# Deformations of torsors

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

Let \(G\) be a group scheme over a scheme \(S\), flat and locally of finite presentation, \(X\to Y\) a torsor under \(G\) over an \(S\)-scheme \(Y\), and \(Y\to Y'\) a first-order thickening of \(S\)-schemes with quasi-coherent ideal \(J\). This lesson classifies the torsors under \(G\) over \(Y'\) that restrict to \(X\) over \(Y\). They are exactly the equivariant deformations of the morphism \(X\to Y\) over \(Y'\), so the deformation theory of the nerves applies, and through descent along the torsor its Ext groups become Ext groups on \(Y\) of the co-Lie complex \(\ell_{X/Y}\). There is an obstruction in \(\operatorname{Ext}^2(\ell_{X/Y},J)\); when it vanishes the deformations form a torsor under \(\operatorname{Ext}^1(\ell_{X/Y},J)\), with automorphism groups \(\operatorname{Ext}^0(\ell_{X/Y},J)\); and the obstruction is the product of the class of the thickening with the Atiyah class of the torsor. Since \(\ell_{X/Y}\) is perfect, these groups are cohomology groups of \(Y\) with coefficients in its dual: over an affine \(Y\) every torsor deforms, and for smooth \(G\) the coefficients are the Lie algebra of \(G\) twisted by the adjoint action.

**Prerequisites.** [*Torsors, co-Lie complexes and the Atiyah class*](torsors-co-lie-complexes-and-the-atiyah-class.md) (all of it; its notation is used throughout); [*Equivariant extensions and deformations*](equivariant-extensions-and-deformations.md) (Definitions 2.2 and 3.1, Propositions 2.3 and 3.2, Theorem 3.3); [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md) (Definitions 2.1, 3.1 and 4.1, Theorem 2.4, Lemmas 3.2–3.4, Remark 3.5, Construction 4.2, Theorem 4.3); [*Equivariant cotangent complexes*](equivariant-cotangent-complexes.md) (Lemma 2.2). From the programme lesson [*The cotangent complex*](course:AG-HP/the-cotangent-complex#2-independence-of-the-resolution): Theorems 2.2 and 6.1. From the programme course *Group schemes*: [*Quotients and torsors*](course:AG-GS/AG-GS-04#8-frames-line-bundles-and-hilbert-90), Theorem 8.1. 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)]; thickenings of affine schemes [[Tag 06AD](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-morphisms.html#more-morphisms-lemma-thickening-affine-scheme)]; fpqc coverings [[Tag 022B](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/topologies.html#topologies-definition-fpqc-covering)]; duals of perfect complexes [[Tag 08DQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/cohomology.html#cohomology-lemma-dual-perfect-complex)]; cohomology of quasi-coherent modules and complexes on affine schemes [[Tag 01XB](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/coherent.html#coherent-lemma-quasi-coherent-affine-cohomology-zero), [Tag 06Z0](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/perfect.html#perfect-lemma-affine-compare-bounded)]; maps out of truncations and split triangles [[Tag 06XS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-negative-exts), [Tag 05QT](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-split)].

## Conventions

The notation is that of [*Torsors, co-Lie complexes and the Atiyah class*](torsors-co-lie-complexes-and-the-atiyah-class.md): \(G\) is a group scheme over \(S\), flat and locally of finite presentation, \(f\colon X\to Y\) a \(G_Y\)-torsor, \(a\colon N(G,X)\to Y\) the augmentation, \(\varphi=N(G,f)\), \(q\colon N(G,Y)=N(G)\times_SY\to Y\) the projection, \(\ell_{X/Y}\) the co-Lie complex and \(\mathrm{at}_n\colon\ell_{X/Y}\to t_{[n}L_{Y/S}[1]\), \(n\leq-1\), the Atiyah class.

An *\(S\)-extension of \(Y\) by \(J\)* is a first-order thickening \(j\colon Y\to Y'\) of \(S\)-schemes whose ideal is identified with a quasi-coherent \(\mathcal O_Y\)-module \(J\), that is, an \(S\)-extension of the scheme \(Y\) by \(J\) in the sense of Definition 2.1 of [*Deformations of diagrams of schemes*](deformations-of-diagrams-of-schemes.md); its class is \(e(j)\in\operatorname{Ext}^1(L_{Y/S},J)\) (Theorem 2.4 there). The scheme \(Y'\) carries the trivial action of \(G\). Since \(J[1]\in D^{\geq-1}\), for \(n\leq-1\) the class factors uniquely as \(L_{Y/S}\to t_{[n}L_{Y/S}\xrightarrow{\bar e_n}J[1]\) [Tag 06XS], and we put
\[
\begin{aligned}&e(j)\cdot\mathrm{at}(X/Y/S)\\&\quad =\bar e_n[1]\circ\mathrm{at}_n\\&\quad \in\operatorname{Ext}^2(\ell_{X/Y},J).\end{aligned}
\]
It does not depend on \(n\leq-1\): the \(\mathrm{at}_n\) are compatible with the truncation maps (Theorem 3.3 (3) there), and so are the \(\bar e_n\), by the uniqueness of the factorization.

## 1. Deformations of a torsor are torsors

**Definition 1.1.** A *deformation of \(X\) over \(Y'\)*, or of \(X\) as a \(G_{Y'}\)-torsor, is a \(G_{Y'}\)-torsor \(f'\colon X'\to Y'\) with a morphism of \(G\)-schemes \(i\colon X\to X'\) such that \(f'i=jf\) and the induced morphism \(X\to X'\times_{Y'}Y\) is an isomorphism. An isomorphism of deformations is an isomorphism of \(G_{Y'}\)-torsors \(h\colon X'_1\to X'_2\) with \(hi_1=i_2\).

**Lemma 1.2.** Let \(j\colon Y\to Y'\) be an \(S\)-extension of \(Y\) by \(J\).

1. If \((X',i,f')\) is an equivariant deformation of \(f\) over \(Y'\) (Definition 3.1 of [*Equivariant extensions and deformations*](equivariant-extensions-and-deformations.md), for \(Y'\) as an equivariant \(S\)-extension of \(Y\)), then \(f'\) is a \(G_{Y'}\)-torsor and \((X',i,f')\) is a deformation of \(X\) over \(Y'\).
2. Conversely, every deformation of \(X\) over \(Y'\) is an equivariant deformation of \(f\) over \(Y'\).

So the groupoid of deformations of \(X\) over \(Y'\) is the groupoid of equivariant deformations of \(f\) over \(Y'\). Here \(Y'\), with the trivial action, is an equivariant \(S\)-extension of \(Y\) by the module \(J\) with the trivial structure: condition 2 of Definition 2.2 there holds by its Proposition 2.3 (1), \(Y\to S\) and \(Y'\to S\) being Tor independent of \(G\) since every \(N_n(G)\) is flat over \(S\). Its ideal on \(N(G,Y)\) is \(q^*J\).

**Proof.** (1) Let \(\mathcal I\) be the ideal of \(i\); by definition \(f'\) induces an isomorphism \(f^*J\to\mathcal I\). By Lemma 3.2 (1) of the lesson on deformations of diagrams, the square formed by \(i\), \(j\), \(f\), \(f'\) is cartesian: \(X\cong X'\times_{Y'}Y\).

*Flatness.* Let \(\operatorname{Spec}C'\subseteq Y'\) and \(\operatorname{Spec}A'\subseteq f'^{-1}(\operatorname{Spec}C')\) be affine opens, \(I=J(\operatorname{Spec}C')\), \(C=C'/I\). Then \(A'/IA'\) is the ring of an affine open of \(X\), flat over \(C\) because \(f\) is flat, and \(\operatorname{Tor}_1^{C'}(C,A')=0\) by Lemma 3.2 (2) there. As \(I^2=0\), \(A'\) is flat over \(C'\) [Tag 051C]. So \(f'\) is flat; it is surjective, having the underlying map of \(f\).

*Covering.* Let \(U\) be an affine open of \(Y'\). Then \(U\cap Y\) is an affine open of \(Y\). As \(\{X\to Y\}\) is an fppf, hence fpqc, covering (Proposition 1.3 (1) of the lesson on torsors), \(U\cap Y\) is the union of the images of finitely many affine opens \(W_k\) of \(X\) [Tag 022B]. The open subschemes \(W'_k\) of \(X'\) with the same underlying sets are first-order thickenings of the \(W_k\), hence affine [Tag 06AD], and \(f'(W'_k)=f(W_k)\). So \(\{X'\to Y'\}\) is an fpqc covering, and \(X'\times_{Y'}X'\to X'\) has the diagonal as a section: condition 2 of the definition of a torsor holds.

*Condition 1.* Consider the triangle \(G\times_SX\xrightarrow{\Phi}X\times_YX\xrightarrow{\mathrm{pr}_2}X\), with composite the projection \(\mathrm{pr}\), and \(\Phi(g,x)=(gx,x)\), an isomorphism. The morphisms \(\Phi'\colon G\times_SX'\to X'\times_{Y'}X'\), \((g,x)\mapsto(gx,x)\), \(\mathrm{pr}_2\) and \(\mathrm{pr}\) on \(X'\) extend it along the first-order thickenings \(G\times_SX\to G\times_SX'\), \(X\times_YX=\) \((X'\times_{Y'}X')\times_{Y'}Y\to\) \(X'\times_{Y'}X'\) and \(i\) [Tag 01JU]. The square formed by the two thickenings of \(X\times_YX\) and \(X\) and the maps \(\mathrm{pr}_2\) is cartesian, because \(X'\times_{Y'}X=X'\times_{Y'}Y\times_YX=X\times_YX\). The functor \(\Phi^*\) is faithful on quasi-coherent modules, \(\Phi\) being an isomorphism (Remark 3.5 there). The projection \(G\times_SX'\to X'\) is a deformation of \(\mathrm{pr}\): it is flat, \(G\) being flat over \(S\), so the ideal of \(G\times_SX\) in \(G\times_SX'\) is the pullback of \(\mathcal I\). By Lemma 3.4 (2) there, \(\Phi'\) is a deformation of \(\Phi\), and by Lemma 3.3 there it is an isomorphism. So \(f'\) is a \(G_{Y'}\)-torsor, and \((X',i,f')\) a deformation of \(X\).

(2) Let \((X',i,f')\) be a deformation of \(X\). Since \(X\cong X'\times_{Y'}Y\), \(i\) is the base change of \(j\), a first-order thickening [Tag 01JU], and the square formed by \(i,j,f,f'\) is cartesian. The map \(f^*J\to\mathcal I\) is an isomorphism by Lemma 3.2 (2) of the lesson on deformations of diagrams, the Tor groups vanishing because \(f'\) is flat (Proposition 1.3 (1) of the lesson on torsors). Condition 2 of Definition 2.2 of the lesson on equivariant extensions holds for \(i\) by its Proposition 2.3 (1): \(jf\) and \(f'\) are Tor independent of \(G\), every \(N_n(G,Y')=N_n(G)\times_SY'\) being flat over \(Y'\). So \((X',i,f')\) is an equivariant \(Y'\)-extension of \(X\) by \(f^*J\), an equivariant deformation of \(f\). Isomorphisms of either kind are isomorphisms of \(G\)-schemes over \(Y'\) compatible with \(i\). \(\square\)

## 2. The deformation theorem

**Lemma 2.1.** Let \(j\colon Y\to Y'\) be an \(S\)-extension of \(Y\) by \(J\). Then \(N(G,Y')=N(G)\times_SY'\) is an \(N(G)\)-extension of \(N(G,Y)\) by \(q^*J\), and under the isomorphism \(q^*L_{Y/S}\cong L_{N(G,Y)/N(G)}\) of Lemma 3.1 (3) of the lesson on torsors, its class \(e(N(G,Y'))\) corresponds to \(q^*e(j)\).

**Proof.** At every level \(N_n(G)\times_SY\to N_n(G)\times_SY'\) is the base change of \(j\) along the flat projection to \(Y'\), a first-order thickening with ideal \(q_n^*J\) [Tag 01JU]; these ideals and the structure maps form \(q^*J\). By Theorem 2.4 of the lesson on deformations of diagrams, \(e(N(G,Y'))\) is the functoriality map \(L_{N(G,Y)/N(G)}\to L_{\mathcal O_{N(G,Y)}/\mathcal O_{N(G,Y')}}\) followed by the conormal truncation \(t_N\), and \(e(j)\) is \(L_{Y/S}\to L_{\mathcal O_Y/\mathcal O_{Y'}}\) followed by \(t_Y\). The projections \(q\) give maps of ring pairs from \(q^{-1}\) of those on \(Y\) to those on \(N(G,Y)\). Functoriality maps compose (Theorem 2.2 of *The cotangent complex*), so the functoriality map of \(L_{N(G,Y)/N(G)}\to L_{\mathcal O_{N(G,Y)}/\mathcal O_{N(G,Y')}}\) composed with \(q^*L_{Y/S}\to L_{N(G,Y)/N(G)}\) equals \(q^*L_{Y/S}\to q^*L_{\mathcal O_Y/\mathcal O_{Y'}}\to L_{\mathcal O_{N(G,Y)}/\mathcal O_{N(G,Y')}}\). The two maps \(q^*L_{\mathcal O_Y/\mathcal O_{Y'}}\to q^*J[1]\), \(t_N\circ(\text{functoriality})\) and \(q^*t_Y\), agree: their source lies in \(D^{\leq-1}\), so they are determined by their effect on \(H^{-1}\) [Tag 06XS], which is in both cases the identification of \(q^*J\) with the ideal (Theorem 6.1 of *The cotangent complex*). \(\square\)

**Theorem 2.2 (deformations of torsors).** Let \(f\colon X\to Y\) be a \(G_Y\)-torsor and \(j\colon Y\to Y'\) an \(S\)-extension of \(Y\) by \(J\). Let \(\omega(f,j)\in\operatorname{Ext}^2(\ell_{X/Y},J)\) correspond to the obstruction \(\omega(\varphi,N(G,Y'))\) of Construction 4.2 of the lesson on deformations of diagrams, under the isomorphisms
\[
\begin{aligned}&\operatorname{Ext}^i(\ell_{X/Y},J)\\&\quad\cong\operatorname{Ext}^i(a^*\ell_{X/Y},a^*J)\\&\quad\cong\operatorname{Ext}^i\bigl(L_{N(G,X)/N(G,Y)},\\&\qquad\varphi^*q^*J\bigr)\end{aligned}
\tag{2.1}
\]
given by Theorems 2.5 and 3.3 (1) of the lesson on torsors.

1. A deformation of \(X\) over \(Y'\) exists if and only if \(\omega(f,j)=0\).
2. If \(\omega(f,j)=0\), the isomorphism classes of deformations of \(X\) over \(Y'\) form a torsor under \(\operatorname{Ext}^1(\ell_{X/Y},J)\), and the automorphism group of each deformation is isomorphic to \(\operatorname{Ext}^0(\ell_{X/Y},J)=\) \(\operatorname{Hom}(H^0(\ell_{X/Y}),J)\).
3. \(\omega(f,j)=e(j)\cdot\mathrm{at}(X/Y/S)\).

**Proof.** By Lemma 1.2 and Proposition 3.2 of the lesson on equivariant extensions, the groupoid of deformations of \(X\) over \(Y'\) is equivalent, through \(N(G,-)\), to the groupoid of deformations of \(\varphi\) over the extension \(N(G,Y')\) of \(N(G,Y)\), whose ideal is the termwise quasi-coherent module \(q^*J\) (Lemma 2.1). Theorem 4.3 (1), (2) of the lesson on deformations of diagrams describes these with \(\operatorname{Ext}^i(L_{N(G,X)/N(G,Y)},\varphi^*q^*J)\), \(i=0,1,2\), and (2.1) identifies them with \(\operatorname{Ext}^i(\ell_{X/Y},J)\); this proves (1) and (2). As \(\ell_{X/Y}\in D^{\leq0}\), \(\operatorname{Ext}^0(\ell_{X/Y},J)=\) \(\operatorname{Hom}(H^0(\ell_{X/Y}),J)\) [Tag 06XS].

(3) \(N(G,Y')\) is an \(N(G)\)-extension of \(N(G,Y)\) through \(N(G,Y')\to N(G)\), so Theorem 4.3 (3) there gives
\[
\begin{aligned}&\omega(\varphi,N(G,Y'))\\&\quad =\varphi^*e(N(G,Y'))[1]\circ K,\end{aligned}
\]
with \(K\) the Kodaira–Spencer class of \(N(G,X)\to N(G,Y)\to N(G)\) and \(\varphi^*\) exact, \(\varphi\) being flat. By Lemma 2.1, through the isomorphism \(\psi\colon\) \(a^*L_{Y/S}=\) \(\varphi^*q^*L_{Y/S}\cong\) \(\varphi^*L_{N(G,Y)/N(G)}\), \(\varphi^*e(N(G,Y'))\circ\psi=\) \(a^*e(j)=\) \(a^*\bar e_n\circ a^*(\mathrm{tr})\), where \(\mathrm{tr}\colon L_{Y/S}\to t_{[n}L_{Y/S}\) is the truncation, \(n\leq-1\). By Theorem 3.3 (3) of the lesson on torsors and Lemma 3.2 there, \(a^*(\mathrm{at}_n)=a^*(\mathrm{tr})[1]\circ\psi^{-1}[1]\circ K\). Hence
\[
\begin{aligned}&\omega(\varphi,N(G,Y'))\\ &\quad =a^*\bar e_n[1]\circ a^*(\mathrm{tr})[1]\circ\psi^{-1}[1]\circ K\\ &\quad =a^*\bigl(\bar e_n[1]\circ\mathrm{at}_n\bigr),\end{aligned}
\]
which corresponds to \(e(j)\cdot\mathrm{at}(X/Y/S)\) under (2.1). \(\square\)

**Remark 2.3.** By Proposition 3.5 of the lesson on torsors, \(\omega(f,j)\) is also the obstruction of Theorem 3.3 of the lesson on equivariant extensions for the equivariant deformations of \(f\), written in \(\operatorname{Ext}^2(t_{[}L^G_{X/Y},f_B^*J)\); both are images of \(\omega(\varphi,N(G,Y'))\). Theorem 2.2 expresses everything on \(Y\) itself.

## 3. Duality and smooth groups

**Proposition 3.1.** Let \(f\colon X\to Y\) be a \(G_Y\)-torsor and \(J\) a quasi-coherent \(\mathcal O_Y\)-module, and put \(\ell^\vee=R\mathcal{H}om(\ell_{X/Y},\mathcal O_Y)\).

1. \(\ell^\vee\) is perfect with tor amplitude in \([0,1]\), and \(\operatorname{Ext}^i(\ell_{X/Y},J)\cong H^i(Y,\ell^\vee\otimes^{\mathbf L}J)\) for every \(i\).
2. If \(Y\) is affine, \(\operatorname{Ext}^2(\ell_{X/Y},J)=0\) and \(\operatorname{Ext}^1(\ell_{X/Y},J)=\) \(\Gamma(Y,H^1(\ell^\vee\otimes^{\mathbf L}J))\). So every \(G_Y\)-torsor over an affine \(Y\) has deformations over every \(S\)-extension of \(Y\) by a quasi-coherent module.
3. If \(G\) is smooth over \(S\), \(\ell^\vee\) is the finite locally free module dual to \(\ell_{X/Y}\), and \(\operatorname{Ext}^i(\ell_{X/Y},J)\cong H^i(Y,\ell^\vee\otimes J)\). For a morphism \(V\to Y\) and a section \(s\) of \(X_V\to V\), let \(\tau_s\colon(\ell_{X/Y})_V\cong\omega_G\otimes\mathcal O_V\) be the isomorphism of Proposition 4.1 of the lesson on torsors for the torsor \(X_V\) over \(V\), through Proposition 3.4 there, and \(\sigma_s=\) \((\tau_s^\vee)^{-1}\colon\) \((\ell^\vee)_V\cong\) \(\operatorname{Lie}(G)\otimes_{\mathcal O_S}\mathcal O_V\), \(\operatorname{Lie}(G)=\omega_G^\vee\). For \(g\in G(V)\), \(\sigma_{gs}=c(\operatorname{int}(g))^\vee\circ\sigma_s\), where \(c(\operatorname{int}(g))^\vee\) is the map on tangent vectors at the unit of \(h\mapsto ghg^{-1}\): \(\ell^\vee\) is \(\operatorname{Lie}(G)\) twisted by \(X\) through the adjoint action. If moreover \(Y\) is affine, deformations of \(X\) over \(Y'\) exist, are unique up to isomorphism, and have the automorphism group \(\Gamma(Y,\ell^\vee\otimes J)\).

**Proof.** (1) \(\ell_{X/Y}\) is perfect with tor amplitude in \([-1,0]\) (Theorem 3.3 (1) of the lesson on torsors), so locally a complex of finite free modules in degrees \(-1,0\), whose dual lies in degrees \(0,1\). By [Tag 08DQ], \(\ell^\vee\) is perfect and \(\operatorname{Hom}(\ell_{X/Y},J[i])=\) \(H^0(Y,\ell^\vee\otimes^{\mathbf L}J[i])=\) \(H^i(Y,\ell^\vee\otimes^{\mathbf L}J)\).

(2) \(K=\ell^\vee\otimes^{\mathbf L}J\) has quasi-coherent cohomology in degrees \(0\) and \(1\): locally it is a complex of quasi-coherent modules in these degrees. On the affine \(Y=\operatorname{Spec}A\), \(K\) is represented by a complex \(M\) of \(A\)-modules with cohomology in degrees \(0,1\), and \(H^i(Y,K)=H^i(M)=\Gamma(Y,H^i(K))\) [Tags 06Z0, 01XB], which vanishes for \(i=2\). With (1) and Theorem 2.2 (1) this gives the assertions.

(3) By Proposition 4.1 (1) of the lesson on torsors, \(\ell_{X/Y}\) is finite locally free, so \(\ell^\vee\) is its dual module and \(\ell^\vee\otimes^{\mathbf L}J=\ell^\vee\otimes J\). Dualizing \(\tau_{gs}=c(\operatorname{int}(g^{-1}))\circ\tau_s\) gives \(\sigma_{gs}=\bigl(c(\operatorname{int}(g^{-1}))^{-1}\bigr)^\vee\circ\sigma_s\). Since \(c\) reverses composition and \(\operatorname{int}(g)\circ\operatorname{int}(g^{-1})=\mathrm{id}\), \(c(\operatorname{int}(g^{-1}))\circ c(\operatorname{int}(g))=\mathrm{id}\), so \(c(\operatorname{int}(g^{-1}))^{-1}=c(\operatorname{int}(g))\). The dual of the map on cotangent vectors at the unit is the map on tangent vectors. On an affine \(Y\), \(H^i(Y,\ell^\vee\otimes J)=0\) for \(i>0\) [Tag 01XB], and Theorem 2.2 gives the last assertion. \(\square\)

When \(G\) is smooth, Theorem 2.2 with Proposition 3.1 (3) is the classical description of the deformations of a torsor by the cohomology of the adjoint bundle, obtained by computations with cocycles; the formula \(\omega(f,j)=e(j)\cdot\mathrm{at}(X/Y/S)\) holds for every flat group scheme locally of finite presentation.

## 4. Exercises

**4.1.** Let \(k\) be a field of characteristic \(p>0\), \(S=\operatorname{Spec}k\), \(A'\) a \(k\)-algebra, \(J\subseteq A'\) an ideal with \(J^2=0\), \(A=A'/J\), \(u\in A^\times\), and \(X=\operatorname{Spec}A[t]/(t^p-u)\) the \(\mu_p\)-torsor over \(Y=\operatorname{Spec}A\) of Exercise 5.2 of the lesson on torsors (that exercise assumed \(A\) smooth only for the Atiyah class). Show that \(\ell_{X/Y}\cong A[1]\oplus A\) in \(D(A)\) and compute \(\operatorname{Ext}^i(\ell_{X/Y},J)\) for \(i=0,1,2\). Show directly that for every \(u'\in A'\) lifting \(u\), \(X'_{u'}=\operatorname{Spec}A'[t]/(t^p-u')\) is a deformation of \(X\) over \(Y'=\operatorname{Spec}A'\), that \(X'_{u'}\) and \(X'_{u''}\) are isomorphic as deformations only if \(u'=u''\), and that the automorphism group of \(X'_{u'}\) is \(\{c\in A':c-1\in J\}\cong J\), acting by \(t\mapsto ct\).

**4.2.** Let \(Y\) be a scheme over \(S\) and \(G=\mathbf G_m\). Show that for every \(\mathbf G_m\)-torsor \(X\) over \(Y\), \(\ell_{X/Y}\cong\mathcal O_Y\), and deduce from Theorem 2.2 that \(X\) deforms over an \(S\)-extension \(Y'\) by \(J\) if and only if a class in \(H^2(Y,J)\) vanishes, the deformations forming a torsor under \(H^1(Y,J)\) with automorphism groups \(H^0(Y,J)\). Through Theorem 8.1 of [*Quotients and torsors*](course:AG-GS/AG-GS-04#8-frames-line-bundles-and-hilbert-90), translate this into a statement about invertible modules on \(Y'\) restricting to a given invertible module on \(Y\).

## 5. Complete solutions

**4.1.** By Exercise 5.2 of the lesson on torsors, whose computation of \(\ell_{X/Y}\) does not use the smoothness of \(A\), \(H^{-1}(\ell_{X/Y})\) and \(H^0(\ell_{X/Y})\) are free \(A\)-modules of rank one. On the affine \(Y\), \(\ell_{X/Y}\) corresponds to an object of \(D(A)\) with these cohomology modules [Tag 06Z0]; the triangle \(H^{-1}[1]\to\ell_{X/Y}\to H^0\) has its third map in \(\operatorname{Ext}^2_A(H^0,H^{-1})=0\), \(H^0\) being free, so it splits [Tag 05QT]: \(\ell_{X/Y}\cong A[1]\oplus A\). Hence \(\operatorname{Ext}^i(\ell_{X/Y},J)=\) \(\operatorname{Ext}^{i-1}_A(A,J)\oplus\operatorname{Ext}^i_A(A,J)\), which is \(J\) for \(i=0\), \(J\) for \(i=1\) and \(0\) for \(i=2\).

A lift \(u'\) of \(u\) is a unit, as \(J\) is nilpotent. The computation of Exercise 5.2 of the lesson on torsors, over \(A'\) instead of \(A\), shows that \(X'_{u'}\) is a \(\mu_{p,Y'}\)-torsor, and \(X'_{u'}\times_{Y'}Y=X\); with the morphism \(X\to X'_{u'}\) given by \(t\mapsto t\) it is a deformation. An isomorphism of deformations \(X'_{u'}\to X'_{u''}\) is given by an \(A'\)-algebra isomorphism \(A'[t]/(t^p-u'')\to A'[t]/(t^p-u')\) compatible with the actions, which preserve the grading by \(\deg t=1\) modulo \(p\) (as in Exercise 5.2 there), and reducing to the identity modulo \(J\). It sends \(t\) to an element of degree \(1\), \(ct\) with \(c\in A'\), and \(c\equiv1\) modulo \(J\). Then \(c^p=1\), because \(c=1+x\) with \(x^2=0\) and \(p=0\) in \(A'\), so \(u''=(ct)^p=c^pt^p=u'\). For \(u''=u'\), every \(c\) with \(c-1\in J\) gives an automorphism, \(c\mapsto c-1\) is an isomorphism of the group \(1+J\) onto \(J\), since \((1+x)(1+y)=1+x+y\). This agrees with Theorem 2.2: no obstruction, automorphism groups \(\operatorname{Ext}^0=J\), and the lifts \(u'\) form a torsor under \(J\), as the deformations do under \(\operatorname{Ext}^1=J\).

**4.2.** \(\mathbf G_m\)-torsors are locally trivial for the Zariski topology (Theorem 8.1 of *Quotients and torsors*, for \(\mathrm{GL}_1\)). \(\mathbf G_m\) is smooth and commutative, so \(c(\operatorname{int}(g^{-1}))=\mathrm{id}\), and by Proposition 4.1 of the lesson on torsors the isomorphisms \(\tau_s\) for local sections \(s\) glue to \(\ell_{X/Y}\cong\omega_{\mathbf G_m}\otimes\mathcal O_Y=\mathcal O_Y\). By Proposition 3.1 (3), \(\operatorname{Ext}^i(\ell_{X/Y},J)=H^i(Y,J)\), and Theorem 2.2 gives the statement. By Theorem 8.1 of *Quotients and torsors*, \(\mathbf G_m\)-torsors over \(Y'\) correspond to invertible \(\mathcal O_{Y'}\)-modules, compatibly with base change. So an invertible module \(\mathcal L\) on \(Y\) extends to an invertible module on \(Y'\) if and only if a class in \(H^2(Y,J)\) vanishes; the extensions, up to isomorphisms restricting to the identity of \(\mathcal L\), form a torsor under \(H^1(Y,J)\), and each has the automorphism group \(H^0(Y,J)\), the automorphisms \(1+x\), \(x\in\Gamma(Y,J)\), of the extended module that restrict to the identity of \(\mathcal L\).

## Sources

The classification of deformations of torsors under flat group schemes locally of finite presentation by the co-Lie complex, with the obstruction as the product of the class of the extension and the Atiyah class, is due to Illusie, in his work on the cotangent complex. For smooth group schemes the description by the cohomology of the adjoint bundle is classical.