Truncated nerves and the level filtration

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

The Ext groups that classify the deformations of a flat group scheme, of a homomorphism or of an equivariant morphism are computed by total complexes with one column for each level of a nerve, and filtering by the column index gives the spectral sequences of the lessons on mixed Hom complexes and on the adjoint co-Lie complex. This lesson identifies that filtration. Dividing out the columns of index greater than \(i\) is restricting to the strict \(i\)-truncation of the nerve: the levels \(0,\ldots,i\) with the face maps between them and without degeneracies. For the nerve of a group the complex is relative to the level \(0\), and its quotients compute the Ext groups of the cotangent complexes of the strict truncations of the nerve. The levels \(1,\ldots,i\) alone do not compute these quotients (Exercise 6.1).

The proof rests on a resolution of a cartesian module on a strict truncated simplicial topos by the modules generated by its levels; at each level its exactness is that of the alternating face complex of a simplex. The filtration then acquires a meaning in deformation theory. The deformations of the strict \(1\)-, \(2\)- and \(3\)-truncations of the nerve of a flat group scheme \(G\) are the deformations of \(G\) as a scheme, the pairs of such a deformation and a lift of the multiplication, and the deformations of \(G\) as a group scheme. So the first graded piece of the obstruction to deforming \(G\) is the obstruction to deforming \(G\) as a scheme, the obstruction lies in the second step of the filtration exactly when \(G\) deforms as a scheme, and in the third exactly when the multiplication lifts to some deformation. The same holds for homomorphisms and for actions of group schemes.

Prerequisites. Strict simplicial objects and Deligne's contractibility theorem (Section 1, Lemma 2.1); Morphisms of fibred topoi and modules on total topoi (Proposition 4.2); Cohomology of total topoi (Definition 1.2, Lemma 1.3); Cohomology of simplicial topoi and cohomological descent (Lemma 1.1, Proposition 1.2, Theorem 2.1); Mixed Hom complexes and Ext groups of equivariant modules (Setting, Lemmas 1.3 and 1.4, Definition 1.5, Section 3); Cohomology of equivariant modules (the conditions (I) and (E), Proposition 5.1); Equivariant sheaves on simplicial topoi (Lemma 1.2, Definition 2.1); The adjoint co-Lie complex of a group scheme (Conventions, Proposition 2.2, Theorem 3.3, Corollaries 3.4 and 3.5); Quasi-coherent and perfect complexes on big fppf sites (Lemma 4.2); Deformations of diagrams of schemes (Definitions 2.1, 3.1 and 4.1, Lemma 3.3, Construction 4.2, Theorem 4.3, Proposition 4.4); Deformations of flat group schemes and of homomorphisms (Conventions, Definition 0.1, Lemma 1.1, Propositions 1.2, 1.3 and 2.2, Theorems 2.1 and 3.1); Equivariant cotangent complexes (Proposition 3.2); Equivariant extensions and deformations (Conventions, Proposition 3.2, Theorem 3.3, Proposition 4.1). From the AI-integrated Stacks project: right adjoints of exact functors preserve injectives [Stacks, Tag 015Z]; Hom complexes into bounded below complexes of injectives [Tag 05TG]; double complexes with exact rows [Tag 0133]; first-quadrant filtrations [Tag 0132]; termwise split sequences and triangles [Tags 014I, 014Q]; cones of maps of triangles [Tag 014A].

Conventions

\(\Delta'\) is the category of nonempty finite ordinals and injective increasing maps, and \(\Delta'_{\leq n}\), for \(0\leq n\leq\infty\), is its full subcategory of the \([k]\) with \(k\leq n\), with \(\Delta'_{\leq\infty}=\Delta'\) (Section 1 of the lesson on strict simplicial objects). The coface \(\delta^j\colon[k-1]\to[k]\) omits \(j\), and \(v_k\colon[0]\to[m]\) is the vertex \(k\).

A strict \(n\)-truncated simplicial topos is a topos \(X\) fibred over \((\Delta'_{\leq n})^{\mathrm{op}}\); for \(n=\infty\) it is a strict simplicial topos. An arrow \(\alpha\colon[k]\to[m]\) of \(\Delta'_{\leq n}\) gives a morphism of topoi \(X(\alpha)\colon X_m\to X_k\). For a topos \(X\) fibred over \(\Delta^{\mathrm{op}}\) or over \((\Delta'_{\leq n})^{\mathrm{op}}\), and \(0\leq i\leq n\), \(\tau_iX\) denotes its restriction to \((\Delta'_{\leq i})^{\mathrm{op}}\), the strict \(i\)-truncation; for \(X\) fibred over \(\Delta^{\mathrm{op}}\), \(\tau_\infty X\) is its restriction along \(\Delta'\subset\Delta\), which forgets the degeneracies. The same notation is used for diagrams of schemes of these types. Restriction along a functor of index categories is the inverse image of a morphism of ringed topoi; it is exact, preserves the polynomial constructions and carries the cotangent complex of a morphism of diagrams to that of the restricted morphism (Conventions of the lesson on the adjoint co-Lie complex).

For a ring \(\mathcal O\) of \(\operatorname{Top}(X)\), an \(\mathcal O\)-module \(E\) is a family of \(\mathcal O_m\)-modules \(E_m\) with structure maps \(E_\alpha\colon X(\alpha)^*E_k\to E_m\) for \(\alpha\colon[k]\to[m]\), satisfying \(E_{\alpha\beta}=E_\alpha\circ X(\alpha)^*(E_\beta)\) (Proposition 4.2 (a) of the lesson on morphisms of fibred topoi, with the canonical isomorphisms of inverse images understood). It is cartesian if every \(E_\alpha\) is an isomorphism; the ring \(\mathcal O\) is a cartesian \(\mathcal O\)-module. The restriction to the level \(k\) has the left adjoint \(k_!\), with \[ (k_!F)_m=\bigoplus_{\alpha\colon[k]\to[m]}X(\alpha)^*F, \] the sum over the arrows of the index category (Proposition 4.2 (b) there); the structure map along \(\beta\colon[m]\to[m']\) carries the summand of \(\alpha\) to that of \(\beta\alpha\).

1. Cartesian resolutions on strict truncated simplicial topoi

1.1. Setting. In this section \(0\leq n\leq\infty\), \(X\) is a strict \(n\)-truncated simplicial topos, and \(\mathcal O\) is a ring of \(\operatorname{Top}(X)\) such that every \(X(\alpha)\) is a flat morphism of ringed topoi \((X_m,\mathcal O_m)\to(X_k,\mathcal O_k)\). Then the module inverse images \(X(\alpha)^*\) are exact, so is \(k_!\), and the restriction to a level carries injective modules to injective modules, being right adjoint to the exact functor \(k_!\) [Tag 015Z].

Lemma 1.2 (the cartesian resolution). Let \(E\) be a cartesian \(\mathcal O\)-module. For \(1\leq k\leq n\) and \(0\leq j\leq k\), let \(\partial_j\colon k_!E_k\to(k-1)_!E_{k-1}\) be the map adjoint to \[ \begin{aligned}&E_k\xrightarrow{\ E_{\delta^j}^{-1}\ }X(\delta^j)^*E_{k-1}\\&\quad\subseteq\bigl((k-1)_!E_{k-1}\bigr)_k,\end{aligned} \] the inclusion of the summand of \(\delta^j\), and let \(\varepsilon\colon0_!E_0\to E\) be the counit of the adjunction. Then the complex \[ \cdots\to2_!E_2\to1_!E_1\to0_!E_0\xrightarrow{\ \varepsilon\ }E\to0, \] with the differentials \(\sum_j(-1)^j\partial_j\), is exact; for \(n<\infty\) it stops at \(n_!E_n\).

Proof. Fix a level \(m\leq n\). The isomorphisms \(E_\alpha\) identify \((k_!E_k)_m=\bigoplus_\alpha X(\alpha)^*E_k\) with \(\bigoplus_{\alpha\in\Delta'([k],[m])}E_m\). By adjunction, the restriction of \(\partial_j\) to the summand of \(\alpha\) at the level \(m\) is the composite of \(X(\alpha)^*(E_{\delta^j}^{-1})\), landing in the summand of \(\alpha\delta^j\), with the structure map of \((k-1)_!E_{k-1}\) along \(\alpha\). Under the identifications it is \[ E_{\alpha\delta^j}\circ X(\alpha)^*(E_{\delta^j}^{-1})\circ E_\alpha^{-1}=\mathrm{id}_{E_m}, \] because \(E_{\alpha\delta^j}=E_\alpha\circ X(\alpha)^*(E_{\delta^j})\). In the same way \(\varepsilon\) is the identity of \(E_m\) on every summand of \(\alpha\colon[0]\to[m]\). So at the level \(m\) the complex is the augmented complex of Lemma 2.1 of the lesson on strict simplicial objects for \(F=[m]\), tensored with \(E_m\): its terms are indexed by the injective maps \([k]\to[m]\), all of which have \(k\leq m\leq n\), and its differential sends \(\alpha\) to \(\sum_j(-1)^j\alpha\delta^j\). That complex is contractible by a homotopy given by integer matrices, and so is its tensor product with \(E_m\). A complex of modules of \(\operatorname{Top}(X)\) is exact when it is exact at every level. \(\square\)

For a complex \(E\) of cartesian modules and a complex \(J\) of \(\mathcal O\)-modules, let \(\mathcal H(E,J)\) be the strict cosimplicial complex, truncated at \(n\), with the terms \(\mathcal H^k(E,J)=\operatorname{Hom}^\bullet(E_k,J_k)\) and the maps \[ \begin{gathered}\alpha_*\colon\operatorname{Hom}^\bullet(E_k,J_k)\to\operatorname{Hom}^\bullet(E_m,J_m),\\ \varphi\longmapsto J_\alpha\circ X(\alpha)^*(\varphi)\circ E_\alpha^{-1},\end{gathered} \] for \(\alpha\colon[k]\to[m]\). The composition rule for the structure maps gives \((\beta\alpha)_*=\beta_*\alpha_*\). Write \(\operatorname{Tot}\mathcal H(E,J)\) for the total complex of the double complex whose columns are the \(\mathcal H^k(E,J)\), \(0\leq k\leq n\), with the differential \(\sum_j(-1)^j\delta^j_*\) in the direction \(k\); for \(n=\infty\) the products in each degree are taken over the columns.

Proposition 1.3. Let \(E\) be a bounded above complex of cartesian modules and \(J\) a bounded below complex of injective modules of \(\operatorname{Top}(X)\). The map \(\operatorname{Hom}^\bullet(E,J)\to\mathcal H^0(E,J)\), \(\varphi\mapsto\varphi_0\), is a map into the kernel of the first differential, and the induced map \[ \operatorname{Hom}^\bullet(E,J)\longrightarrow\operatorname{Tot}\mathcal H(E,J) \] is a quasi-isomorphism. So \(\operatorname{Tot}\mathcal H(E,J)\) represents \(R\operatorname{Hom}(E,J)\).

Proof. Fix degrees \(p\), \(q\). By adjunction \(\operatorname{Hom}(k_!E^p_k,J^q)=\operatorname{Hom}(E^p_k,J^q_k)\), and \(\operatorname{Hom}(\partial_j,J^q)\) is \(\delta^j_*\): the adjoint of \(\varphi\colon E^p_{k-1}\to J^q_{k-1}\) is \(J^q_\beta\circ X(\beta)^*(\varphi)\) on the summand of \(\beta\), and composing with \(\partial_j\) gives \(J^q_{\delta^j}\circ X(\delta^j)^*(\varphi)\circ(E^p_{\delta^j})^{-1}\) at the level \(k\). Since \(J^q\) is injective, \(\operatorname{Hom}(-,J^q)\) carries the exact complex of Lemma 1.2 for \(E^p\) to an exact complex \[ \begin{aligned}&0\to\operatorname{Hom}(E^p,J^q)\to\operatorname{Hom}(E^p_0,J^q_0)\\&\quad\to\operatorname{Hom}(E^p_1,J^q_1)\to\cdots,\end{aligned} \] whose first map is \(\varphi\mapsto\varphi_0\). Now apply [Tag 0133] to \(\operatorname{Hom}^\bullet(E,J)\) and the double complex with the terms \(\prod_{q-p=a}\operatorname{Hom}(E^p_k,J^q_k)\) in bidegree \((a,k)\). Since \(E^p=0\) for \(p>p_0\) and \(J^q=0\) for \(q<q_0\), the term vanishes unless \(a\geq q_0-p_0\), so for each total degree only finitely many terms are nonzero and each is a finite product; the columns are exact in degrees \(k\geq1\) with the kernel \(\operatorname{Hom}^a(E,J)\) in degree \(0\), these properties being stable under finite products. \(\square\)

Corollary 1.4 (cohomology). Let \(C\) be a bounded below complex of abelian sheaves of \(\operatorname{Top}(X)\) whose terms have totally acyclic levels (Definition 1.2 of the lesson on cohomology of total topoi). Then \(R\Gamma(\operatorname{Top}(X),C)\) is represented by the total complex of the strict cosimplicial complex \([k]\mapsto\Gamma(X_k,C_k)\), \(k\leq n\), with the alternating sums of the cofaces.

Proof. Take \(\mathcal O=\mathbf Z\), for which every \(X(\alpha)\) is flat, and \(E=\mathbf Z\), a cartesian module with \(\operatorname{Hom}(\mathbf Z,J_k)=\Gamma(X_k,J_k)\). For a bounded below complex \(J\) of injectives, Proposition 1.3 is the statement. For \(C\), choose a quasi-isomorphism \(C\to J\) into such a complex. At each level the cone of \(C_k\to J_k\) is a bounded below exact complex of totally acyclic sheaves, the levels of injectives being injective (1.1), so \(\Gamma(X_k,-)\) of it is exact (Lemma 1.3 (5) there). The map of double complexes is then a quasi-isomorphism on every column, and the column filtration is finite in each total degree; so the map of total complexes is a quasi-isomorphism [Tag 0132]. \(\square\)

In particular, for the punctual topos at every level, a functor \(V\colon\Delta'_{\leq n}\to\operatorname{Ab}\) is an abelian sheaf on the total topos, every abelian group being totally acyclic on the punctual topos, and the derived limit of a bounded below complex of such functors is the total complex of the truncated alternating complexes \(V^q_0\to V^q_1\to\cdots\to V^q_n\).

Corollary 1.5 (truncations). In 1.1, let \(E\) be a bounded above complex of cartesian modules, \(J\) a bounded below complex of injective modules, and \(K=\operatorname{Tot}\mathcal H(E,J)\), with the subcomplexes \(F^iK\) of the columns \(k\geq i\).

1. For \(i\leq n\), \(K/F^{i+1}K\), the total complex of the columns \(k\leq i\), represents \(R\operatorname{Hom}_{\operatorname{Top}(\tau_iX)}(E|_{\tau_iX},J|_{\tau_iX})\), and under Proposition 1.3 the projections \(K\to K/F^{i+1}K\to K/F^{i}K\) are the restriction maps.
2. \(F^iK/F^{i+1}K=\mathcal H^i(E,J)[-i]\) represents \(R\operatorname{Hom}_{X_i}(E_i,J_i)[-i]\), and \(F^{i}K\) is a cocone of the restriction map \(R\operatorname{Hom}_{\operatorname{Top}(X)}(E,J)\to\) \(R\operatorname{Hom}_{\operatorname{Top}(\tau_{i-1}X)}(E|,J|)\).
3. Let \(Y\) be a topos fibred over \(\Delta^{\mathrm{op}}\) with a ring for which every \(Y(\alpha)\) is flat, \(E'\) a bounded above complex of cartesian modules and \(J'\) a bounded below complex of injective modules of \(\operatorname{Top}(Y)\). Then \(R\operatorname{Hom}_{\operatorname{Top}(Y)}(E',J')\) is represented by the total complex of the cosimplicial complex \([k]\mapsto\operatorname{Hom}^\bullet(E'_k,J'_k)\) with the alternating sums of the cofaces, and the restriction map to \(\operatorname{Top}(\tau_\infty Y)\) is an isomorphism.

Proof. (1) The restrictions \(E|_{\tau_iX}\) and \(J|_{\tau_iX}\) are a complex of cartesian modules and a complex with injective levels. Choose a quasi-isomorphism \(J|_{\tau_iX}\to J'\) into a bounded below complex of injective modules of \(\operatorname{Top}(\tau_iX)\). It induces a map \(\mathcal H(E|,J|)\to\mathcal H(E|,J')\), a quasi-isomorphism on every column because \(\operatorname{Hom}^\bullet(E_k,-)\) preserves quasi-isomorphisms between bounded below complexes of injective \(\mathcal O_k\)-modules [Tag 05TG], hence a quasi-isomorphism of total complexes, the column filtration being finite in each degree. Proposition 1.3 for \(\tau_iX\) and \(J'\) identifies the target with \(R\operatorname{Hom}(E|,J|)\), and \(\mathcal H(E|,J|)\) is the truncation of \(\mathcal H(E,J)\) to the columns \(k\leq i\). The restriction of a map \(\varphi\colon E\to J\) has the same component \(\varphi_0\), so the square formed by the restriction map, the maps of Proposition 1.3 and the projection commutes; in the same way for \(\tau_{i-1}X\subset\tau_iX\).

(2) The first statement is [Tag 05TG] for the injective levels \(J^q_i\). The sequence \(0\to F^iK\to K\to K/F^iK\to0\) is termwise split, so it gives a distinguished triangle [Tags 014I, 014Q] in which, by (1), the second map is the restriction map.

(3) For a cartesian module \(E'\) of \(\operatorname{Top}(Y)\), the complex with the terms \(k_!E'_k\), here formed with all maps \([k]\to[m]\) of \(\Delta\), and the alternating sums of the faces \(\partial_j\) defined as in Lemma 1.2, is a resolution of \(E'\): at the level \(m\) it is the augmented chain complex of the simplicial set \(\Delta[m]\) tensored with \(E'_m\), contractible as in the proof of Lemma 1.1 of the lesson on cohomology of simplicial topoi. The proof of Proposition 1.3 then gives the first statement, with the map \(\varphi\mapsto\varphi_0\). The restrictions of \(E'\) and \(J'\) to \(\tau_\infty Y\) are a complex of cartesian modules and a complex with injective levels, and as in (1) the total complex of \(\mathcal H(E'|,J'|)\) represents their derived Hom. It is the same double complex, the codegeneracies playing no role in the alternating sums, and the restriction map commutes with \(\varphi\mapsto\varphi_0\). \(\square\)

2. The level filtration of a mixed Hom complex

In this section \(X\) is a good pseudo-category with a ring \(\mathcal O\) of \(B'X\) satisfying the conditions (I) and (E), as in Section 3 of the lesson on mixed Hom complexes, with its notation: the transports \(a\), \(b\) of modules of \(BX\), the nerve \(\operatorname{ner}\) and the second nerve \(\operatorname{ner}^{\mathrm o}\), the cartesian rings \(\mathcal O_n\), and for \(E\in D^-(BX)\) and \(F\in D^+(BX)\) the double complex (3.2) there, which we write \(\mathcal G\), built from a bounded above complex representing \(E\) and a bounded below complex \(I\) of injective modules representing \(F\). Let \(K=\operatorname{Tot}(\mathcal G)\), with the subcomplexes \(F^iK\) of the columns \(n\geq i\); by Theorem 3.3 there, \(K\) represents \(R\operatorname{Hom}_{BX}(E,F)\). The module inverse images along the maps of \(X\) are the inverse images of sheaves, the rings being cartesian, so they are exact and 1.1 applies to the total topos of the nerve, with the ring \(\operatorname{ner}\mathcal O\), and to all its truncations.

Theorem 2.1. In this situation:

1. For \(0\leq i\leq\infty\), \(K/F^{i+1}K\) represents the derived limit over \(\Delta'_{\leq i}\) of the restriction of \(R\operatorname{Hom}^!(\operatorname{ner}^{\mathrm o}E,\operatorname{ner}F)\) (Definition 1.5 of the lesson on mixed Hom complexes), and the projections \(K\to K/F^{i+1}K\to K/F^iK\) are the restriction maps of derived limits. In particular \(K/F^1K=\mathcal G^{\bullet,0}\) represents \(R\operatorname{Hom}_{X_0}(UE,UF)\), and \(K\to K/F^1K\) is the map induced by the forgetful functor \(U\).
2. \(F^iK/F^{i+1}K=\mathcal G^{\bullet,i}[-i]\) represents \(R\operatorname{Hom}_{\mathcal O_i}(X(v_0)^*E,X(v_i)^*F)[-i]\).
3. Suppose that the structures of the terms of the complex representing \(E\) are invertible, so that their nerves are cartesian modules of \(\operatorname{Top}(X)\). Then for \(0\leq i\leq\infty\), \(K/F^{i+1}K\) represents \(R\operatorname{Hom}_{\operatorname{Top}(\tau_iX)}(\operatorname{ner}E,\operatorname{ner}F)\), the projections are the restriction maps, and \(K\) itself represents \(R\operatorname{Hom}_{\operatorname{Top}(X)}(\operatorname{ner}E,\operatorname{ner}F)\).

Proof. (1) By (1.1) of the lesson on mixed Hom complexes, the cosimplicial complex \(\operatorname{Hom}^!(\operatorname{ner}^{\mathrm o}E,\operatorname{ner}I)\) represents \(R\operatorname{Hom}^!\), the levels of \(\operatorname{ner}I\) being injective by (I). Its restriction to \(\Delta'_{\leq i}\) is a bounded below complex of functors, whose derived limit is the total complex of its truncated alternating complexes (the remark after Corollary 1.4); that total complex is \(K/F^{i+1}K\). The restriction maps of derived limits along \(\Delta'_{\leq i-1}\subset\Delta'_{\leq i}\) are the projections, by Corollary 1.5 (1) for the punctual levels. For the last statement, \(\mathcal G^{\bullet,0}=\operatorname{Hom}^\bullet(UE,UI)\) with \(UI\) a bounded below complex of injectives by (I), and the map (3.1) of the lesson on mixed Hom complexes sends \(u\) to a family whose component in degree \(0\) is \(u\) (Lemma 3.1 there).

(2) By [Tag 05TG], the column \(\mathcal G^{\bullet,i}=\operatorname{Hom}^\bullet(X(v_0)^*E,X(v_i)^*I)\) represents that derived Hom, \(X(v_i)^*I\) being a bounded below complex of injective \(\mathcal O_i\)-modules.

(3) For a module \(L\) of \(BX\) with invertible structure \(a\), the nerve has the levels \(X(v_n)^*L\) and, along \(\alpha\colon[m]\to[n]\), the structure map \(a_{\alpha(m)n}\colon X(v_{\alpha(m)})^*L\to X(v_n)^*L\), an isomorphism. For a module \(M\) of \(BX\) with structure \(b\), the maps \[ \begin{gathered}\operatorname{Hom}\bigl(X(v_n)^*L,X(v_n)^*M\bigr)\\ \longrightarrow\operatorname{Hom}\bigl(X(v_0)^*L,X(v_n)^*M\bigr),\quad\psi\\ \longmapsto\psi\circ a_{0n},\end{gathered} \] are bijective, and they form an isomorphism from \(\mathcal H(\operatorname{ner}L,\operatorname{ner}M)\) to \(\operatorname{Hom}^!(\operatorname{ner}^{\mathrm o}L,\operatorname{ner}M)\). Indeed, for \(\alpha\colon[m]\to[n]\), the map of \(\mathcal H\) is \(\psi\mapsto b_{\alpha(m)n}\circ X(\alpha)^*(\psi)\circ a_{\alpha(m)n}^{-1}\), that of \(\operatorname{Hom}^!\) is \(\varphi\mapsto b_{\alpha(m)n}\circ X(\alpha)^*(\varphi)\circ a_{0\alpha(0)}\), and the cocycle identities of the transports (Lemma 1.2 of the lesson on equivariant sheaves) give \(a_{\alpha(m)n}^{-1}\circ a_{0n}=\) \(a_{0\alpha(m)}=\) \(X(\alpha)^*(a_{0m})\circ a_{0\alpha(0)}\). Hence \(\mathcal G\) is isomorphic to the double complex \(\mathcal H(\operatorname{ner}E,\operatorname{ner}I)\) of Section 1, by isomorphisms compatible with the truncations. The modules \(\operatorname{ner}I^q\) have injective levels, and \(\operatorname{ner}E\) is a bounded above complex of cartesian modules. As in the proof of Corollary 1.5 (1), for every \(i\) a quasi-isomorphism from the restriction of \(\operatorname{ner}I\) to a bounded below complex of injective modules of \(\operatorname{Top}(\tau_iX)\) shows, with Proposition 1.3, that the truncation of \(\mathcal H(\operatorname{ner}E,\operatorname{ner}I)\) to the columns \(k\leq i\) represents \(R\operatorname{Hom}_{\operatorname{Top}(\tau_iX)}(\operatorname{ner}E,\operatorname{ner}F)\), compatibly with restrictions. For \(\operatorname{Top}(X)\) itself use Corollary 1.5 (3), with \(\operatorname{ner}I\) replaced by an injective resolution on \(\operatorname{Top}(X)\) as in (1). \(\square\)

Corollary 2.2 (relative cohomology of truncations). In this situation, let \(M\in D^+(BX)\) and let \(K\) be the total complex of the cosimplicial complex \([n]\mapsto\Gamma(X_n,X(v_n)^*I)\) for a bounded below complex \(I\) of injective modules representing \(M\), so that \(F^1K\) represents the relative cohomology \(R\Gamma(BX/X_0,M)\), a cocone of the forgetful map \(R\Gamma(BX,M)\to R\Gamma(X_0,UM)\). For \(1\leq i\leq\infty\), \(F^1K/F^{i+1}K\) represents a cocone of the restriction map \(R\Gamma(\operatorname{Top}(\tau_iX),\operatorname{ner}M)\to R\Gamma(X_0,UM)\) to the level \(0\), compatibly with the restriction maps.

Proof. Apply Theorem 2.1 (3) to \(E=\mathcal O\), whose structure is invertible, with \(\operatorname{Hom}(\mathcal O_n,X(v_n)^*I)=\Gamma(X_n,X(v_n)^*I)\) (the remark after Corollary 3.5 of the lesson on mixed Hom complexes). The sequences \(0\to\) \(F^1K/F^{i+1}K\to\) \(K/F^{i+1}K\to\) \(K/F^1K\to\) \(0\) are termwise split, and by Theorem 2.1 (1) and (3) their second maps are the restriction maps to the level \(0\) [Tags 014I, 014Q]. \(\square\)

So the quotients of the relative complex \(F^1K\) are cohomology groups of the strict truncations \(\tau_iX\) relative to their level \(0\), not of the restrictions of \(X\) to the levels \(1,\ldots,i\) (Exercise 6.1).

3. The nerve of a flat group scheme

3.0. Setting. \(S\) is a scheme, \(G\) a group scheme over \(S\), flat and locally of finite presentation, \(N=N(G)\) its nerve and \(f=f_G\colon N\to S\), with the notation of the lesson on the adjoint co-Lie complex: \(T\) the big fppf topos of \(S\), \(B=B[G,S]^{\mathrm o}\), \(\varepsilon\), \(\ell^{\mathrm{ad}}_G\), and \(i_0\colon S_{\mathrm{Zar}}\to\operatorname{Top}(N_{\mathrm{Zar}})\) the level \(0\), with the left adjoint \(i_{0!}\) of its inverse image, \((i_{0!}E)_n=\bigoplus_{k=0}^n\pi_n^*E\). Let \(X\) be the simplicial topos of the nerve of \([G,S]^{\mathrm o}\), with the levels \(T/G^n\), so that \(\operatorname{Top}(X)=\operatorname{Top}(N_{\mathrm{fppf}})\) and \(BX=B\) (proof of Corollary 3.5 of that lesson); it satisfies (I) and (E) (Proposition 5.1 of the lesson on cohomology of equivariant modules). Since \(G\) is a group, the structure of every module of \(B\) is invertible, with inverse given by the inversion of \(G\). For a quasi-coherent \(\mathcal O_S\)-module \(I\), \(f^*I\) is the cartesian module with the levels \(\pi_n^*I\), and \(\varepsilon^*I\) the module of \(B\) with the trivial structure, so that \(\operatorname{ner}(\varepsilon^*I)=\varepsilon^*f^*I\). For \(1\leq i\leq\infty\), \(L_{\tau_iN/S}\) is the cotangent complex of the morphism \(\tau_iN\to S\), the restriction of \(L_{N/S}\).

Let \(K\) be the complex \(\operatorname{Tot}(\mathcal G)\) of Section 2 for \(E=\ell^{\mathrm{ad}}_G\) and \(F=\varepsilon^*I\). By Corollary 3.5 (1) of the lesson on the adjoint co-Lie complex, \(F^1K\) represents the relative derived Hom \(R\operatorname{Hom}_{B/S}(\ell^{\mathrm{ad}}_G,\varepsilon^*I)\), and \(F^kK/F^{k+1}K\) represents \(R\operatorname{Hom}_{G^k}(\pi_k^*\ell_G,\pi_k^*I)[-k]\) for \(k\geq1\), Ext groups of the Zariski topos of \(G^k\) (Corollary 3.5 (3) there).

Theorem 3.1. There are isomorphisms in the derived category of abelian groups \[ \begin{aligned}&\theta_i \colon R\operatorname{Hom}\bigl(L_{\tau_iN/S},f^*I\bigr)[-1]\\&\quad \xrightarrow{\ \sim\ }F^1K/F^{i+1}K,\end{aligned} \] for \(1\leq i\leq\infty\), such that \(\theta_{i'}\circ\rho=\mathrm{pr}\circ\theta_i\) for \(i'\leq i\), where \(\rho\) is the restriction map from \(\tau_iN\) to \(\tau_{i'}N\) and \(\mathrm{pr}\) the projection. The restriction map from \(N\) to \(\tau_\infty N\) is an isomorphism, so \(\theta_\infty\circ\rho\) identifies \(R\operatorname{Hom}(L_{N/S},f^*I)[-1]\) with \(F^1K\), and the restriction maps from \(N\) to the \(\tau_iN\) with the projections. In particular \(\operatorname{Ext}^j(L_{\tau_iN/S},f^*I)\cong\) \(H^{j+1}(F^1K/F^{i+1}K)\).

Proof. Passage to the fppf topoi. Lemma 4.2 (2) of the lesson on big fppf sites, for the diagrams \(\tau_iN\) and \(N\), with the truncation index \(0\) as in the proof of Corollary 3.5 (3) of the lesson on the adjoint co-Lie complex, gives isomorphisms \(R\operatorname{Hom}(L_{\tau_iN/S},f^*I)\cong\) \(R\operatorname{Hom}(L\varepsilon^*L_{\tau_iN/S},\varepsilon^*f^*I)\), compatible with the restrictions, all constructions being levelwise.

A cocone model. Represent \(\ell^{\mathrm{ad}}_G\) by a bounded above complex of modules of \(B\), as in the definition of \(K\), and let \(Q\) be its nerve, a bounded above complex of cartesian modules of \(\operatorname{Top}(X)\) representing \(\operatorname{ner}\ell^{\mathrm{ad}}_G\). The adjunction map \(\epsilon_Q\colon i_{0!}i_0^*Q\to Q\) is a map of complexes; let \(Q'\) be its cocone, the complex \(\operatorname{cone}(\epsilon_Q)[-1]\). Applying \(L\varepsilon^*\) to the triangle (3.1) of the lesson on the adjoint co-Lie complex gives a distinguished triangle \[ \begin{aligned}&L\varepsilon^*L_{N/S}\to i_{0!}i_0^*\operatorname{ner}\ell^{\mathrm{ad}}_G\\&\quad\xrightarrow{\ \epsilon\ }\operatorname{ner}\ell^{\mathrm{ad}}_G\to L\varepsilon^*L_{N/S}[1]\end{aligned} \] whose second map is the adjunction map (proof of Corollary 3.5 (2) there). The triangle \(Q'\to i_{0!}i_0^*Q\to Q\to Q'[1]\) is distinguished, and the identifications of the second and third vertices carry \(\epsilon\) to \(\epsilon_Q\), adjunction maps being natural. So there is an isomorphism \(\varphi\colon L\varepsilon^*L_{N/S}\cong Q'\) in the derived category of \(\operatorname{Top}(X)\) compatible with the triangles [Tag 014A]. Fix it. Restriction to \(\tau_iX\) is exact and carries \(i_{0!}\), \(\epsilon_Q\) and cocones to the corresponding objects for \(\tau_iX\), the vertices \([0]\to[n]\) being injective maps; so the restriction of \(\varphi\) is an isomorphism \(\varphi_i\colon L\varepsilon^*L_{\tau_iN/S}\cong Q'|_{\tau_iX}\), and these are compatible with all restrictions.

Computation. Let \(I\) be the bounded below complex of injective modules of \(B\) representing \(\varepsilon^*I\) used for \(K\), and let \(\operatorname{ner}I|_{\tau_iX}\to J_i\) be a quasi-isomorphism into a bounded below complex of injective modules of \(\operatorname{Top}(\tau_iX)\). The complex \(\operatorname{Hom}^\bullet(Q'|,J_i)\) is the cone of \[ \begin{aligned}&\operatorname{Hom}^\bullet(Q|,J_i)\to\operatorname{Hom}^\bullet(i_{0!}Q_0,J_i)\\&\quad=\operatorname{Hom}^\bullet(Q_0,J_{i,0}),\end{aligned} \] composition with \(\epsilon_Q\), so \(\operatorname{Hom}^\bullet(Q'|,J_i)[-1]\) is its cocone; by adjunction this map is \(\varphi\mapsto\varphi_0\). The map \(\varphi\mapsto\varphi_0\) of Proposition 1.3, \(\operatorname{Hom}^\bullet(Q|,J_i)\to\operatorname{Tot}\mathcal H(Q|,J_i)\), is a quasi-isomorphism compatible with the projections to \(\operatorname{Hom}^\bullet(Q_0,J_{i,0})\). The maps \(\mathcal H(Q|,\operatorname{ner}I|)\to\mathcal H(Q|,J_i)\) and \(\operatorname{Hom}^\bullet(Q_0,UI)\to\operatorname{Hom}^\bullet(Q_0,J_{i,0})\) are quasi-isomorphisms, as in the proof of Corollary 1.5 (1). Hence \(R\operatorname{Hom}(Q'|_{\tau_iX},\operatorname{ner}\varepsilon^*I)[-1]\) is represented by the cocone of the projection \[ \operatorname{Tot}_{k\leq i}\mathcal H(Q,\operatorname{ner}I)\longrightarrow\mathcal H^0(Q,\operatorname{ner}I), \] and by the proof of Theorem 2.1 (3) the source is isomorphic to \(K/F^{i+1}K\) and the target to \(K/F^1K\), compatibly with the projection. This projection is surjective in each degree with the kernel \(F^1K/F^{i+1}K\), and the inclusion of the kernel into the cocone is a quasi-isomorphism. Every step is compatible with the restriction maps and the projections, since the double complex \(\mathcal H(Q,\operatorname{ner}I)\) does not depend on \(i\), and the restriction of a map of complexes has the same component in degree \(0\). This defines the \(\theta_i\).

The full nerve. By Corollary 1.5 (3), the same computation on \(\operatorname{Top}(X)\), with an injective resolution of \(\operatorname{ner}I\) on \(\operatorname{Top}(X)\), gives the same total complex, and the restriction to \(\tau_\infty X\) induces the identity of it; the restriction of \(\varphi\) to \(\tau_\infty X\) is \(\varphi_\infty\). \(\square\)

So \(F^1K/F^{i+1}K\) computes the Ext groups of the cotangent complex of the strict \(i\)-truncation of the nerve, and \(F^kK/F^{k+1}K\) is the relative part contributed by the level \(k\).

Corollary 3.2 (the level filtration). For every \(j\) and \(k\geq2\) put \[ \begin{aligned}&F^k\operatorname{Ext}^j(L_{N/S},f^*I)\\&\quad=\ker\bigl(\operatorname{Ext}^j(L_{N/S},f^*I)\\&\qquad\to\operatorname{Ext}^j(L_{\tau_{k-1}N/S},f^*I)\bigr),\end{aligned} \] and \(F^1=\operatorname{Ext}^j(L_{N/S},f^*I)\).

1. There is a convergent spectral sequence \[ \begin{aligned}E_1^{k,q}&=\operatorname{Ext}^q_{G^k}(\pi_k^*\ell_G,\pi_k^*I)\ \ (k\geq1),\\ E_1^{k,q}&=0\ \ (k\leq0),\\ E_1^{k,q}&\Longrightarrow\operatorname{Ext}^{k+q-1}(L_{N/S},f^*I),\end{aligned} \] whose filtration on the abutment is \(F^\bullet\), with \(E_\infty^{k,q}=\) \(F^k\operatorname{Ext}^{k+q-1}/F^{k+1}\operatorname{Ext}^{k+q-1}\); its rows are the complexes of Corollary 3.5 (3) of the lesson on the adjoint co-Lie complex.
2. \(F^{j+2}\operatorname{Ext}^j(L_{N/S},f^*I)=0\). In particular \(\operatorname{Ext}^2(L_{N/S},f^*I)=\) \(F^1\supseteq\) \(F^2\supseteq\) \(F^3\supseteq\) \(F^4=\) \(0\), and the restriction map \(\operatorname{Ext}^j(L_{N/S},f^*I)\to\) \(\operatorname{Ext}^j(L_{\tau_iN/S},f^*I)\) is bijective for \(j\leq i-2\) and injective for \(j=i-1\).
3. \(L_{\tau_1N/S}\) is the extension by zero of \(L_{G/S}\) from the level \(1\), so \(\operatorname{Ext}^j(L_{\tau_1N/S},f^*I)=\) \(\operatorname{Ext}^j_G(L_{G/S},\pi^*I)\), and the edge map \(\operatorname{Ext}^j(L_{N/S},f^*I)\to E_1^{1,j}\) is the restriction to the level \(1\), followed by the isomorphism \(\operatorname{Ext}^j_G(L_{G/S},\pi^*I)\cong E_1^{1,j}\) given by \(\theta_1\).

Proof. (1) Filter \(F^1K\) by the column index. In each total degree the filtration is finite: the columns have \(E_1\)-terms in degrees \(q\geq0\), \(\ell_G\) having tor amplitude in \([-1,0]\), and the double complex has finitely many nonzero terms in each total degree. By Theorem 3.1, \(H^{j+1}(F^1K)\to H^{j+1}(F^1K/F^kK)\) is the restriction map to \(\tau_{k-1}N\), so its kernel, the image of \(H^{j+1}(F^kK)\), is \(F^k\operatorname{Ext}^j\). The terms \(E_1^{k,q}=H^{k+q}(F^kK/F^{k+1}K)\) are those of Corollary 3.5 (3) there.

(2) The complex \(F^{i+1}K\) has cohomology only in degrees \(\geq i+1\), since \(E_1^{k,q}=0\) for \(q<0\). In the long exact sequence of \(0\to\) \(F^{i+1}K\to\) \(F^1K\to\) \(F^1K/F^{i+1}K\to\) \(0\), the map \(H^{j+1}(F^1K)\to H^{j+1}(F^1K/F^{i+1}K)\), which is the restriction map by Theorem 3.1, is injective when \(H^{j+1}(F^{i+1}K)=0\), that is, for \(j+1\leq i\), and surjective when moreover \(H^{j+2}(F^{i+1}K)=0\), that is, for \(j+2\leq i\). With \(i=j+1\) the injectivity is \(F^{j+2}=0\).

(3) The level \(0\) of \(L_{\tau_1N/S}\) is \(L_{S/S}\), which is acyclic. The left adjoint \(1_!\) of the restriction to the level \(1\) has \((1_!F)_1=F\) and \((1_!F)_0=0\), there being no injective map \([1]\to[0]\). So the counit \(1_!(L_{G/S})\to L_{\tau_1N/S}\) is the identity at the level \(1\) and \(0\to L_{S/S}\) at the level \(0\), a quasi-isomorphism. The functor \(1_!\) is exact and its right adjoint preserves injectives, so \(R\operatorname{Hom}(1_!L_{G/S},f^*I)=\) \(R\operatorname{Hom}_G(L_{G/S},\pi^*I)\). The edge map is the map \(H^{j+1}(F^1K)\to H^{j+1}(F^1K/F^2K)\), which Theorem 3.1 identifies with the restriction to \(\tau_1N\). \(\square\)

Corollary 3.3 (homomorphisms). Let \(u\colon F\to G\) be a homomorphism of group schemes over \(S\), flat and locally of finite presentation, and \(K_u\) the complex of Section 2 for the classifying topos \(B_F\), \(E=u^*\ell^{\mathrm{ad}}_G\) and \(F=\varepsilon^*I\) (Theorem 3.1 (3) of the lesson on deformations of flat group schemes). Then Theorem 3.1 and Corollary 3.2 (1), (2) hold with \(R\operatorname{Hom}(LN(u)^*L_{N(G)/S},f_F^*I)\) and its restrictions to the \(\tau_iN(F)\) in place of \(R\operatorname{Hom}(L_{N/S},f^*I)\), \(K_u\) in place of \(K\), and \(E_1^{k,q}=\operatorname{Ext}^q_{F^k}(\pi_k^*\ell_G,\pi_k^*I)\) for \(k\geq1\).

Proof. The proof of Theorem 3.1 applies to the distinguished triangle \(LN(u)^*L_{N(G)/S}\to\) \(i_{0!}\ell_G\to\) \(LN(u)^*\ell_{P_G/N(G)}\to\) of the proof of Theorem 3.1 (3) of the lesson on deformations of flat group schemes, whose second map is the adjunction map, with \(Q\) the nerve of a complex representing \(u^*\ell^{\mathrm{ad}}_G\); the structure of the modules of \(B_F\) is invertible, \(F\) being a group. The \(E_1\)-terms are those of Theorem 3.1 (3) there. \(\square\)

4. Deformations of truncated nerves

4.0. Setting. In this section \(S\to T\) is a morphism of schemes and \((S',i,s')\) a \(T\)-extension of \(S\) by a quasi-coherent \(\mathcal O_S\)-module \(I\), as in the Conventions of the lesson on deformations of flat group schemes, whose notation we use. For a diagram \(Z\) of flat \(S\)-schemes indexed by \((\Delta'_{\leq i})^{\mathrm{op}}\) or by \(\Delta^{\mathrm{op}}\), with the structure morphism \(g\colon Z\to S\), a deformation of \(Z\) over \(S'\) is a deformation of \(g\) over the extension \(S'\) of \(S\) (Definition 4.1 of the lesson on deformations of diagrams), and \(\omega(Z)=\omega(g,S')\in\operatorname{Ext}^2(L_{Z/S},g^*I)\) is the obstruction of Construction 4.2 there. By Theorem 4.3 there, \(Z\) has a deformation if and only if \(\omega(Z)=0\); the isomorphism classes then form a torsor under \(\operatorname{Ext}^1(L_{Z/S},g^*I)\), and the automorphism groups are \(\operatorname{Ext}^0(L_{Z/S},g^*I)\). For the nerve of a group scheme, \(\omega(N)=\omega(G,i)\).

Proposition 4.1 (restriction). Let \(Z\) be as in 4.0, indexed by \(\Delta^{\mathrm{op}}\) or by \((\Delta'_{\leq i})^{\mathrm{op}}\), and \(i'\leq i\).

1. Restriction to \(\tau_{i'}Z\) carries \(\omega(Z)\) to \(\omega(\tau_{i'}Z)\).
2. Restriction carries deformations of \(Z\) to deformations of \(\tau_{i'}Z\) and isomorphisms to isomorphisms. On isomorphism classes it is equivariant for the torsor structures, through the restriction map on \(\operatorname{Ext}^1\), and on automorphism groups it is the restriction map on \(\operatorname{Ext}^0\).

Proof. (1) Restriction is the inverse image of a morphism of ringed topoi, exact and compatible with the polynomial constructions. It carries the rings \(\mathcal A'\), \(\mathcal A\), \(\mathcal B\) of Construction 4.2 of the lesson on deformations of diagrams, the transitivity triangle, the object \(D\) and the map \(\tilde c\) to those for \(\tau_{i'}Z\), as in the proof of Proposition 2.2 of the lesson on deformations of flat group schemes, so it carries \(\omega(Z)\) to \(\omega(\tau_{i'}Z)\).

(2) A deformation is an \(S'\)-extension whose ideal map \(g^*I\to\mathcal I\) is an isomorphism; restriction is exact and keeps these properties. By the proof of Theorem 4.3 of the lesson on deformations of diagrams, the deformations of \(Z\) are the extensions of \(\mathcal A'\)-algebras of \(\mathcal B\) by \(M=g^*I\) whose structure map on \(\ker(\mathcal A'\to\mathcal A)\) is the canonical map \(c\), and the action of \(\operatorname{Ext}^1(L_{Z/S},M)\) on their classes \(\kappa\in\operatorname{Ext}^1(L_{\mathcal B/\mathcal A'},M)\) is through composition with the functoriality map \(L_{\mathcal B/\mathcal A'}\to L_{\mathcal B/\mathcal A}=L_{Z/S}\) (Theorem 10.4 of The cotangent complex). Restriction carries extensions to extensions, and it carries the class \(\kappa\) of an extension, a composite of a functoriality map and a conormal truncation, to the class of the restricted extension, because it preserves both, as in the proof of Proposition 3.5 (1) of the lesson on deformations of flat group schemes. It commutes with composition with the functoriality map. Automorphisms of a deformation are \(e\mapsto e+D(\bar e)\) for derivations \(D\) of \(\mathcal B\) over \(\mathcal A\) into \(M\), that is, elements of \(\operatorname{Ext}^0(L_{Z/S},M)\), and restriction restricts the derivation. \(\square\)

Proposition 4.2 (the truncations of the nerve of a group). Let \(G\) be a group scheme over \(S\), flat and locally of finite presentation, and \(N=N(G)\). Write \(N_2=G\times_SG\), with the faces \(d_2(g_1,g_2)=g_1\), \(d_0(g_1,g_2)=g_2\) and \(d_1(g_1,g_2)=g_2g_1\).

1. The functor \(Z'\mapsto(Z'_1,\kappa_1)\) is an equivalence from the groupoid of deformations of \(\tau_1N\) over \(S'\) onto the groupoid of deformations of \(G\) as a scheme: pairs \((G'_1,\iota)\) of a flat \(S'\)-scheme \(G'_1\), locally of finite presentation, and a morphism \(\iota\colon G\to G'_1\) over \(i\) inducing an isomorphism \(G\cong G'_1\times_{S'}S\), with the isomorphisms of \(S'\)-schemes compatible with the \(\iota\).
2. The deformations of \(\tau_2N\) correspond, by an equivalence of groupoids, to the triples \((G'_1,\iota,\mu')\) of a deformation of \(G\) as a scheme and an \(S'\)-morphism \(\mu'\colon G'_1\times_{S'}G'_1\to G'_1\) with \(\mu'\circ(\iota\times\iota)=\iota\circ d_1\); an isomorphism of triples is an isomorphism \(h\) of deformations of \(G\) as a scheme with \(\mu''\circ(h\times h)=h\circ\mu'\). The triple of a deformation \(Z'\) is \((Z'_1,\kappa_1,d'_1)\), read through the isomorphism \((d'_2,d'_0)\colon Z'_2\cong Z'_1\times_{S'}Z'_1\).
3. For \(3\leq i\leq\infty\), restriction is an equivalence from the groupoid of deformations of \(N\), that is, of deformations of \(G\) as a group scheme (Proposition 1.2 (3) of the lesson on deformations of flat group schemes), onto the groupoid of deformations of \(\tau_iN\). Restriction from \(\tau_3N\) to \(\tau_2N\) is fully faithful, and its essential image consists of the triples with \(\mu'\circ(\mu'\times\mathrm{id})=\mu'\circ(\mathrm{id}\times\mu')\) on \(G'_1\times_{S'}G'_1\times_{S'}G'_1\), which for \(\mu'=d_1\) is the identity \(g_3(g_2g_1)=(g_3g_2)g_1\).

Proof. (1) Let \(Z'\) be a deformation of \(\tau_1N\). Its level \(0\) is a deformation of \(\mathrm{id}_S\) over \(S'\), hence an isomorphism \(Z'_0\cong S'\) (Lemma 3.3 of the lesson on deformations of diagrams), and its two faces \(Z'_1\to Z'_0\) are then both the structure morphism, being morphisms over \(S'\). Its level \(1\) is a deformation of \(G\to S\), that is, by Lemma 1.1 of the lesson on deformations of flat group schemes, a flat \(S'\)-scheme locally of finite presentation with \(G\cong Z'_1\times_{S'}S\) through \(\kappa_1\). Conversely, a deformation of \(G\) as a scheme, with \(S'\) at the level \(0\) and the structure morphism as both faces, is a diagram of type \((\Delta'_{\leq1})^{\mathrm{op}}\), which has no relations, and an \(S'\)-extension of \(\tau_1N\) whose ideal map is an isomorphism at both levels. Isomorphisms correspond, the level \(0\) of an isomorphism being the identity of \(S'\).

(2) Let \(Z'\) be a deformation of \(\tau_2N\). As in (1), \(Z'_0\cong S'\). The commutative square of \(\Delta'_{\leq2}\) formed by \(\delta^0\delta^1=\delta^2\delta^0\colon[0]\to[2]\) has as its image under \(N\) the square formed by \(d_0,d_2\colon N_2\to N_1\) and the faces \(d_1,d_0\colon N_1\to N_0=S\), which is cartesian. By Proposition 4.4 of the lesson on deformations of diagrams, for the morphism \(\tau_2N\to S\), its image under \(Z'\) is cartesian: \((d'_2,d'_0)\colon Z'_2\to Z'_1\times_{S'}Z'_1\) is an isomorphism. Through it, \(d'_1\) is an \(S'\)-morphism \(\mu'\) lifting \(d_1\). Conversely, given a triple, put \(Z'_0=S'\), \(Z'_1=G'_1\), \(Z'_2=G'_1\times_{S'}G'_1\), with the two projections as \(d'_2,d'_0\), \(\mu'\) as \(d'_1\), and the structure morphisms as the faces into \(Z'_0\). The relations \(d'_id'_j=d'_{j-1}d'_i\) for \(i<j\) concern maps \(Z'_2\to Z'_0=S'\) over \(S'\), so they hold. This diagram, with \(\iota\), \(\iota\times\iota\) and \(i\), is an \(S'\)-extension of \(\tau_2N\): the level \(2\) is the base change of the flat \(S'\)-scheme \(G'_1\) along itself, with \(G'_1\times_{S'}G'_1\times_{S'}S\cong N_2\). Its ideal maps are isomorphisms, the levels being flat (Lemma 1.1 there). Isomorphisms correspond: an isomorphism of deformations of \(\tau_2N\) is the identity at the level \(0\), an isomorphism \(h\) at the level \(1\), and at the level \(2\) it is \(h\times h\), being compatible with \(d'_0\) and \(d'_2\); compatibility with \(d'_1\) is \(\mu''\circ(h\times h)=h\circ\mu'\).

(3) Equivalence. By Corollary 3.2 (2), the restriction maps \(\operatorname{Ext}^j(L_{N/S},f^*I)\to\) \(\operatorname{Ext}^j(L_{\tau_iN/S},f^*I)\) are bijective for \(j\leq1\) and injective for \(j=2\), as \(i\geq3\). If \(\tau_iN\) has a deformation, then \(\omega(\tau_iN)=0\), so \(\omega(N)=0\) by Proposition 4.1 (1) and injectivity, and \(N\) has a deformation. The map on isomorphism classes is then a map of torsors equivariant along a group isomorphism (Proposition 4.1 (2)), hence bijective. For two deformations \(Z'\), \(Z''\) of \(N\), the set of isomorphisms \(Z'\to Z''\) is empty exactly when their classes differ, and otherwise a torsor under the automorphism group of \(Z'\); restriction is equivariant along the isomorphism on \(\operatorname{Ext}^0\), so it is bijective on isomorphisms. So restriction is an equivalence.

The image in \(\tau_2N\). Let \(Z'\) be a deformation of \(\tau_3N\), with the triple \((Z'_1,\kappa_1,\mu')\) of its restriction. As in (2), the square of \(N\) formed by \(d_3,d_0\colon N_3\to N_2\) and \(d_0,d_2\colon N_2\to N_1\), the image of \(\delta^3\delta^0=\delta^0\delta^2\), is cartesian, so its image under \(Z'\) is cartesian, and \(Z'_3\cong Z'_2\times_{Z'_1}Z'_2\cong Z'_1\times_{S'}Z'_1\times_{S'}Z'_1\) through the edges \(\{0,1\}\), \(\{1,2\}\), \(\{2,3\}\) of \([3]\), the face \(d'_0\) being the projection to the last two factors and \(d'_3\) to the first two. The relations \(d'_0d'_1=d'_0d'_0\) and \(d'_2d'_1=d'_1d'_3\) then determine \(d'_1\) through the isomorphism \((d'_2,d'_0)\) of the level \(2\): \(d'_1(g_1,g_2,g_3)=(\mu'(g_1,g_2),g_3)\). In the same way \(d'_0d'_2=d'_1d'_0\) and \(d'_2d'_2=d'_2d'_3\) give \(d'_2(g_1,g_2,g_3)=(g_1,\mu'(g_2,g_3))\). The relation \(d'_1d'_1=d'_1d'_2\) then reads \(\mu'(\mu'(g_1,g_2),g_3)=\mu'(g_1,\mu'(g_2,g_3))\), which is the stated associativity. Conversely, for an associative \(\mu'\), these formulas define a diagram of type \((\Delta'_{\leq3})^{\mathrm{op}}\): of the six relations \(d'_id'_j=d'_{j-1}d'_i\) at the level \(3\), four were used to define \(d'_1\), \(d'_2\), the relation \(d'_0d'_3=d'_2d'_0\) holds for the projections, and the last is associativity; the relations into the level \(0\) hold over \(S'\). As in (2) it is a deformation of \(\tau_3N\), and isomorphisms are determined by their level \(1\), which gives full faithfulness. \(\square\)

Consequently, for a deformation \((G'_1,\iota)\) of \(G\) as a scheme, an associative \(S'\)-morphism \(\mu'\) lifting \(d_1\) comes from a group law: by (3) the triple is isomorphic to the triple of a deformation \(G'\) of \(G\) as a group scheme, so \(\mu'(a,b)=ba\) for the group structure on \(G'_1\) transported from \(G'\). Unit and inverses need not be lifted separately.

Proposition 4.3 (homomorphisms). Let \((F',\iota_F)\) and \((G',\iota_G)\) be deformations of the group schemes \(F\) and \(G\), \(u\colon F\to G\) a homomorphism, and \(1\leq i\leq\infty\). Call a lift of \(u\) on \(\tau_i\) a morphism of diagrams \(\tau_iN(F')\to\tau_iN(G')\) over \(S'\) whose composite with \(\tau_iN(\iota_F)\) is \(\tau_iN(\iota_G)\circ\tau_iN(u)\).

1. The lifts of \(u\) on \(\tau_1\) are given by their level \(1\), which is any \(S'\)-morphism \(u'_1\colon F'\to G'\) with \(u'_1\iota_F=\iota_Gu\).
2. For \(i\geq2\), the lifts of \(u\) on \(\tau_i\) are the \(\tau_iN(u')\) for the extensions \(u'\colon F'\to G'\) of \(u\), homomorphisms of group schemes.
3. Restriction to \(\tau_i\) carries the obstruction \(\omega(u,F',G')\) of Theorem 3.1 of the lesson on deformations of flat group schemes to the obstruction, constructed in the same way, for the lifts on \(\tau_i\). For \(i=1\) this is the obstruction \(\omega(u,F',G')_1\in\) \(\operatorname{Ext}^1_F(Lu^*L_{G/S},\pi_F^*I)\) to lifting \(u\) to a morphism of schemes \(F'\to G'\).

Proof. (1) At the level \(0\) a lift is a morphism \(S'\to S'\) over \(S'\), the identity; at the level \(1\) it is \(u'_1\), and the compatibilities with the faces into the level \(0\) hold over \(S'\). (2) At the level \(2\) a lift is compatible with \(d'_0\) and \(d'_2\), so through the isomorphisms of Proposition 4.2 (2) it is \(u'_1\times u'_1\); compatibility with \(d'_1\) says that \(u'_1\) is multiplicative. Conversely a homomorphism \(u'\) gives \(N(u')\) and its restrictions, and at the levels \(\geq3\) a lift is determined by its level \(1\) in the same way, by the cartesian squares of Proposition 4.2 (3). (3) In the proof of Theorem 3.1 of the lesson on deformations of flat group schemes, the class \(o=\kappa(\mathcal E')\circ\rho-c\circ\kappa(\mathcal E)\) is built from rings, extension classes, functoriality maps and the canonical map \(c\), all preserved by the exact restriction functor, which also commutes with the functoriality map \(w\) used to pass from \(S'\) to \(S\). At the level \(1\) the construction is Lemma 5.1 of the lesson on equivariant extensions for the morphism of schemes \(u\), whose lifts are the morphisms \(F'\to G'\) of (1). \(\square\)

Proposition 4.4 (actions of a group scheme). Let \(G\) be a group scheme over \(S\), flat and locally of finite presentation, \(f\colon X\to Y\) and \(g\colon Y\to Z\) morphisms of \(G\)-schemes and \((Y',j,g')\) an equivariant \(Z\)-extension of \(Y\) by \(J\), as in Theorem 3.3 of the lesson on equivariant extensions, with \(\varphi=N(G,f)\). Write \(a\colon G\times_SX\to X\) for the action, so that \(N_1(G,X)=G\times_SX\) has the faces \(d_1=\mathrm{pr}_X\) and \(d_0=a\). Let \(\omega_i\) be the obstruction of Construction 4.2 of the lesson on deformations of diagrams for \(\tau_i\varphi\colon\tau_iN(G,X)\to\tau_iN(G,Y)\) over \(\tau_iN(G,Y')\).

1. The deformations of \(\tau_0\varphi\) over \(\tau_0N(G,Y')\) are the deformations of \(f\) over \(Y'\), and \(\omega_0=\omega(f,Y')\).
2. The deformations of \(\tau_1\varphi\) correspond, by an equivalence of groupoids, to the pairs \((X',a')\) of a deformation \((X',\iota,f')\) of \(f\) over \(Y'\) and a morphism \(a'\colon G\times_SX'\to X'\) with \(a'\circ(\mathrm{id}\times\iota)=\iota\circ a\) and \(f'\circ a'=a_{Y'}\circ(\mathrm{id}\times f')\), where \(a_{Y'}\) is the action on \(Y'\); isomorphisms are the isomorphisms of deformations of \(f\) compatible with the \(a'\).
3. For \(2\leq i\leq\infty\), restriction is an equivalence from the groupoid of equivariant deformations of \(f\) over \(Y'\) onto the groupoid of deformations of \(\tau_i\varphi\) (through Proposition 3.2 of that lesson). Restriction from \(\tau_2\) to \(\tau_1\) is fully faithful, with the essential image formed by the pairs whose \(a'\) satisfies \(a'(g_2,a'(g_1,x))=a'(g_2g_1,x)\).
4. Restriction to \(\tau_i\) carries \(\omega(f,j)\), read as \(\omega(\varphi,N(G,Y'))\), to \(\omega_i\). The groups \(\operatorname{Ext}^k(L_{\tau_i\varphi},\tau_i\varphi^*\underline J)\) are computed by the total complex of the columns \(p\leq i\) of the double complex of Proposition 4.1 of that lesson, compatibly with the restrictions, and the restriction maps from \(N(G,X)\) to \(\tau_iN(G,X)\) are bijective on \(\operatorname{Ext}^k\) for \(k\leq i-1\) and injective for \(k=i\).

Proof. (4) The first statement is proved as Proposition 4.1 (1). For the second, let \(n\) be an integer. Lemma 4.2 (2) of the lesson on big fppf sites and Proposition 3.2 (2) of the lesson on equivariant cotangent complexes, restricted to \(\tau_i\), identify \(\operatorname{Ext}^k(L_{\tau_i\varphi},\tau_i\varphi^*\underline J)\), for \(k\leq-n\), with the Ext groups on \(\operatorname{Top}(\tau_iN(G,X)_{\mathrm{fppf}})\) from the nerve of \(t_{[n}L^G_{X/Y}\) to the nerve of \(f_B^*J\) (Conventions of the lesson on equivariant extensions), compatibly with the restrictions, all identifications being levelwise. The structures of modules of \(B[G,X]^{\mathrm o}\) are invertible, \(G\) being a group, so Theorem 2.1 (3) computes these derived Hom complexes by the truncations of the double complex of Proposition 4.1 of the lesson on equivariant extensions, whose \(E_1\)-terms vanish in degrees \(<0\), the complex \(t_{[n}L\varepsilon^*L_{X/Y}\) lying in degrees \(\leq0\). The last statement follows as in Corollary 3.2 (2).

(1) The level \(0\) of \(\tau_0\varphi\) is \(f\), and Construction 4.2 at one level is that for the morphism of schemes.

(2) Since \(G\) is a group, the ideal of \(N(G,Y')\), the nerve of \(J\), is cartesian, so \(N(G,Y')\) is a deformation of \(N(G,Y)\). At the level \(1\), the square formed by \(d_1=\mathrm{pr}_X\colon G\times_SX\to X\), its analogue for \(Y\), \(\varphi_1\) and \(f\) is cartesian, so by Proposition 4.4 of the lesson on deformations of diagrams its image under a deformation \(\Phi'\) of \(\tau_1\varphi\) is cartesian: \(\Phi'_1\) is \(G\times_SX'\to G\times_SY'\), with \(X'=\Phi'_0\) a deformation of \(f\) and \(d'_1\) the projection. The face \(d'_0\) is then a morphism \(a'\) as stated, the compatibility with \(f'\) expressing that \(\Phi'\) is a morphism of diagrams over \(\tau_1N(G,Y')\). There are no relations in \(\Delta'_{\leq1}\). Conversely such a pair defines a deformation, and isomorphisms correspond as in Proposition 4.2.

(3) As in Proposition 4.2 (3), with (4), Theorem 4.3 of the lesson on deformations of diagrams for \(\varphi\) and \(\tau_i\varphi\), Proposition 4.1 (2), and Proposition 3.2 of the lesson on equivariant extensions. At the level \(2\), \(N_2(G,X)=G\times_SG\times_SX\) has the faces \(d_2(g_1,g_2;x)=(g_1;x)\), \(d_1(g_1,g_2;x)=(g_2g_1;x)\) and \(d_0(g_1,g_2;x)=(g_2;a(g_1,x))\). A deformation of \(\tau_2\varphi\) has the level \(2\) equal to \(G\times_SG\times_SX'\), by the cartesian square of \(d_2\) and \(\mathrm{pr}_X\), with \(d'_2(g_1,g_2;x)=(g_1;x)\) given by that square, \(d'_1(g_1,g_2;x)=(g_2g_1;x)\) forced by the relation \(d'_1d'_1=d'_1d'_2\) and by the map to \(N(G)\), and \(d'_0(g_1,g_2;x)=(g_2;a'(g_1,x))\) forced by the relation \(d'_1d'_0=d'_0d'_2\) and by the map to \(N(G)\). The remaining relation \(d'_0d'_1=d'_0d'_0\) is \(a'(g_2g_1,x)=a'(g_2,a'(g_1,x))\). \(\square\)

5. The graded pieces of the obstructions

For a complex \(C\) with a decreasing filtration \(F^\bullet C\), finite in each degree, put \(F^kH^m(C)=\operatorname{Im}\bigl(H^m(F^kC)\to H^m(C)\bigr)\), and \[ \begin{aligned}Z^{k}&=\operatorname{Im}\bigl(H^m(F^kC)\\&\qquad\to H^m(F^kC/F^{k+1}C)\bigr),\\ B^{k}&=\operatorname{Im}\bigl(\partial\colon H^{m-1}(C/F^kC)\\&\qquad\to H^m(F^kC/F^{k+1}C)\bigr),\end{aligned} \] where \(\partial\) is the connecting map of \(0\to F^k/F^{k+1}\to C/F^{k+1}\to C/F^k\to0\). The quotient \(Z^k/B^k\) is the term \(E_\infty^{k,m-k}\) of the spectral sequence of the filtered complex, a subquotient of \(E_1^{k,m-k}=H^m(F^kC/F^{k+1}C)\).

Lemma 5.1. \(B^k\subseteq Z^k\), and there is a canonical isomorphism \(F^kH^m(C)/F^{k+1}H^m(C)\cong Z^k/B^k\). For \(x\in F^kH^m(C)\), its class in \(Z^k/B^k\) is the class of any \(y\in H^m(F^kC/F^{k+1}C)\) with the same image as \(x\) in \(H^m(C/F^{k+1}C)\).

Proof. The connecting map \(\partial\) is the composite of the connecting map \(\partial_1\colon H^{m-1}(C/F^kC)\to H^m(F^kC)\) of \(0\to F^k\to C\to C/F^k\to0\) with the projection, by naturality of connecting maps; so \(B^k\subseteq Z^k\). For \(x\in F^kH^m\), choose \(z\in H^m(F^kC)\) with image \(x\), and send \(x\) to the class of the image \(\bar z\) of \(z\) in \(H^m(F^k/F^{k+1})\). The choices of \(z\) differ by elements of \(\ker(H^m(F^kC)\to H^m(C))=\operatorname{Im}\partial_1\), whose images lie in \(B^k\), so the map is well defined; it is surjective onto \(Z^k/B^k\). If \(\bar z=\partial(w)\), then \(z-\partial_1(w)\) has image \(x\) and maps to zero in \(H^m(F^k/F^{k+1})\), so it comes from \(H^m(F^{k+1}C)\) and \(x\in F^{k+1}H^m\); conversely an element of \(F^{k+1}H^m\) has a lift \(z\) coming from \(H^m(F^{k+1}C)\), with \(\bar z=0\). Finally, \(y\) and \(\bar z\) have the same image in \(H^m(C/F^{k+1})\), so \(y-\bar z\) lies in the kernel of \(H^m(F^k/F^{k+1})\to H^m(C/F^{k+1})\), which is \(B^k\). \(\square\)

For the filtered complex \(F^1K\) of Section 3 and \(m=3\), where \(F^1K/F^2K\) is the column \(1\) shifted by \(-1\), \(B^2\) is the image of the differential \(d_1\colon E_1^{1,1}\to E_1^{2,1}\). The group \(H^2(F^1K/F^3K)\) maps onto the kernel of \(d_1\) on \(E_1^{1,1}\), with the kernel \(E_1^{2,0}/d_1(E_1^{1,0})\), so \(B^3\subseteq E_1^{3,0}\) is generated by \(d_1(E_1^{2,0})\) and by the images under \(\partial\) of lifts of the elements of \(\ker(d_1\colon E_1^{1,1}\to E_1^{2,1})\). In the same way, for the complex of Corollary 3.3 and \(m=2\), \(B^2=d_1(E_1^{1,0})\); for the complex of Proposition 4.4, whose columns start at \(0\), and \(m=2\), \(B^1=d_1(E_1^{0,1})\), and \(B^2\) is generated by \(d_1(E_1^{1,0})\) and by the images under \(\partial\) of lifts of the elements of \(\ker(d_1\colon E_1^{0,1}\to E_1^{1,1})\).

Theorem 5.2 (deformations of a group scheme). In 4.0, let \(G\) be a group scheme over \(S\), flat and locally of finite presentation, \(\omega=\omega(G,i)\in\operatorname{Ext}^2(L_{N/S},f^*I)\), and \(F^\bullet\) the filtration of Corollary 3.2, so that \(\operatorname{Ext}^2=F^1\supseteq F^2\supseteq F^3\supseteq0\).

1. The image of \(\omega\) in \(F^1/F^2\subseteq E_1^{1,2}=\operatorname{Ext}^2_G(L_{G/S},\pi^*I)\) is the obstruction \(\omega(f_1,S')\) to deforming \(G\) as a scheme over \(S'\) (Proposition 2.2 of the lesson on deformations of flat group schemes).
2. \(\omega\in F^2\) if and only if \(G\) has a deformation as a scheme; \(\omega\in F^3\) if and only if there is a triple \((G'_1,\iota,\mu')\) as in Proposition 4.2 (2); \(\omega=0\) if and only if there is such a triple with \(\mu'\) associative, and then \(G\) has a deformation as a group scheme.
3. For \(k=2,3\) and \(\omega\in F^k\), the component of \(\omega\) in \(F^k/F^{k+1}=Z^k/B^k\) is the class of any \(y\in E_1^{k,3-k}=H^3(F^kK/F^{k+1}K)\) whose image in \(H^3(F^1K/F^{k+1}K)\) corresponds to \(\omega(\tau_kN)\in\operatorname{Ext}^2(L_{\tau_kN/S},f^*I)\) under \(\theta_k\).

Proof. (1) By Corollary 3.2 (3), the map \(F^1\to E_1^{1,2}\) is the restriction to the level \(1\), which carries \(\omega\) to \(\omega(f_1,S')\) by Proposition 2.2 of the lesson on deformations of flat group schemes; the component of \(\omega\) in \(F^1/F^2=Z^1\) is this image, as \(B^1=0\), \(E_1^{0,q}\) being zero.

(2) By definition \(\omega\in F^{k+1}\) if and only if its restriction to \(\tau_kN\), which is \(\omega(\tau_kN)\) by Proposition 4.1 (1), vanishes, that is, if and only if \(\tau_kN\) has a deformation. Proposition 4.2 describes these deformations for \(k=1,2\); for \(k=3\), \(F^4=0\) and Proposition 4.2 (3) gives the last statement.

(3) Lemma 5.1 for \(C=F^1K\) and \(m=3\), with Theorem 3.1 and Proposition 4.1 (1). \(\square\)

Proposition 5.3 (homomorphisms). In the situation of Proposition 4.3, let \(\omega=\) \(\omega(u,F',G')\in\) \(\operatorname{Ext}^1(LN(u)^*L_{N(G)/S},f_F^*I)\), filtered as in Corollary 3.3, so that \(\operatorname{Ext}^1=F^1\supseteq F^2\supseteq0\). The image of \(\omega\) in \(F^1/F^2\subseteq E_1^{1,1}=\operatorname{Ext}^1_F(\pi_F^*\ell_G,\pi_F^*I)\) is the obstruction \(\omega(u,F',G')_1\) to lifting \(u\) to a morphism of schemes. So \(\omega\in F^2\) if and only if \(u\) lifts to an \(S'\)-morphism \(F'\to G'\), and \(\omega=0\) if and only if it lifts to a homomorphism; for \(\omega\in F^2\), its component in \(F^2=Z^2/B^2\) is the class of any \(y\in E_1^{2,0}\) whose image in \(\operatorname{Ext}^1\) of the restriction to \(\tau_2N(F)\) is the restricted obstruction.

Proof. As for Theorem 5.2, with Corollary 3.3, Proposition 4.3, and Lemma 5.1 for \(m=2\); here \(F^3\operatorname{Ext}^1=0\) by Corollary 3.2 (2), and the edge map is the restriction to the level \(1\), as in Corollary 3.2 (3). \(\square\)

Proposition 5.4 (actions of a group scheme). In the situation of Proposition 4.4, filter \(\operatorname{Ext}^2(t_[L^G_{X/Y},f_B^*J)\) by the kernels \(F^{k}\) of the restrictions to \(\tau_{k-1}N(G,X)\), \(k\geq1\), with \(F^0=\operatorname{Ext}^2\), so that \(F^0\supseteq F^1\supseteq F^2\supseteq F^3=0\). Then the image of \(\omega(f,j)\) in \(F^0/F^1\subseteq E_1^{0,2}\) is \(\omega(f,Y')\); \(\omega(f,j)\in F^1\) if and only if \(f\) has a deformation over \(Y'\); \(\omega(f,j)\in F^2\) if and only if there is a pair \((X',a')\) as in Proposition 4.4 (2); and \(\omega(f,j)=0\) if and only if there is such a pair with \(a'\) associative. For \(k=1,2\) and \(\omega(f,j)\in F^k\), its component in \(F^k/F^{k+1}\) is the class of any element of \(E_1^{k,2-k}\) whose image in \(\operatorname{Ext}^2(L_{\tau_k\varphi},\tau_k\varphi^*\underline J)\) is \(\omega_k\).

Proof. Proposition 4.1 (3) of the lesson on equivariant extensions for the first statement, Proposition 4.4 for the others, and Lemma 5.1. \(\square\)

6. Exercises

Exercise 6.1. Let \(\Gamma\) be the category with the objects \([1]\), \([2]\) and, besides the identities, the three maps \(\delta^0,\delta^1,\delta^2\colon[1]\to[2]\), the full subcategory of \(\Delta'\) on these objects. For a functor \(V\colon\Gamma\to\operatorname{Ab}\), show that the derived limit of \(V\) is represented by the complex \(V_1\oplus V_2\to V_2^3\), \((x,y)\mapsto(V(\delta^j)x-y)_{j}\), in degrees \(0\) and \(1\). For \(V_1=0\) and \(V_2=A\), compare its first cohomology with that of the truncated alternating complex \(V_1\to V_2\).

Exercise 6.2. In 3.0, suppose that \(G\) is étale over \(S\). Show that \(\operatorname{Ext}^j(L_{\tau_iN/S},f^*I)=0\) for all \(i\) and \(j\), and deduce from Proposition 4.2 that every deformation of \(G\) as a scheme carries a unique lift of the multiplication, which is associative.

Exercise 6.3. In 1.1 with \(n=1\), let \(E\) be a cartesian module and \(J\) an injective module. Write out the exact sequence of Lemma 1.2 at the levels \(0\) and \(1\), and deduce that \(\operatorname{Hom}(E,J)\) is the kernel of \(\delta^0_*-\delta^1_*\colon\operatorname{Hom}(E_0,J_0)\to\operatorname{Hom}(E_1,J_1)\) and that this map is surjective. For an abelian sheaf \(M\) of \(\operatorname{Top}(X)\) whose two levels are injective, deduce that \(H^1(\operatorname{Top}(X),M)\) is the cokernel of \(\delta^0_*-\delta^1_*\colon\Gamma(X_0,M_0)\to\Gamma(X_1,M_1)\), and that \(H^q(\operatorname{Top}(X),M)=0\) for \(q\geq2\).

Exercise 6.4. In 4.0, let \(S=\operatorname{Spec}A\), \(T=\operatorname{Spec}\mathbf Z\), \(S'=\operatorname{Spec}A[\epsilon]/(\epsilon^2)\), so that \(I=A\epsilon\), and \(G=\mathbf G_{a,S}\). On \(G'_1=\mathbf G_{a,S'}\) put \(\mu'(x,y)=x+y+\epsilon xy\). Show that \(\mu'\) is an associative lift of \(d_1\), find the unit and the inverse of the resulting group law, and show that \((G'_1,\mu')\) is isomorphic to \(\mathbf G_{a,S'}\) with its addition, as a deformation of \(G\), if and only if \(2\) is invertible in \(A\).

7. Complete solutions

6.1. A functor on \(\Gamma\) is a representation of the quiver with two vertices and three arrows. For a functor \(W\) and an object \(c\), let \(c_!W_c\) be the functor generated by \(W_c\) at \(c\): \(([1])_!A\) has the values \(A\) and \(A^3\), \(([2])_!A\) the values \(0\) and \(A\). Every functor \(V\) has the exact sequence \[ \begin{aligned}0&\to([2])_!(V_1^3)\to([1])_!V_1\oplus([2])_!V_2\\&\quad\to V\to0,\end{aligned} \] where the first map sends the component \(j\) of \(V_1^3\) at the level \([2]\) to \((e_j,-V(\delta^j))\), \(e_j\) the \(j\)-th inclusion; exactness is checked at both objects. Maps out of \(c_!A\) are maps out of \(A\) at \(c\), so applying \(\operatorname{Hom}(-,W)\) for an injective \(W\) gives the exact sequence \(0\to\) \(\operatorname{Hom}(V,W)\to\) \(\operatorname{Hom}(V_1,W_1)\oplus\operatorname{Hom}(V_2,W_2)\to\) \(\operatorname{Hom}(V_1,W_2)^3\to\) \(0\). With \(V=\mathbf Z\) at both objects this shows that the derived limit of \(W'\), computed by an injective resolution, is the total complex of \(W'_1\oplus W'_2\to(W'_2)^3\), \((x,y)\mapsto(W'(\delta^j)x-y)_j\), whose terms are exact in \(W'\); so the formula holds for \(V\) itself. For \(V_1=0\), \(V_2=A\), the map is \(y\mapsto(-y,-y,-y)\), so \(H^0=0\) and \(H^1=A^3/\Delta(A)\cong A^2\). The truncated alternating complex \(0\to A\) has \(H^1=A\). So restricting to the levels \(1\) and \(2\) does not compute the column \(2\) alone; Corollary 2.2 identifies the quotients of the relative complex with the strict truncations relative to the level \(0\) instead.

6.2. For \(G\) étale, \(\ell_G=0\) (Exercise 4.3 of the lesson on the adjoint co-Lie complex), so all the terms \(E_1^{k,q}\) vanish, and Theorem 3.1 gives \(R\operatorname{Hom}(L_{\tau_iN/S},f^*I)=0\). By Corollary 2.3 of the lesson on deformations of flat group schemes, \(G\) has a deformation \(G'\) as a group scheme, and its restriction to \(\tau_2N\) is the triple with \(\mu'\) the multiplication, read as in Proposition 4.2 (2). By Theorem 4.3 of the lesson on deformations of diagrams and the vanishing of the groups \(\operatorname{Ext}^j(L_{\tau_iN/S},f^*I)\), the deformations of \(\tau_1N\) and of \(\tau_2N\) exist, form a single isomorphism class, and have no automorphisms other than the identity. Let \((G'_1,\iota)\) be a deformation of \(G\) as a scheme, and \(\mu'\), \(\mu''\) two lifts of \(d_1\) to it. By Proposition 4.2 (2) the two triples are deformations of \(\tau_2N\), hence isomorphic by an isomorphism whose level \(1\) is an automorphism of the deformation \((G'_1,\iota)\) of \(\tau_1N\), hence the identity; so \(\mu'=\mu''\). A lift exists, coming from the restriction of \(G'\), transported to \(G'_1\) by the isomorphism of deformations of \(\tau_1N\), and it is associative by Proposition 4.2 (3).

6.3. For \(n=1\), Lemma 1.2 gives the exact sequence \(0\to1_!E_1\to0_!E_0\to E\to0\), with the differential \(\partial_0-\partial_1\). At the level \(0\) it reads \(0\to0\to E_0\to E_0\to0\); at the level \(1\), \(0\to E_1\to E_1\oplus E_1\to E_1\to0\), the first map being \(x\mapsto(x,-x)\) and the second the sum, after the identifications of the proof. Since \(J\) is injective, \(\operatorname{Hom}(-,J)\) gives the exact sequence \(0\to\) \(\operatorname{Hom}(E,J)\to\) \(\operatorname{Hom}(E_0,J_0)\xrightarrow{\delta^0_*-\delta^1_*}\) \(\operatorname{Hom}(E_1,J_1)\to\) \(0\), by the computation of \(\operatorname{Hom}(\partial_j,J)\) in the proof of Proposition 1.3. For \(M\), injective levels are totally acyclic (Lemma 1.3 (1) of the lesson on cohomology of total topoi), so by Corollary 1.4 \(R\Gamma(\operatorname{Top}(X),M)\) is represented by the two-term complex \(\Gamma(X_0,M_0)\to\Gamma(X_1,M_1)\) with the differential \(\delta^0_*-\delta^1_*\), the alternating sum of the cofaces. Its cohomology is the kernel in degree \(0\), the cokernel in degree \(1\), and zero above.

6.4. Modulo \(\epsilon\), \(\mu'(x,y)\) is \(x+y\), and \(d_1(g_1,g_2)=g_2+g_1\) is the addition, so \(\mu'\) lifts \(d_1\); it is an \(S'\)-morphism given by a polynomial. Using \(\epsilon^2=0\), \(\mu'(\mu'(x,y),z)=\) \(x+y+z+\epsilon(xy+xz+yz)=\) \(\mu'(x,\mu'(y,z))\). The unit is \(0\), since \(\mu'(x,0)=x\), and the inverse of \(x\) is \(-x+\epsilon x^2\), since \(x+(-x+\epsilon x^2)+\epsilon x(-x+\epsilon x^2)=0\); by the consequence of Proposition 4.2 (3) these exist without being lifted separately. An isomorphism of deformations of \(G\) as a group scheme is an \(S'\)-automorphism \(h\) of \(\mathbf G_{a,S'}\) reducing to the identity, so \(h(x)=x+\epsilon c(x)\) with \(c\in A[x]\), such that \(h(\mu'(x,y))=h(x)+h(y)\), that is, \[ \begin{aligned}&c(x+y)-c(x)-c(y)\\&\quad =-xy\quad\text{in }A[x,y].\end{aligned} \] Write \(c=\sum_kc_kx^k\). The part of degree \(2\) of the left side is \(2c_2xy\), and every term of degree \(k\) comes from \(c_k\). So a solution requires \(2c_2=-1\), and then \(2\) is invertible in \(A\). Conversely, if \(2\) is invertible, \(c(x)=-x^2/2\) is a solution. So for \(A=\mathbf Z\) or \(A=\mathbf F_2\), \(\mu'\) defines a deformation of \(\mathbf G_{a,S}\) as a group scheme that is not isomorphic to the trivial one, although its underlying scheme is \(\mathbf G_{a,S'}\).

Sources

The spectral sequences of the nerve of a group scheme and of the nerve of an action, their filtrations by the restrictions to the truncated nerves, and the interpretation of the graded pieces of the deformation obstructions are due to Illusie, in his work on the cotangent complex. For smooth groups the theory recovers the description of deformations of group schemes by Hochschild cohomology in the third exposé of the seminar on group schemes of Demazure and Grothendieck. The homological algebra used here is that of the Stacks project.