# Deformations of flat group schemes and of homomorphisms

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

Let \(S\subseteq S'\) be a first-order thickening with a quasi-coherent ideal \(I\), and \(G\) a group scheme over \(S\), flat and locally of finite presentation. A deformation of \(G\) is a flat group scheme \(G'\) over \(S'\) with \(G'\times_{S'}S\cong G\). Its nerve deforms the morphism \(N(G)\to S\) of simplicial schemes, and conversely every deformation of this morphism is the nerve of a deformation of \(G\): the fibre products that express the group law survive deformation. So deforming a group law is deforming a diagram of schemes. The deformation theory of diagrams then gives an obstruction in \(\operatorname{Ext}^2(L_{N(G)/S},I)\), a classification by \(\operatorname{Ext}^1\) and automorphisms \(\operatorname{Ext}^0\), with the obstruction the product of the class of the thickening and a Kodaira–Spencer class. The triangle of [*The adjoint co-Lie complex of a group scheme*](the-adjoint-co-lie-complex-of-a-group-scheme.md) expresses these groups as relative Ext groups of the adjoint co-Lie complex. For a smooth affine group over an affine base they are the cohomology groups of a complex of cochains of \(G\) with values in its Lie algebra tensored with \(I\). The same method treats homomorphisms: extending \(u\colon G\to H\) to given deformations of \(G\) and \(H\), and deforming \(G\) together with \(u\) over a given deformation of \(H\); it also compares the obstructions for two deformations of \(H\).

**Prerequisites.** [*The adjoint co-Lie complex of a group scheme*](the-adjoint-co-lie-complex-of-a-group-scheme.md) (Propositions 2.2 and 2.5, Theorem 3.3, Corollaries 3.4 and 3.5, Proposition 3.8, Proposition 3.10, Corollary 3.12, Exercise 4.3); [*The co-Lie complex of a group scheme*](the-co-lie-complex-of-a-group-scheme.md) (Proposition 1.5); [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md) (Conventions, Propositions 1.2 and 1.3, Definitions 2.1, 3.1 and 4.1, Theorem 2.4, Lemmas 3.2–3.4, Construction 4.2, Theorem 4.3, Proposition 4.4); [*Equivariant extensions and deformations*](equivariant-extensions-and-deformations.md) (Lemma 5.1 and the proof of Proposition 5.2); [*Décalage, nerves and the homotopy type of a category*](decalage-nerves-and-the-homotopy-type-of-a-category.md) (Section 2.3, Propositions 2.4 and 2.6); [*Equivariant cotangent complexes*](equivariant-cotangent-complexes.md) (Proposition 1.5); [*Quasi-coherent and perfect complexes on big fppf sites*](quasi-coherent-and-perfect-complexes-on-big-fppf-sites.md) (Theorem 3.1, Lemma 4.2); [*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md) (Theorem 4.3). From the programme lesson [*The cotangent complex*](course:AG-HP/the-cotangent-complex#extensions-of-sheaves-of-algebras): Theorem 2.2, and Theorems 10.3 and 10.4 with the class (10.6). From the AI-integrated Stacks project: closed immersions and fibre products [[Stacks, Tag 01JU](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/schemes.html#schemes-lemma-fibre-product-immersion)]; the local flatness criterion for nilpotent ideals [[Tag 051C](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/algebra.html#algebra-lemma-what-does-it-mean)]; properties of morphisms over a thickening [[Tag 06AG](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-deform-property)]; closed immersions as proper monomorphisms [[Tag 04XV](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/etale.html#etale-lemma-characterize-closed-immersion)]; cohomology of quasi-coherent modules on affine schemes [[Tag 01XB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-quasi-coherent-affine-cohomology-zero)]; duals of perfect complexes [[Tag 08DQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html#cohomology-lemma-dual-perfect-complex)]; maps out of objects in negative degrees and long exact sequences [[Tag 06XS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-negative-exts), [Tag 0149](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-representable-homological)].

## Conventions

\(S\to T\) is a morphism of schemes, and \((S',i,s')\) is a \(T\)-extension of \(S\) by a quasi-coherent \(\mathcal O_S\)-module \(I\) (Definition 2.1 of the lesson on deformations of diagrams): \(i\colon S\to S'\) is a first-order thickening with ideal \(I\), and \(s'\colon S'\to T\) restricts to the structure morphism of \(S\). Its class is \(e(i)\in\operatorname{Ext}^1(L_{S/T},I)\) (Theorem 2.4 there). The underlying spaces of first-order thickenings are identified.

Group schemes are flat and locally of finite presentation over their base unless stated otherwise. The nerve \(N(G)\) of a group scheme \(G\) over \(S\) is that of the internal category of \((\mathrm{Sch})\) with the object \(S\) and the arrows \(G\) (Section 2.3 of [*Décalage, nerves and the homotopy type of a category*](decalage-nerves-and-the-homotopy-type-of-a-category.md)), with \(N_n(G)=G^n\), the \(n\)-fold fibre product over \(S\), and \(f_G\colon N(G)\to S\) is its structure morphism, a morphism of diagrams of type \(\Delta^{\mathrm{op}}\to\{\ast\}\). For an \(\mathcal O_S\)-module \(J\), \(f_G^*J\) is the cartesian module with the levels \(\pi_n^*J\), \(\pi_n\colon G^n\to S\). A homomorphism \(u\colon G\to H\) gives the \(\Delta^{\mathrm{op}}\)-morphism \(N(u)\) with the components \(u^n\). The notation \(B\), \(\varepsilon\), \(\ell_{P/N}\), \(\ell^{\mathrm{ad}}_G\) and relative Ext groups are those of the lesson on the adjoint co-Lie complex; for a homomorphism \(u\colon G\to H\) we write \(B_G\), \(P_H\), and so on.

**Definition 0.1.** A *deformation of \(G\) over \(S'\)* is a pair \((G',\iota)\) of a group scheme \(G'\) over \(S'\), flat and locally of finite presentation, and a morphism of schemes \(\iota\colon G\to G'\) over \(i\), compatible with the multiplications, which induces an isomorphism of group schemes \(G\cong G'\times_{S'}S\). An *isomorphism* of deformations \((G'_1,\iota_1)\to(G'_2,\iota_2)\) is an isomorphism of group schemes \(h\colon G'_1\to G'_2\) over \(S'\) with \(h\iota_1=\iota_2\). For deformations \((G',\iota_G)\) of \(G\) and \((H',\iota_H)\) of \(H\), an *extension* of a homomorphism \(u\colon G\to H\) is a homomorphism \(u'\colon G'\to H'\) over \(S'\) with \(u'\iota_G=\iota_Hu\).

## 1. Group schemes over a thickening and their nerves

**Lemma 1.1 (flatness over a first-order thickening).** Let \(X'\to S'\) be a morphism of schemes such that \(X=X'\times_{S'}S\) is flat over \(S\). Then \(X'\) is flat over \(S'\) if and only if \(X'\to S'\) is a deformation of \(X\to S\) (Definition 3.1 of the lesson on deformations of diagrams). In this case, if \(X\) is locally of finite presentation over \(S\), so is \(X'\) over \(S'\).

**Proof.** The square formed by \(X\to X'\), \(S\to S'\) and the two structure morphisms is cartesian, so by Lemma 3.2 (2) of the lesson on deformations of diagrams, \(X'\to S'\) is a deformation exactly when \(\operatorname{Tor}_1^{C'}(C'/I,A')=0\) for all affine opens \(\operatorname{Spec}C'\) of \(S'\) and \(\operatorname{Spec}A'\) of \(X'\) over it, with \(I\) the ideal of \(S\) in \(C'\). These groups vanish if \(X'\) is flat over \(S'\). Conversely, if they vanish, \(A'/IA'\), the ring of an affine open of \(X\), is flat over \(C'/I\), and \(I^2=0\), so \(A'\) is flat over \(C'\) [Tag 051C]. The last assertion is [Tag 06AG], for the thickenings \(X\subseteq X'\) and \(S\subseteq S'\) over \(S\subseteq S'\). \(\square\)

**Proposition 1.2 (deformations of group schemes and of their nerves).**

1. Let \((G',\iota)\) be a deformation of \(G\) over \(S'\). Then \(N(G')\), with the closed immersions \(\iota^n\colon G^n\to G'^n\) and the structure morphism \(f_{G'}\colon N(G')\to S'\), is a deformation of \(f_G\colon N(G)\to S\) over \(S'\) (Definition 4.1 of the lesson on deformations of diagrams).
2. Every deformation \((X',\kappa,f')\) of \(f_G\) over \(S'\) is isomorphic, compatibly with the maps from \(N(G)\) and to \(S'\), to \(N(G')\) for a deformation \((G',\iota)\) of \(G\).
3. The functor \((G',\iota)\mapsto N(G')\) is an equivalence from the groupoid of deformations of \(G\) over \(S'\) onto the groupoid of deformations of \(f_G\) over \(S'\).

**Proof.** (1) The morphism \(\iota\) is a functor of internal categories of \((\mathrm{Sch})\), so the \(\iota^n\) form a morphism of simplicial schemes \(N(\iota)\), and \(f_{G'}N(\iota)=if_G\). Since \(G'^n\times_{S'}S\cong G^n\) through \(\iota^n\), each \(\iota^n\) is the base change of \(i\) along \(G'^n\to S'\), a first-order thickening [Tag 01JU]. The scheme \(G'^n\) is flat over \(S'\), so by Lemma 1.1 each \(G'^n\to S'\) is a deformation of \(G^n\to S\): the map \(f_G^*I\to\mathcal I\) to the ideal of \(N(G)\) in \(N(G')\) is an isomorphism at every level, hence an isomorphism.

(2) The thickening \(S'\) is a deformation of the one-object diagram \(S\), its ideal being trivially cartesian. By Proposition 4.4 of the lesson on deformations of diagrams, every commutative square of \(\Delta^{\mathrm{op}}\) whose image under \(N(G)\) is cartesian has a cartesian image under \(X'\).

(a) The level \(0\) of \(f'\) is a deformation of \(\mathrm{id}_S\), its ideal map being the level \(0\) of an isomorphism. By Lemma 3.3 there it is an isomorphism \(X'_0\cong S'\).

(b) For \(n\geq2\), the square formed by \(d_n\colon N_n(G)\to N_{n-1}(G)\), the last edge \(N_n(G)\to N_1(G)\), the last vertex of \(N_{n-1}(G)\) and the first vertex of \(N_1(G)\) is the image of a commutative square of \(\Delta^{\mathrm{op}}\), and it is cartesian by Proposition 2.4 ((1)\(\Rightarrow\)(4)) of the lesson on décalage. So its image under \(X'\) is cartesian, and \(X'\), as a simplicial object of the category of \(S'\)-schemes through \(f'\), satisfies condition (4) of that proposition. Hence \(X'\) is isomorphic to the nerve of an internal category of \((\mathrm{Sch}/S')\) with the object \(X'_0\cong S'\), the final object: a monoid scheme \(G'\) over \(S'\) with the underlying scheme \(X'_1\).

(c) The square formed by the faces \(d_1,d_2\colon N_2(G)\to N_1(G)\) and twice the first vertex \(N_1(G)\to N_0(G)\) is the image of the commutative square of the edges \((0,2)\), \((0,1)\) and the vertex \(0\), and it is cartesian because \(G\) is a group (Proposition 2.6 ((1)\(\Rightarrow\)(4)) there). So it is cartesian for \(X'\), and by Proposition 2.6 ((4)\(\Rightarrow\)(1)) \(G'\) is a groupoid with one object, a group scheme over \(S'\).

(d) The level \(1\) of the ideal map is an isomorphism, so \(X'_1\to S'\) is a deformation of \(G\to S\). By Lemma 1.1, \(G'\) is flat and locally of finite presentation over \(S'\). The square formed by \(\kappa_1\), \(i\) and the structure morphisms is cartesian (Lemma 3.2 (1) of the lesson on deformations of diagrams), so \(\kappa_1\) induces an isomorphism \(G\cong G'\times_{S'}S\). The map \(\kappa\colon N(G)\to X'\cong N(G')\) is a morphism of simplicial schemes, so by the full faithfulness of the nerve functor (Proposition 2.4 of the lesson on décalage, in \((\mathrm{Sch})\)), \(\iota=\kappa_1\) is compatible with the multiplications. So \((G',\iota)\) is a deformation of \(G\), and \(X'\cong N(G')\) compatibly with \(\kappa\), \(N(\iota)\) and the maps to \(S'\).

(3) Isomorphisms of deformations of \(f_G\) are isomorphisms of simplicial \(S'\)-schemes compatible with the maps from \(N(G)\). By the full faithfulness of the nerve functor in \((\mathrm{Sch}/S')\), an isomorphism \(N(G'_1)\to N(G'_2)\) over \(S'\) is \(N(h)\) for a unique isomorphism of group schemes \(h\), and compatibility with the maps from \(N(G)\) is \(h\iota_1=\iota_2\) at the level \(1\), and then at all levels. Together with (2), this proves (3). \(\square\)

**Proposition 1.3 (homomorphisms).** Let \((G',\iota_G)\) and \((H',\iota_H)\) be deformations of \(G\) and \(H\) over \(S'\), and \(u\colon G\to H\) a homomorphism.

1. The morphisms of simplicial \(S'\)-schemes \(N(G')\to N(H')\) whose composite with \(N(\iota_G)\) is \(N(\iota_H)N(u)\) are the \(N(u')\) for the extensions \(u'\) of \(u\).
2. Up to isomorphism, the deformations of the morphism of diagrams \(N(u)\colon N(G)\to N(H)\) over \(N(H')\) (Definition 4.1 of the lesson on deformations of diagrams) are the \((N(G'),N(\iota_G),N(u'))\) for the pairs of a deformation \((G',\iota_G)\) of \(G\) and an extension \(u'\colon G'\to H'\) of \(u\). Their isomorphisms are the isomorphisms \(h\) of deformations of \(G\) with \(u'_2h=u'_1\).

**Proof.** (1) This is the full faithfulness of the nerve functor in \((\mathrm{Sch}/S')\) (Proposition 2.4 of the lesson on décalage), compatibility with the maps from \(N(G)\) being checked at the level \(1\).

(2) By Proposition 1.2 (1), \(N(H')\) is an extension of \(N(H)\) whose ideal \(f_H^*I\) is cartesian, a deformation. Let \((X',\kappa,f')\) be a deformation of \(N(u)\) over \(N(H')\). Every level of \(f'\) is a deformation of a level of \(N(u)\), and every \(N_n(H')\to S'\) is a deformation of \(H^n\to S\) (Lemma 1.1); by Lemma 3.4 (1) of the lesson on deformations of diagrams their composites are deformations of the \(G^n\to S\). So \(X'\), with \(\kappa\) and the composite \(X'\to N(H')\to S'\), is a deformation of \(f_G\) over \(S'\), and Proposition 1.2 identifies it with \(N(G')\) for a deformation \((G',\iota_G)\) of \(G\). By (1), \(f'=N(u')\) for an extension \(u'\) of \(u\). Conversely, \((N(G'),N(\iota_G),N(u'))\) is an \(N(H')\)-extension of \(N(G)\), and its ideal map is an isomorphism by the second statement of Lemma 3.4 (1) there, \(N(G')\to S'\) and \(N(H')\to S'\) being deformations. Isomorphisms of deformations of \(N(u)\) are isomorphisms of extensions of \(N(G)\) compatible with the maps to \(N(H')\); by Proposition 1.2 (3) and (1) they are the stated \(h\). \(\square\)

## 2. Deformations of flat group schemes

Write \(N=N(G)\), \(f=f_G\), and let \(\omega(G,i)=\omega(f,S')\in\operatorname{Ext}^2(L_{N/S},f^*I)\) be the obstruction of Construction 4.2 of the lesson on deformations of diagrams, for the morphism \(f\) and the extension \(S'\) of \(S\).

**Theorem 2.1 (deformations of flat group schemes).**

1. \(G\) has a deformation over \(S'\) if and only if \(\omega(G,i)=0\).
2. If \(\omega(G,i)=0\), the set of isomorphism classes of deformations of \(G\) over \(S'\) is a torsor under \(\operatorname{Ext}^1(L_{N/S},f^*I)\), and the automorphism group of every deformation is isomorphic to \(\operatorname{Ext}^0(L_{N/S},f^*I)=\) \(\operatorname{Hom}(L_{N/S},f^*I)\).
3. \(\omega(G,i)=f^*e(i)[1]\circ K(N/S/T)\), where \(K(N/S/T)\in\) \(\operatorname{Ext}^1(L_{N/S},Lf^*L_{S/T})\) is the Kodaira–Spencer class of \(N\to S\to T\) and \(f^*e(i)\) is as in Theorem 4.3 (3) of the lesson on deformations of diagrams.
4. For every \(j\), \(\operatorname{Ext}^j(L_{N/S},f^*I)\cong\) \(\operatorname{Ext}^{j+1}_{B/S}(\ell^{\mathrm{ad}}_G,\varepsilon^*I)\), and these groups are the abutment of a spectral sequence with \(E_1^{0,k}=0\) and
\[
\begin{aligned}E_1^{n,k}&=\operatorname{Ext}^k_{G^n}(\pi_n^*\ell_G,\pi_n^*I)\\&=H^k\bigl(G^n,\pi_n^*I\otimes^{\mathbf L}(\pi_n^*\ell_G)^\vee\bigr)\end{aligned}
\]
for \(n\geq1\). So the obstruction lies in \(\operatorname{Ext}^3_{B/S}\), the classes form a torsor under \(\operatorname{Ext}^2_{B/S}\) and the automorphisms form \(\operatorname{Ext}^1_{B/S}\).

**Proof.** The ideal \(I\) of \(S'\) is quasi-coherent, so Theorem 4.3 of the lesson on deformations of diagrams applies to \(f\), which has the type \(\Delta^{\mathrm{op}}\to\{\ast\}\), and to the extension \(S'\) of \(S\). By Proposition 1.2 (3), its deformations of \(f\) over \(S'\) and their isomorphisms are those of \(G\). This gives (1) and (2), and (3) is Theorem 4.3 (3) there with \(Z=T\). (4) is Corollary 3.5 (3) of the lesson on the adjoint co-Lie complex. \(\square\)

**Proposition 2.2 (the levels of the obstruction).** For every \(n\geq1\), restriction to the level \(n\) carries \(\omega(G,i)\) to the obstruction \(\omega(f_n,S')\in\operatorname{Ext}^2(L_{G^n/S},\pi_n^*I)\) of Theorem 4.3 of the lesson on deformations of diagrams for the morphism of schemes \(f_n=\pi_n\colon G^n\to S\). In particular, the restriction of \(\omega(G,i)\) to the level \(1\) vanishes if and only if \(G\) has a deformation over \(S'\) as a flat scheme, that is, a flat \(S'\)-scheme \(X'\) with \(X'\times_{S'}S\cong G\).

**Proof.** Restriction to the level \(n\) is the inverse image of a morphism of ringed topoi. It is exact and preserves the polynomial constructions, so it carries the rings \(\mathcal A'\), \(\mathcal A\), \(\mathcal B\) of Construction 4.2 there, the transitivity triangle of \(\mathcal A'\to\mathcal A\to\mathcal B\) (Proposition 1.3 there) and the derived tensor product \(D\) to those of the level \(n\); it carries the map \(\tilde c\colon D\to f^*I[1]\), which induces the identity on \(H^{-1}\), to the corresponding map at the level \(n\). So it carries \(\omega(f,S')\) to \(\omega(f_n,S')\). By Theorem 4.3 there, \(\omega(f_1,S')=0\) exactly when \(G\to S\) has a deformation over \(S'\), and by Lemma 1.1 these deformations are the flat \(S'\)-schemes \(X'\) with \(X'\times_{S'}S\cong G\). \(\square\)

**Corollary 2.3 (étale groups).** If \(G\) is étale over \(S\), it has a deformation over \(S'\), unique up to unique isomorphism.

**Proof.** By Exercise 4.3 of the lesson on the adjoint co-Lie complex, \(L_{N/S}=0\), so all the groups of Theorem 2.1 vanish. \(\square\)

**Proposition 2.4 (smooth affine groups over affine bases).** Let \(S\) be affine and \(G\) smooth and affine over \(S\), with \(\operatorname{Lie}(G)=\mathcal Hom(\omega_G,\mathcal O_S)\). For \(n\geq1\) put
\[
C^n=\Gamma\bigl(G^n,\pi_n^*(\operatorname{Lie}(G)\otimes I)\bigr),
\]
the cochain complex \(C^{\geq1}\) having the differentials of the row \(E_1^{\bullet,0}\) of the spectral sequence of Theorem 2.1 (4). Then
\[
\operatorname{Ext}^j(L_{N/S},f^*I)\cong H^{j+1}(C^{\geq1})
\]
for all \(j\). So the obstruction to deforming \(G\) lies in \(H^3(C^{\geq1})\), the deformations form a torsor under \(H^2(C^{\geq1})\), and the automorphism group of a deformation is \(H^1(C^{\geq1})=\ker(C^1\to C^2)\).

**Proof.** By Proposition 1.5 of the lesson on the co-Lie complex, \(\ell_G\cong\omega_G[0]\) with \(\omega_G\) finite locally free, so \((\pi_n^*\omega_G)^\vee=\pi_n^*\operatorname{Lie}(G)\), and \(\operatorname{Lie}(G)\otimes I=\mathcal Hom(\omega_G,I)\). The terms of Theorem 2.1 (4) are \(E_1^{n,k}=H^k(G^n,\pi_n^*(\operatorname{Lie}(G)\otimes I))\) [Tag 08DQ]. As \(G^n\) is affine and the module is quasi-coherent, they vanish for \(k\geq1\) [Tag 01XB], and \(E_1^{n,0}=C^n\). So the spectral sequence has a single nonzero row, it degenerates at \(E_2\), and its abutment in degree \(j+1\) is \(E_2^{j+1,0}=H^{j+1}(C^{\geq1})\). In degree \(1\) there is no \(C^0\), so \(H^1(C^{\geq1})\) is the kernel of \(C^1\to C^2\). \(\square\)

## 3. Homomorphisms

**Theorem 3.1 (extending homomorphisms).** Let \((G',\iota_G)\) and \((H',\iota_H)\) be deformations of \(G\) and \(H\) over \(S'\), and \(u\colon G\to H\) a homomorphism.

1. There is a class \(\omega(u,G',H')\in\) \(\operatorname{Ext}^1(LN(u)^*L_{N(H)/S},f_G^*I)\) whose vanishing is necessary and sufficient for the existence of an extension \(u'\colon G'\to H'\) of \(u\).
2. If \(\omega(u,G',H')=0\), the extensions of \(u\) form a torsor under \(\operatorname{Ext}^0(LN(u)^*L_{N(H)/S},f_G^*I)\).
3. For every \(j\), \(\operatorname{Ext}^j(LN(u)^*L_{N(H)/S},f_G^*I)\cong\) \(\operatorname{Ext}^{j+1}_{B_G/S}(u^*\ell^{\mathrm{ad}}_H,\varepsilon^*I)\), where \(u^*\ell^{\mathrm{ad}}_H\in D^b(B_G)\) is the object, unique up to unique isomorphism, whose nerve is \(L\varepsilon^*LN(u)^*\ell_{P_H/N(H)}\): the restriction of \(\ell^{\mathrm{ad}}_H\) along \(u\) (Proposition 2.5 of the lesson on the adjoint co-Lie complex). These groups are the abutment of a spectral sequence with \(E_1^{0,k}=0\) and \(E_1^{n,k}=\operatorname{Ext}^k_{G^n}(\pi_n^*\ell_H,\pi_n^*I)\) for \(n\geq1\).

**Proof.** (1), (2). By Proposition 1.3 (1), the extensions of \(u\) are the morphisms of simplicial \(S'\)-schemes \(N(G')\to N(H')\) whose composite with \(N(\iota_G)\) is \(N(\iota_H)N(u)\). Work in \(\operatorname{Top}(N(G)_{\mathrm{Zar}})\), with the rings
\[
\begin{gathered}\mathcal Z =f_G^{-1}\mathcal O_{S'},\quad\mathcal A\\ =N(u)^{-1}\mathcal O_{N(H)},\\ \mathcal B =\mathcal O_{N(G)},\quad\mathcal E\\ =N(u)^{-1}\mathcal O_{N(H')},\quad\mathcal E'\\ =\mathcal O_{N(G')},\end{gathered}
\]
all on the spaces of \(N(G)\), the thickenings being homeomorphisms. Then \(\mathcal E\) is an extension of \(\mathcal Z\)-algebras of \(\mathcal A\) by \(N=N(u)^{-1}f_H^*I\), and \(\mathcal E'\) one of \(\mathcal B\) by \(M=f_G^*I\); let \(c\colon N\to M\) be the canonical map. As in the proof of Proposition 5.2 of [*Equivariant extensions and deformations*](equivariant-extensions-and-deformations.md), the morphisms of the statement are the lifts of \(N(u)^\sharp\colon\mathcal A\to\mathcal B\) in the sense of Lemma 5.1 there: a map of rings over the identity of the spaces is a morphism of locally ringed spaces at every level, units being detected modulo the nilpotent ideals. Such a map of \(\mathcal Z\)-algebras induces \(c\) on the kernels automatically, the kernels being generated by the image of \(I\). By Lemma 5.1 (3) there, a lift exists if and only if
\[
o=\kappa(\mathcal E')\circ\rho-c\circ\kappa(\mathcal E)=0
\]
in \(\operatorname{Ext}^1_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)\), and the lifts then form a torsor under \(\operatorname{Ext}^0_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)\). Here \(\operatorname{Ext}^j_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)=\) \(\operatorname{Ext}^j(LN(u)^*L_{N(H)/S'},M)\), as \(N(u)^{-1}\) preserves the polynomial constructions and \(L_{\mathcal A/\mathcal Z}\otimes^{\mathbf L}_{\mathcal A}\mathcal B=LN(u)^*L_{N(H)/S'}\) (proof of Proposition 1.3 of the lesson on deformations of diagrams).

It remains to replace \(S'\) by \(S\). Apply \(LN(u)^*\) to the transitivity triangle of \(N(H)\to S\to S'\):
\[
\begin{aligned}&\quad LN(u)^*f_H^*L_{S/S'}\\ &\quad \xrightarrow{\ \rho'\ }LN(u)^*L_{N(H)/S'}\\ &\quad \quad \xrightarrow{\ w\ }LN(u)^*L_{N(H)/S}\\ &\quad \to LN(u)^*f_H^*L_{S/S'}[1].\end{aligned}
\]
As \(L_{S/S'}\) lies in \(D^{\leq-1}\), \(\operatorname{Hom}(LN(u)^*f_H^*L_{S/S'},M)=0\) [Tag 06XS], so the long exact sequence [Tag 0149] shows that composition with \(w\) is an isomorphism on \(\operatorname{Ext}^0(-,M)\) and, on \(\operatorname{Ext}^1(-,M)\), injective with image the classes killed by composition with \(\rho'\). We show \(o\circ\rho'=0\). Let \(\mathcal O=f_G^{-1}\mathcal O_S\), the ring map \(\mathcal Z\to\mathcal O\) having the kernel \(f_G^{-1}I\). The structure map \(\mathcal Z\to\mathcal E\) gives a map of extensions from \(\mathcal Z\) to the pullback \(\mathcal E\times_{\mathcal A}\mathcal O\), with the canonical coefficient map \(c_N\colon f_G^{-1}I\to N\), so \(\mathcal E\times_{\mathcal A}\mathcal O\cong(c_N)_*\mathcal Z\) by Lemma 5.1 (1) there. By Lemma 5.1 (2) there, \(\kappa(\mathcal E)\circ\rho'_0=c_N\circ\kappa(\mathcal Z)\), where \(\rho'_0\colon L_{\mathcal O/\mathcal Z}\to L_{\mathcal A/\mathcal Z}\) is the functoriality map; and \(\kappa(\mathcal Z)\colon L_{\mathcal O/\mathcal Z}\to f_G^{-1}I[1]\) is the conormal truncation \(t\), by the definition (10.6) of the class. In the same way \(\kappa(\mathcal E')\circ\rho\circ\rho'_0=c_M\circ t\), with \(c_M=c\circ c_N\) the canonical map to \(M\). Hence \(o\circ\rho'_0=c_Mt-cc_Nt=0\). After \(-\otimes^{\mathbf L}\mathcal B\), \(\rho'_0\) becomes \(\rho'\), so \(o\circ\rho'=0\). Let \(\omega(u,G',H')\) be the class with \(\omega\circ w=o\). Then (1) and (2) hold.

(3) Apply \(LN(u)^*\) to the triangle (3.1) of the lesson on the adjoint co-Lie complex for \(H\). The functor \(N(u)^*\) carries \(i_{0!}\) for \(N(H)\) to \(i_{0!}\) for \(N(G)\), levelwise, compatibly with the adjunction maps. The object \(LN(u)^*\ell_{P_H/N(H)}\) is quasi-cartesian, with the levels \(L(u^n)^*\pi_n^*\ell_H=\pi_n^*\ell_H\), and its level \(0\) is \(\ell_H\). This gives a distinguished triangle
\[
\begin{aligned}&\quad LN(u)^*L_{N(H)/S} \to i_{0!}\ell_H\\ &\quad \quad \to LN(u)^*\ell_{P_H/N(H)}\\ &\quad \to LN(u)^*L_{N(H)/S}[1]\end{aligned}
\]
whose second map is the adjunction map. As in Proposition 2.2 (2) of the lesson on the adjoint co-Lie complex, \(L\varepsilon^*LN(u)^*\ell_{P_H/N(H)}\) is the nerve of an object \(u^*\ell^{\mathrm{ad}}_H\) of \(D^b(B_G)\), unique up to unique isomorphism (Proposition 1.5 (3) of [*Equivariant cotangent complexes*](equivariant-cotangent-complexes.md)). The proof of Corollary 3.5 (2), (3) there then applies word for word to this triangle, and the spectral sequence is that of Corollary 3.5 (1) there, with the levels \(\pi_n^*L\varepsilon^*\ell_H\) of the nerve of \(u^*\ell^{\mathrm{ad}}_H\). \(\square\)

By Proposition 3.8 of the lesson on the adjoint co-Lie complex, the relative Ext groups of Theorems 2.1 (4) and 3.1 (3) are relative cohomology groups of the classifying topoi with coefficients in the duals: \(\operatorname{Ext}^{j+1}_{B/S}(\ell^{\mathrm{ad}}_G,\varepsilon^*I)\cong\) \(H^{j+1}\bigl(B/S,(\ell^{\mathrm{ad}}_G)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\), and \(\operatorname{Ext}^{j+1}_{B_G/S}(u^*\ell^{\mathrm{ad}}_H,\varepsilon^*I)\cong\) \(H^{j+1}\bigl(B_G/S,(u^*\ell^{\mathrm{ad}}_H)^\vee\otimes^{\mathbf L}\varepsilon^*I\bigr)\), the underlying objects \(L\varepsilon^*\ell_G\) and \(L\varepsilon^*\ell_H\) being perfect (Theorem 3.1 of [*Quasi-coherent and perfect complexes on big fppf sites*](quasi-coherent-and-perfect-complexes-on-big-fppf-sites.md)).

**Proposition 3.2 (the identity).** Let \(G=H\), \(u=\mathrm{id}_G\), and \((G',\iota')\), \((G'',\iota'')\) two deformations of \(G\). Then \(\omega(\mathrm{id}_G,G',G'')\in\operatorname{Ext}^1(L_{N/S},f^*I)\) is the difference of the classes of \(G'\) and \(G''\) in the torsor of Theorem 2.1 (2): the element that carries the class of \(G''\) to that of \(G'\).

**Proof.** In Theorem 4.3 of the lesson on deformations of diagrams, the torsor structure is that of Theorem 10.4 of *The cotangent complex*: the deformations are the extensions of \(\mathcal Z\)-algebras of \(\mathcal B=\mathcal O_N\) by \(M=f^*I\) with the structure map \(c\) on \(f^{-1}I\), their classes \(\kappa\) lie in the fibre over \(c\) of the map \(r\) of (10.9) there, and \(\operatorname{Ext}^1(L_{N/S},M)\) acts through the injective map of (10.9). In the proof of Theorem 3.1, with \(u=\mathrm{id}\), \(\rho=\mathrm{id}\) and \(c=\mathrm{id}\), we have \(o=\kappa(\mathcal O_{N(G')})-\kappa(\mathcal O_{N(G'')})\), and the map \(w\) there is that injective map. So \(\omega\) is the element of \(\operatorname{Ext}^1(L_{N/S},M)\) carrying the class of \(N(G'')\) to that of \(N(G')\), and Proposition 1.2 transfers this to \(G''\) and \(G'\). \(\square\)

In particular \(\mathrm{id}_G\) extends to a homomorphism \(G'\to G''\) exactly when \(G'\) and \(G''\) are isomorphic deformations, as it must, an extension of the identity being an isomorphism of deformations (Lemma 3.3 of the lesson on deformations of diagrams).

**Theorem 3.3 (deforming a homomorphism with a given target).** Let \((H',\iota_H)\) be a deformation of \(H\) over \(S'\), \(u\colon G\to H\) a homomorphism, and \(\omega(u,H')=\) \(\omega(N(u),N(H'))\in\) \(\operatorname{Ext}^2(L_{N(G)/N(H)},f_G^*I)\) the obstruction of Construction 4.2 of the lesson on deformations of diagrams for the morphism \(N(u)\) and the extension \(N(H')\) of \(N(H)\).

1. There is a pair of a deformation \((G',\iota_G)\) of \(G\) and an extension \(u'\colon G'\to H'\) of \(u\) if and only if \(\omega(u,H')=0\).
2. If \(\omega(u,H')=0\), the isomorphism classes of such pairs form a torsor under \(\operatorname{Ext}^1(L_{N(G)/N(H)},f_G^*I)\), and the automorphism group of each pair is isomorphic to \(\operatorname{Ext}^0(L_{N(G)/N(H)},f_G^*I)\).
3. \(\omega(u,H')=\) \(N(u)^*e(N(H'))[1]\circ\) \(K(N(G)/N(H)/T)\), where \(e(N(H'))\in\operatorname{Ext}^1(L_{N(H)/T},f_H^*I)\) is the class of \(N(H')\) as a \(T\)-extension of \(N(H)\).
4. If \(H=S\) is the trivial group, \(N(H)\) is the constant simplicial scheme \(S\), and (1)–(3) are Theorem 2.1 (1)–(3).

**Proof.** The ideal \(f_H^*I\) of \(N(H')\) is termwise quasi-coherent, and \(N(u)^*f_H^*I=f_G^*I\). By Proposition 1.3 (2), the deformations of \(N(u)\) over \(N(H')\), with their isomorphisms, are the pairs of the statement. So (1), (2) and (3) are Theorem 4.3 of the lesson on deformations of diagrams, with \(Z=T\) in (3): \(N(H')\) is a \(T\)-extension of \(N(H)\) through \(N(H')\to S'\to T\). For \(H=S\), \(N(H)=N(S)\) has all levels \(S\) and all maps the identity, \(N(H')\) is the constant simplicial scheme \(S'\), and the morphism \(N(u)\) is \(f_G\). \(\square\)

**Corollary 3.4 (relative cohomology form).** In Theorem 3.3, let \(C_u\in D^b(B_G)\) be the object of Proposition 3.10 (2) of the lesson on the adjoint co-Lie complex for the homomorphism \(u\colon G\to H\), and \(K=C_u^\vee\), the first vertex of a distinguished triangle \(K\to(\ell^{\mathrm{ad}}_G)^\vee\to(u^*\ell^{\mathrm{ad}}_H)^\vee\to K[1]\) whose second map is the dual of the functoriality map \(\lambda_u\). Then \(\operatorname{Ext}^j(L_{N(G)/N(H)},f_G^*I)\cong\) \(H^{j+1}(B_G/S,K\otimes^{\mathbf L}\varepsilon^*I)\) for every \(j\): the obstruction \(\omega(u,H')\) is a class in \(H^3(B_G/S,K\otimes^{\mathbf L}\varepsilon^*I)\), the pairs of Theorem 3.3 (2) form a torsor under \(H^2(B_G/S,K\otimes^{\mathbf L}\varepsilon^*I)\), and the automorphism group of a pair is isomorphic to \(H^1(B_G/S,K\otimes^{\mathbf L}\varepsilon^*I)\).

**Proof.** Corollary 3.12 (2) of the lesson on the adjoint co-Lie complex, with \(f_G^*I\) the cartesian module with the levels \(\pi_n^*I\). \(\square\)

**Proposition 3.5 (two deformations of the target).** Let \((H',\iota')\) and \((H'',\iota'')\) be deformations of \(H\) over \(S'\), and let \(d\in\operatorname{Ext}^1(L_{N(H)/S},f_H^*I)\) be the element that carries the class of \(H''\) to that of \(H'\) in the torsor of Theorem 2.1 (2) for \(H\); by Proposition 3.2, \(d=\omega(\mathrm{id}_H,H',H'')\). Let \(u\colon G\to H\) be a homomorphism, and put \(N(u)^*d=\tau\circ LN(u)^*(d)\colon\) \(LN(u)^*L_{N(H)/S}\to f_G^*I[1]\), where \(\tau\colon LN(u)^*f_H^*I[1]\to f_G^*I[1]\) is the map inducing the identity on \(H^{-1}\).

1. For every deformation \((G',\iota_G)\) of \(G\), \(\omega(u,G',H')-\omega(u,G',H'')=\) \(-N(u)^*d\) in \(\operatorname{Ext}^1(LN(u)^*L_{N(H)/S},f_G^*I)\).
2. \(\omega(u,H')-\omega(u,H'')=\) \((N(u)^*d)[1]\circ\) \(K(N(G)/N(H)/S)\) in \(\operatorname{Ext}^2(L_{N(G)/N(H)},f_G^*I)\): the difference of the obstructions of Theorem 3.3 is the image of \(N(u)^*d\) under the connecting map of the transitivity triangle of \(N(G)\to N(H)\to S\).

**Proof.** *The classes of \(H'\) and \(H''\).* For a deformation \(H'\) of \(H\), let \(\kappa(H')\in\operatorname{Ext}^1(L_{N(H)/S'},f_H^*I)\) be the class of \(N(H')\) as an \(S'\)-extension of \(N(H)\) (Theorem 2.4 of the lesson on deformations of diagrams), the class used in the proof of Proposition 3.2. As there, \(\kappa(H')-\kappa(H'')=d\circ w_H\), where \(w_H\colon L_{N(H)/S'}\to L_{N(H)/S}\) is the functoriality map, the second map of the transitivity triangle of \(N(H)\to S\to S'\). The class \(e(N(H'))\in\operatorname{Ext}^1(L_{N(H)/T},f_H^*I)\) of Theorem 3.3 (3) is the conormal truncation composed with the functoriality map \(L_{N(H)/T}\to L_{\mathcal O_{N(H)}/\mathcal O_{N(H')}}\). This map is the composite of the functoriality map \(\rho_H\colon L_{N(H)/T}\to L_{N(H)/S'}\) with the functoriality map that enters \(\kappa(H')\), functoriality maps being compatible with composition (Theorem 2.2 of *The cotangent complex*). So \(e(N(H'))=\kappa(H')\circ\rho_H\), and
\[
\begin{aligned}&e(N(H'))-e(N(H''))\\&\quad=d\circ w_H\circ\rho_H=d\circ\psi_H,\end{aligned}
\]
where \(\psi_H=w_H\rho_H\colon L_{N(H)/T}\to L_{N(H)/S}\) is the functoriality map.

(1) In the proof of Theorem 3.1 for \(u\), \(G'\) and \(H'\), the class \(o\) is \(\kappa(\mathcal E')\circ\rho-c\circ\kappa(\mathcal E)\), where \(\mathcal E=N(u)^{-1}\mathcal O_{N(H')}\) and \(\mathcal E'=\mathcal O_{N(G')}\). The functor \(N(u)^{-1}\) is exact and preserves the polynomial constructions and the conormal truncation, so \(\kappa(\mathcal E)=N(u)^{-1}\kappa(H')\), and likewise for \(H''\), while \(\mathcal E'\) does not change. Hence the classes \(o\) for \(H'\) and for \(H''\) differ by \(-c\circ N(u)^{-1}(d)\circ N(u)^{-1}(w_H)\). In the identification of \(\operatorname{Ext}^1_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)\) with \(\operatorname{Ext}^1(LN(u)^*L_{N(H)/S'},M)\) made in that proof, a map of \(\mathcal A\)-modules corresponds to its \(\mathcal B\)-linear extension. The \(\mathcal B\)-linear extension of \(c\) is the identity of \(M=f_G^*I\), so \(c\circ N(u)^{-1}(d)\) corresponds to \(N(u)^*d\), and \(N(u)^{-1}(w_H)\) to the map \(w\) of that proof. So the two classes \(o\) differ by \(-N(u)^*d\circ w\). Since \(o=\omega(u,G',H')\circ w\), and composition with \(w\) is injective on \(\operatorname{Ext}^1\), (1) follows.

(2) The map \(e\mapsto N(u)^*e\) of Theorem 4.3 (3) of the lesson on deformations of diagrams is additive. So Theorem 3.3 (3) and the first paragraph give
\[
\begin{aligned}&\omega(u,H')-\omega(u,H'')\\&\quad=N(u)^*(d\circ\psi_H)[1]\circ K_T\\&\quad=(N(u)^*d)[1]\\&\qquad\circ LN(u)^*(\psi_H)[1]\circ K_T,\end{aligned}
\]
with \(K_T=K(N(G)/N(H)/T)\). On \(\operatorname{Top}(N(G)_{\mathrm{Zar}})\), let \(\mathcal Z_T\) and \(\mathcal Z_S\) be the inverse images of \(\mathcal O_T\) and \(\mathcal O_S\), \(\mathcal A=N(u)^{-1}\mathcal O_{N(H)}\) and \(\mathcal B=\mathcal O_{N(G)}\). The map of triples of rings \((\mathcal Z_T,\mathcal A,\mathcal B)\to(\mathcal Z_S,\mathcal A,\mathcal B)\) that is the identity on \(\mathcal A\) and \(\mathcal B\) gives, by the naturality of the transitivity triangle (Proposition 1.3 of the lesson on deformations of diagrams), a morphism from the triangle of \(N(G)\to N(H)\to T\) to that of \(N(G)\to N(H)\to S\). As in the proof of Theorem 4.3 (3) there, its components are \(LN(u)^*(\psi_H)\), the functoriality map \(L_{N(G)/T}\to L_{N(G)/S}\), the identity of \(L_{N(G)/N(H)}\) and \(LN(u)^*(\psi_H)[1]\). Its third square gives \(LN(u)^*(\psi_H)[1]\circ K_T=\) \(K(N(G)/N(H)/S)\). \(\square\)

## 4. Exercises

**4.1.** Let \(k\) be a field, \(S=\operatorname{Spec}k\), \(S'=\operatorname{Spec}k[\epsilon]\) with \(\epsilon^2=0\), and \(G=\mathbf G_a^2=\operatorname{Spec}k[x,y]\). On \(G'=\operatorname{Spec}k[\epsilon][x,y]\) put
\[
\begin{aligned}&(x_1,y_1)(x_2,y_2)\\&\quad=(x_1+x_2,\ y_1+y_2+\epsilon x_1y_2).\end{aligned}
\]
Show that this is a group law, with unit \((0,0)\) and inverse \((x,y)^{-1}=(-x,-y+\epsilon xy)\), that \(G'\) with the obvious \(\iota\) is a deformation of \(G\), and that \(G'\) is not commutative. Deduce that \(\operatorname{Ext}^1(L_{N(G)/S},f^*I)\neq0\) for \(I=k\epsilon\).

**4.2.** In Theorem 3.3, suppose that \(u\) is a closed immersion. Show that \(u'\) is a closed immersion for every pair \((G',u')\), so that Theorem 3.3 classifies the deformations of the closed subgroup scheme \(G\) of \(H\) to closed subgroup schemes of \(H'\).

**4.3.** Let \(G\) be étale over \(S\), and let \((H',\iota_H)\) be a deformation of a group scheme \(H\). Show that the obstruction of Theorem 3.3 lies in \(\operatorname{Ext}^2(L_{N(G)/N(H)},f_G^*I)\cong\) \(\operatorname{Ext}^1(LN(u)^*L_{N(H)/S},f_G^*I)\), and that the pairs of Theorem 3.3 correspond to the extensions of \(u\) to the unique deformation of \(G\) given by Corollary 2.3.

**4.4.** In Proposition 3.5 (1), take \(G=H\), \(u=\mathrm{id}_H\) and \(G'=H''\). Show that \(\omega(\mathrm{id}_H,H'',H')=-d\), and check this against Proposition 3.2.

## 5. Complete solutions

**4.1.** Associativity: both \(((a,b)(c,d))(e,f)\) and \((a,b)((c,d)(e,f))\) have the first coordinate \(a+c+e\) and the second coordinate \(b+d+f+\epsilon(ad+af+cf)\), using \(\epsilon^2=0\). The unit laws are clear, and \((x,y)(-x,-y+\epsilon xy)=\) \((0,\ \epsilon xy-\epsilon xy+\epsilon x(-y+\epsilon xy))\), which is \((0,0)\); likewise \((-x,-y+\epsilon xy)(x,y)=\) \((0,\ \epsilon xy-\epsilon xy)=\) \((0,0)\). The multiplication, unit and inverse are given by polynomials over \(k[\epsilon]\), so \(G'\) is a group scheme over \(S'\), and it is flat, being an affine plane over \(S'\), and of finite presentation. Modulo \(\epsilon\) the law is that of \(\mathbf G_a^2\), so the closed immersion \(\iota\colon G\to G'\) given by \(\epsilon=0\) is compatible with the multiplications and \(G'\times_{S'}S=G\). Finally \((x_1,y_1)(x_2,y_2)-(x_2,y_2)(x_1,y_1)=\) \((0,\epsilon(x_1y_2-x_2y_1))\neq\) \(0\). The trivial deformation \(G\times_SS'\) is commutative, and an isomorphism of deformations is an isomorphism of group schemes, so \(G'\) is not isomorphic to it. So the torsor of Theorem 2.1 (2) has at least two elements, and \(\operatorname{Ext}^1(L_{N(G)/S},f^*I)\neq0\).

**4.2.** A closed immersion is a proper monomorphism and conversely [Tag 04XV]. Apply [Tag 06AG] to the thickenings \(G\subseteq G'\) and \(H\subseteq H'\) over \(S\subseteq S'\): \(G'\) is flat over \(S'\), \(G=G'\times_{S'}S\) and \(H=H'\times_{S'}S\). Since \(u\) is proper and a monomorphism, so is \(u'\), and \(u'\) is a closed immersion.

**4.3.** By the transitivity triangle of \(N(G)\to N(H)\to S\), \(LN(u)^*L_{N(H)/S}\to\) \(L_{N(G)/S}\to\) \(L_{N(G)/N(H)}\to\) \(LN(u)^*L_{N(H)/S}[1]\), with \(L_{N(G)/S}=0\) (Exercise 4.3 of the lesson on the adjoint co-Lie complex), the third map is an isomorphism \(L_{N(G)/N(H)}\cong LN(u)^*L_{N(H)/S}[1]\) [Tag 05QR], which gives the isomorphism of Ext groups. By Corollary 2.3, \(G\) has a deformation \(G'\), unique up to unique isomorphism, and every pair of Theorem 3.3 has \(G'\) as its first member, up to a unique isomorphism. So the pairs correspond to the extensions of \(u\) to \(G'\to H'\), classified by Theorem 3.1.

**4.4.** The identity of \(H''\) is an extension of \(\mathrm{id}_H\) from \(H''\) to \(H''\), so \(\omega(\mathrm{id}_H,H'',H'')=0\) by Theorem 3.1 (1). For \(u=\mathrm{id}_H\), \(N(u)\) is the identity of \(N(H)\), so \(LN(u)^*\) is the identity functor, \(\tau\) is the identity and \(N(u)^*d=d\). Proposition 3.5 (1) with \(G'=H''\) gives \(\omega(\mathrm{id}_H,H'',H')=\) \(\omega(\mathrm{id}_H,H'',H')-\) \(\omega(\mathrm{id}_H,H'',H'')=-d\). By Proposition 3.2, applied to the deformations \(H''\) and \(H'\) in this order, \(\omega(\mathrm{id}_H,H'',H')\) is the element that carries the class of \(H'\) to that of \(H''\). The element \(d\) carries the class of \(H''\) to that of \(H'\), so \(-d\) carries the class of \(H'\) to that of \(H''\), and the group \(\operatorname{Ext}^1(L_{N(H)/S},f_H^*I)\) acts freely; the two descriptions agree.

## Sources

The deformation theory of flat group schemes and of their homomorphisms through the nerve, with the obstructions in relative cohomology of the classifying topos, is due to Illusie, in his work on the cotangent complex. For smooth groups it recovers the theory of Hochschild cohomology of group schemes in the third exposé of the seminar on group schemes of Demazure and Grothendieck.