(L_{N(A,F)/N(A,G)}\cong p^*C\), where \(C\) is the third vertex of a triangle on \(\ell(N(A,u))\colon\ell_{N(A,G)}\to\ell_{N(A,F)}\), with perfect levels. The proof of Proposition 1.6 of the lesson on commutative group schemes applies to every object with perfect levels, so the chain of Theorem 4.1 gives \(\operatorname{RHom}(L\varepsilon^*p^*C,M_{N(A,F)})\cong\) \(R\Gamma(\operatorname{Top}^{\mathrm o}(\mathcal X_F),p^{\mathrm o*}(C^!\otimes^{\mathbf L}M_{N(A,S)}))\), with \(C^!=R\underline{\operatorname{Hom}}^{\mathrm o}(L\varepsilon^*C,\mathcal O_{N(A,S)})\). The functor \(R\underline{\operatorname{Hom}}^{\mathrm o}(L\varepsilon^*(-),\mathcal O)\) carries the triangle on \(\ell(N(A,u))\) to a distinguished triangle \(C^!\to\ell^!_{N(A,F)}\to\ell^!_{N(A,G)}\to C^![1]\) whose second map is \(\ell^!(N(A,u))\). By Proposition 3.2 (1) the latter is \(\operatorname{ner}^{\mathrm o}\Psi({}_A\ell^\vee(u))\), and \(\operatorname{ner}^{\mathrm o}\Psi\) is exact, so \(C^!\cong\operatorname{ner}^{\mathrm o}\Psi(K)\) [Stacks, Tag 014A]. As in Proposition 3.3, \(C^!\otimes^{\mathbf L}M\cong\operatorname{ner}^{\mathrm o}\Psi(K\otimes^{\mathbf L}_{\mathcal O}M)\), \(K\otimes^{\mathbf L}_{\mathcal O}M\) being bounded, and Theorem 4.3 of the lesson on Ext groups concludes. For the last statement, the proof of Proposition 3.5 gives a distinguished triangle in \(D^b(BA)\) with the vertices \(\Psi({}_A\ell^\vee_E)\), \(\Psi({}_A\ell^\vee_F)\), \(\Psi({}_A\ell^\vee_G)\) and the second map \(\Psi({}_A\ell^\vee(u))\). So \(\Psi(K)\cong\Psi({}_A\ell^\vee_E)\) [Stacks, Tag 014A]. Since \(\Psi\) commutes with cohomology (proof of Proposition 3.2 (3)) and \(\Psi(\bar\varepsilon^*H)\cong C_\bullet(H(1))\) vanishes only for \(H=0\), \(K\) lies in \(D^{[0,1]}\), where \(\Psi\) is fully faithful (Proposition 3.4 of the lesson on Ext groups); so \(K\cong{}_A\ell^\vee_E\). \(\square\)

The object \(K\) is determined only up to isomorphism, so the isomorphism of Theorem 4.2 depends on the choice of the triangle; it is natural in \(M\) for a fixed choice.

For a homomorphism \(u\colon F\to G\), the morphism \(N(A,u)\) induces at every level the localization of the topos \(T/N(A,G)_\lambda\) at \(N(A,F)_\lambda\). For \(K\in D^-(\operatorname{Top}(N(A,G)_{\mathrm{fppf}}))\) and \(N\in D^+(\mathcal O)\), the *relative* \(\operatorname{RHom}\), written \(\operatorname{RHom}(G/F;K,N)\), is the cone, shifted by \(-1\), of the restriction map
\[
\begin{aligned}&\operatorname{Hom}^\bullet(K',J_{N(A,G)})\\&\quad\to\operatorname{Hom}^\bullet(u^*K',J_{N(A,F)}),\end{aligned}
\]
where \(J\) is a bounded below complex of injective \(\mathcal O\)-modules of \(T\) resolving \(N\) and \(K'\) is a bounded above complex representing \(K\) whose terms are sums of modules \(i_!Q\) with \(Q\) flat, as in the proof of Theorem 2.3 of the lesson on the duality formula. The terms of \(u^*K'\) are of the same form, restriction to a localization commuting with \(i_!\) and preserving flatness, so both complexes compute \(\operatorname{RHom}\) (proof of Theorem 2.3 there). Let \(\operatorname{cone}(u)\) be the complex \(F\xrightarrow{u}G\) with \(G\) in degree \(0\).

**Theorem 4.3 (relative Ext groups).** For a homomorphism \(u\colon F\to G\) of \(A\)-module schemes and \(M\in D^b(\mathcal O)\) there is an isomorphism, natural in \(M\),
\[
\begin{aligned}&\operatorname{RHom}\bigl(G/F;L\varepsilon^*L_{N(A,G)/N(A,S)},M\bigr)\\&\quad\cong\operatorname{RHom}_A\bigl(\operatorname{cone}(u),({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M)|_A\bigr).\end{aligned}
\]

**Proof.** The isomorphism of Theorem 2.3 of the lesson on the duality formula is induced by the identification of complexes \(\Gamma(\operatorname{Top}^{\mathrm o}(X),\underline{\operatorname{Hom}}^{\mathrm o}(L,J_X))=\) \(\operatorname{Hom}^\bullet_{\operatorname{Top}(X)}(L,J_X)\) of Lemma 1.2 there, for a suitable bounded above complex \(L\). It commutes with restriction along \(N(A,u)\), both sides being restricted levelwise. So the relative \(\operatorname{RHom}\) is the relative cohomology of Proposition 4.4 of the lesson on Ext groups, the cone of the restriction map shifted by \(-1\), of \(R\underline{\operatorname{Hom}}^{\mathrm o}(L\varepsilon^*L_{N(A,G)/N(A,S)},M_{N(A,G)})\) on \(\mathcal Y_G=\operatorname{Top}^{\mathrm o}(\mathcal X_G)\) relative to \(\mathcal Y_F\). By the proof of Theorem 4.1 for the identity of \(G\), the object of which it is taken is isomorphic to \(q^*\operatorname{ner}^{\mathrm o}\Psi(M')\), with \(M'={}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M\), and relative cohomology is a functor of this object. Proposition 4.4 there, for \(Y=\Psi(M')|_Q\), gives the result. \(\square\)

**Corollary 4.4 (Zariski Ext groups).** Let \(u\colon F\to G\) be a homomorphism of \(A\)-module schemes and \(I\) a quasi-coherent \(\mathcal O_S\)-module, and let \(p^*I\) be the module of \(\operatorname{Top}(N(A,F)_{\mathrm{Zar}})\) with the levels \(p_\lambda^*I\). Then for every \(j\),
\[
\begin{aligned}&\operatorname{Ext}^j\bigl(Lu^*L_{N(A,G)/N(A,S)},p^*I\bigr)\\&\quad\cong\operatorname{Ext}^j_A\bigl(F,({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I)|_A\bigr),\end{aligned}
\]
the left side being formed on the Zariski topos, and similarly for Theorems 4.2 and 4.3.

**Proof.** The module \(p^*I\) is termwise quasi-coherent, in degree \(0\). By Lemma 4.2 (2) of the lesson on big fppf sites, \(\operatorname{RHom}(K,p^*I)\cong\operatorname{RHom}(L\varepsilon^*K,\varepsilon^*p^*I)\) for \(K\in D^-\), \(\varepsilon^*p^*I\) being the truncation \(t_{[0}L\varepsilon^*p^*I\). At every level \(\varepsilon^*p_\lambda^*I\) is the restriction of \(\varepsilon^*I\), the pullbacks of quasi-coherent modules composing; so \(\varepsilon^*p^*I=(\varepsilon^*I)_{N(A,F)}\), and Theorem 4.1 applies with \(M=\varepsilon^*I\). \(\square\)

## 5. Exercises

**5.1.** Let \(S=\operatorname{Spec}R\), \(A=\mathcal O\), the ring scheme \(\mathbf A^1_S\), and \(G=\mathbf G_a^r\), an \(A\)-module scheme by scalar multiplication. Show that \(\ell^!_G\cong\mathcal O^r[0]\), that \(a\in A\) acts on \(H^0(\ell^!_G)=\mathcal O^r\) by multiplication by \(a\), and deduce that \({}_A\ell^\vee_G\cong\bar\varepsilon^*H\), where \(H=\mathcal O^r\) is a module over \(A\otimes\mathcal O=\mathcal O\otimes\mathcal O\) through the product \(\mathcal O\otimes\mathcal O\to\mathcal O\).

**5.2.** Let \(p\) be a prime, \(S\) an \(\mathbb F_p\)-scheme, \(A=\mathbb Z\) the constant ring scheme, and \(0\to\alpha_p\to\mathbf G_a\xrightarrow{F}\mathbf G_a\to0\) the exact sequence of Example 3.3 of the lesson on the co-Lie complex, with \(F(x)=x^p\). Using Proposition 3.4, compute \({}_{\mathbb Z}\ell^\vee(F)\) and a fibre \(K\) as in Theorem 4.2, and check that \(K\cong{}_{\mathbb Z}\ell^\vee_{\alpha_p}\), as Theorem 4.2 asserts.

**5.3.** Let \(p\) be a prime, \(A\) the constant ring scheme \(\mathbb F_p\), and suppose \(p\mathcal O=0\), that is, \(S\) is an \(\mathbb F_p\)-scheme. Show that \(H^{-1}(R_{\mathcal O})\cong\mathcal O\) and \(H^0(R_{\mathcal O})=\mathcal O\), so that \(R_{\mathcal O}\to A\otimes\mathcal O\) is not a quasi-isomorphism.

## 6. Complete solutions

**5.1.** \(G\) is smooth with \(\omega_G\) free on \(dx_1,\ldots,dx_r\), so \(\ell_G=\mathcal O_S^r[0]\) and \(\ell^!_G=R\underline{\operatorname{Hom}}(\mathcal O^r,\mathcal O)=\mathcal O^r[0]\), with the dual basis. For a section \(a\), \(a_G(x)=ax\) pulls \(dx_i\) back to \(a\,dx_i\) at the unit, so \(\ell(a_G)\) and its dual \(\ell^!(a_G)\) are multiplication by \(a\). By Proposition 3.2 (3), \({}_A\ell^\vee_G\) has cohomology only in degree \(0\), equal to \(\mathcal O^r\) with \(a\otimes b\) acting by \(ab\). An object of \(D(R_{\mathcal O})\) with cohomology only in degree \(0\) is \(\bar\varepsilon^*\) of its \(H^0\) (Exercise 4.2 of the lesson on differential graded modules, \(R_{\mathcal O}\) being in nonpositive degrees with \(H^0(R_{\mathcal O})=A\otimes\mathcal O\)).

**5.2.** By Proposition 3.4, \(R_{\mathcal O}\simeq\mathcal O\) and \({}_{\mathbb Z}\ell^\vee_G=\ell^!_G\) for every \(G\), and \({}_{\mathbb Z}\ell^\vee(F)=\ell^!(F)\). The group \(\mathbf G_a\) is smooth with \(\ell_{\mathbf G_a}=\mathcal O_S\,dx\) in degree \(0\), and \(F^*(dx)=d(x^p)=px^{p-1}dx=0\), so \(\ell(F)=0\) and \(\ell^!(F)\colon\mathcal O\to\mathcal O\) is \(0\) in degree \(0\). A fibre of the zero map \(\mathcal O\to\mathcal O\) is \(K=\mathcal O\oplus\mathcal O[-1]\), with \(H^0(K)=\mathcal O\) and \(H^1(K)=\mathcal O\). On the other hand, by Example 3.3 there \(\ell_{\alpha_p}\cong[\mathcal O_S\xrightarrow{0}\mathcal O_S]\) in degrees \(-1\) and \(0\), so \(\ell^!_{\alpha_p}\) is \([\mathcal O\xrightarrow{0}\mathcal O]\) in degrees \(0\) and \(1\), which is \(\mathcal O\oplus\mathcal O[-1]\). So \(K\cong\ell^!_{\alpha_p}={}_{\mathbb Z}\ell^\vee_{\alpha_p}\), as Theorem 4.2 asserts for the faithfully flat \(F\) with kernel \(\alpha_p\).

**5.3.** \(R_{\mathcal O}=P_A\otimes\mathcal O\), where \(P_A=(J\to Q)\) is a flat presentation of \(A=\mathbb F_p\) (Proposition 2.4 of the lesson on the stable ring and Lemma 1.1 there). Hence \(H^i(R_{\mathcal O})=\operatorname{Tor}^{\mathbb Z}_{-i}(\mathbb F_p,\mathcal O)\), computed by the flat resolution \(\mathbb Z\xrightarrow{p}\mathbb Z\) of \(\mathbb F_p\): \(H^0=\mathcal O/p\mathcal O=\mathcal O\) and \(H^{-1}=\{x\in\mathcal O:px=0\}=\mathcal O\). Since \(H^{-1}\neq0\) while \(A\otimes\mathcal O=\mathcal O\) is concentrated in degree \(0\), the map \(R_{\mathcal O}\to A\otimes\mathcal O\) is not a quasi-isomorphism.

## Sources

The Lie complex of a module scheme with coefficients in the derived tensor product of the ring scheme and the structure ring, its tensor products, its behaviour under change of rings and in exact sequences, and the identification of the Ext groups of the cotangent complex of the spectral nerve with Ext groups of modules are due to Illusie, in his work on the cotangent complex; he notes there that the derived tensor product is needed in place of the ordinary one in his earlier work on deformations of Barsotti–Tate groups. The proof given here that the second nerve is fully faithful on the spectral pseudo-categories uses only the localizations of the décalage.