Deforming the target of a morphism of diagrams

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

Let \(Z\) be a first-order thickening of a diagram of schemes \(Z_o\), and let \(X_o\xrightarrow{f_o}Y_o\xrightarrow{g_o}Z_o\) be morphisms of diagrams with \(g_o\) flat. Suppose that \(X_o\) has a flat deformation \(X\) over \(Z\). When does \(Y_o\) have a flat deformation \(Y\) over \(Z\) to which \(f_o\) extends? For ringed topoi this is answered in Deforming a morphism together with its target, by relative Ext groups of the arrow topos of \(f_o\). This lesson shows that the answer is the same for diagrams of schemes, and applies it, through nerves, to three problems:

• equivariant deformations: given an equivariant flat deformation \(X'\to Z'\) of a composite \(X\to Y\to Z\) of morphisms of \(G\)-schemes, deform \(Y\) equivariantly over \(Z'\) together with \(X\to Y\);
• group schemes: given a homomorphism \(u\colon F\to G\) of flat group schemes and a deformation \(F'\) of \(F\), deform \(G\) together with an extension \(F'\to G'\) of \(u\);
• module schemes: the same for module schemes over a ring scheme, where the groups are Ext groups over the ring scheme of the cone of the homomorphism with coefficients in the Lie complex of the target, and where a given deformation of \(G\) reduces the obstruction to the obstruction to extending \(u\), modulo the choice of that deformation.

In each case there is an obstruction in a relative \(\operatorname{Ext}^2\) of the nerves; when it vanishes the solutions form a torsor under the relative \(\operatorname{Ext}^1\), and the automorphisms of a solution form the relative \(\operatorname{Ext}^0\).

Prerequisites. Deforming a morphism together with its target (1.3, Definitions 1.1 and 1.4, Lemmas 1.2 and 6.1, Theorems 3.2 and 4.1, Propositions 5.1 and 5.2); The arrow topos and relative Ext groups (Definition 3.1, Proposition 3.2, Exercise 5.3); Deformations of diagrams of schemes (Conventions, Definitions 1.1, 2.1, 3.1 and 4.1, Construction 2.2, Proposition 2.3); Equivariant cotangent complexes (Lemmas 1.3 and 3.1, Proposition 3.2 (1)); Equivariant extensions and deformations (Lemma 2.1, Definitions 2.2 and 3.1, Proposition 2.3); Deformations of flat group schemes and of homomorphisms (Conventions, Definition 0.1, Lemma 1.1, Propositions 1.2 and 1.3); Deformations of module schemes (Conventions, Definition 0.1, Lemma 1.1, Proposition 1.2, Theorem 2.2); Lie complexes of module schemes (Section 4: Theorem 4.3 and Corollary 4.4); Quasi-coherent and perfect complexes on big fppf sites (Lemma 4.2). From the programme lesson The cotangent complex: Proposition 5.1 and Theorem 10.4. From the AI-integrated Stacks project: first-order thickenings and base change [Stacks, Tag 01JU]; properties of morphisms over a thickening [Tag 06AG]; closed immersions as proper monomorphisms [Tag 04XV]; composition of derived pullbacks [Tag 0D6D]; the third isomorphism of triangles [Tag 014A].

Conventions

Diagrams of schemes, morphisms of diagrams of a given type, their Zariski topoi \(\operatorname{Top}(X_{\mathrm{Zar}})\), termwise quasi-coherent and cartesian modules, and the cotangent complex \(L_{X/Y}\) of a morphism of diagrams are those of the lesson on deformations of diagrams. A morphism of diagrams \(f\colon X\to Y\) is a morphism of ringed topoi \(\operatorname{Top}(X_{\mathrm{Zar}})\to\operatorname{Top}(Y_{\mathrm{Zar}})\) (Conventions there), and \(L_{X/Y}\) is the cotangent complex of this morphism of ringed topoi. For a morphism of diagrams \(f\colon X\to Y\), a module \(F\) on \(Y\) with \(Lf^*F\) bounded below and \(M\in D^-(Y_{\mathrm{Zar}})\), \(\operatorname{Ext}^i(Y/X;M,F)\) is the relative Ext group of Definition 3.1 of the arrow topos lesson for the morphism of ringed topoi \(\operatorname{Top}(X_{\mathrm{Zar}})\to\operatorname{Top}(Y_{\mathrm{Zar}})\); in the flat problems below \(Lf_o^*(g_o^*K)=h_o^*K\), by the proof of Theorem 3.2 (1) of the lesson on deforming a morphism together with its target. A morphism of diagrams is flat if all its components are flat. A module on a Zariski topos of a diagram is flat over a ring of that topos if and only if all its levels are, since tensor products and exactness are computed levelwise.

1. Diagrams of schemes

1.1. The data. Let \(Z_o\colon\Lambda_Z\to(\mathrm{Sch})\) be a diagram and \((Z,k)\) an extension of \(Z_o\) (Definition 2.1 of the lesson on deformations of diagrams) with a termwise quasi-coherent ideal \(K\), so that \(k\colon Z_o\to Z\) is a \(\Lambda_Z\)-morphism whose components are first-order thickenings. Let \(f_o\colon X_o\to Y_o\) be a morphism of diagrams of type \(u\colon\Lambda_X\to\Lambda_Y\), \(g_o\colon Y_o\to Z_o\) one of type \(v\colon\Lambda_Y\to\Lambda_Z\), and \(h_o=g_of_o\), of type \(vu\). Let \(X\) be a \(Z\)-extension of \(X_o\) for \(kh_o\), with the \(\Lambda_X\)-morphism \(i\colon X_o\to X\), the morphism \(h\colon X\to Z\) of type \(vu\), the ideal \(\mathcal I\) and the map \(w\colon h_o^*K\to\mathcal I\) induced by \(h\).

The underlying topoi of \(Z\) and \(Z_o\), and of \(X\) and \(X_o\), are identified, the spaces of the levels being identified. So \(\operatorname{Top}(Z_{\mathrm{Zar}})\) is a square-zero thickening of the ringed topos \(\operatorname{Top}(Z_{o,\mathrm{Zar}})\) with kernel \(K\), and \(\operatorname{Top}(X_{\mathrm{Zar}})\) is a \(Z\)-extension of \(\operatorname{Top}(X_{o,\mathrm{Zar}})\) by \(\mathcal I\) in the sense of Definition 1.1 of the lesson on deforming a morphism together with its target (Construction 2.2 of the lesson on deformations of diagrams). These ringed topoi and the morphisms of ringed topoi defined by \(f_o\), \(g_o\), \(k\), \(i\) and \(h\) form the data 1.3 of that lesson, with \(S=Z\).

Definition 1.2. Suppose that \(g_o\) is flat. The diagram \(X\) is a flat deformation of \(X_o\) over \(Z\) if \(h\) is flat and \(w\) is an isomorphism. A solution of the flat problem is a \(Z\)-extension \((Y,j,g)\) of \(Y_o\) for \(kg_o\), with \(g\) flat and the map \(v_Y\colon g_o^*K\to\mathcal J\) to its ideal \(\mathcal J\) an isomorphism, together with a morphism of diagrams \(f\colon X\to Y\) of type \(u\) such that \(f\circ i=j\circ f_o\) and \(g\circ f=h\). An isomorphism of solutions \((Y,f)\to(Y',f')\) is an isomorphism \(\theta\colon Y\to Y'\) of \(Z\)-extensions with \(\theta\circ f=f'\).

In a solution, \(g\) is a deformation of \(g_o\) over \(Z\) in the sense of Definition 4.1 of the lesson on deformations of diagrams, its ideal map \(v_Y\) being an isomorphism, and so is \(f\) over \(Y\): its ideal map \(f_o^*\mathcal J\to\mathcal I\) is \(w\circ f_o^*(v_Y)^{-1}\).

Lemma 1.3. In 1.1, suppose that \(g_o\) is flat and that \(X\) is a flat deformation of \(X_o\) over \(Z\).

1. A solution \((Y,f)\) of Definition 1.2 gives a solution of the flat problem for the ringed topoi of 1.1 (Definition 1.4 of the lesson on deforming a morphism together with its target): the ringed topos \(\operatorname{Top}(Y_{\mathrm{Zar}})\), whose underlying topos is that of \(Y_o\), with the morphism of ringed topoi defined by \(f\).
2. Every solution of that flat problem for ringed topoi is isomorphic to the one given by a solution of Definition 1.2, and isomorphisms of solutions correspond. So isomorphism classes and automorphism groups of solutions are the same in both senses.

Proof. (1) By Construction 2.2 of the lesson on deformations of diagrams, \(\mathcal O_Y\) is an extension of \(g_o^{-1}\mathcal O_Z\)-algebras of \(\mathcal O_{Y_o}\) by \(\mathcal J\) on \(\operatorname{Top}(Y_{o,\mathrm{Zar}})\), and the ring maps \(f_\lambda^\sharp\) form a ring map \(f_o^{-1}\mathcal O_Y\to\mathcal O_X\); this is a morphism of ringed topoi with the underlying morphism of \(f_o\). The equalities \(f\circ i=j\circ f_o\) and \(g\circ f=h\) hold levelwise, hence for the ringed topoi. The ring \(\mathcal O_Y\) is flat over \(g_o^{-1}\mathcal O_Z\), every level of \(g\) being flat, and \(v_Y\) is an isomorphism.

(2) Let \((\mathcal E,\phi)\) be a solution of the flat problem for ringed topoi: an extension \(\mathcal E\) of \(g_o^{-1}\mathcal O_Z\)-algebras of \(\mathcal O_{Y_o}\) by \(g_o^*K\), flat over \(g_o^{-1}\mathcal O_Z\), and a ring map \(\phi\colon f_o^{-1}\mathcal E\to\mathcal O_X\) with the stated compatibilities. As \(g_o^*K\) is termwise quasi-coherent, Proposition 2.3 (2) of the lesson on deformations of diagrams, for the morphism \(kg_o\colon Y_o\to Z\), gives a \(Z\)-extension \((Y,j,g)\) of \(Y_o\) by \(g_o^*K\) with \(\mathcal O_Y=\mathcal E\); every level of \(g\) is flat, the levels of \(\mathcal E\) being flat. For an object \(\lambda\) of \(\Lambda_X\), the level \(\lambda\) of \(\phi\) is a ring map \(f_{o,\lambda}^{-1}\mathcal O_{Y_{u(\lambda)}}\to\mathcal O_{X_\lambda}\), which defines a morphism of ringed spaces \(f_\lambda\colon X_\lambda\to Y_{u(\lambda)}\) with the underlying map of \(f_{o,\lambda}\). It is a morphism of locally ringed spaces, hence of schemes: if a section \(s\) of \(f_{o,\lambda}^{-1}\mathcal O_{Y_{u(\lambda)}}\) has an invertible image, its image in \(\mathcal O_{X_{o,\lambda}}\) is \(f_{o,\lambda}^\sharp(\bar s)\), with \(\bar s\) the image of \(s\) in \(f_{o,\lambda}^{-1}\mathcal O_{Y_{o,u(\lambda)}}\); so \(\bar s\) is invertible, \(f_{o,\lambda}\) being a morphism of schemes, and \(s\) is invertible by Lemma 6.1 (1) of the lesson on deforming a morphism together with its target, the kernel of \(f_{o,\lambda}^{-1}\mathcal O_{Y_{u(\lambda)}}\to f_{o,\lambda}^{-1}\mathcal O_{Y_{o,u(\lambda)}}\) having square zero. The compatibility of \(\phi\) with the structure maps says that \(f_\mu X_\alpha=Y_{u(\alpha)}f_\lambda\) for \(\alpha\colon\lambda\to\mu\), so the \(f_\lambda\) form a morphism of diagrams \(f\) of type \(u\), and \(f\circ i=j\circ f_o\), \(g\circ f=h\) hold because they hold for the ringed topoi. By Proposition 2.3 (1) there, isomorphisms of \(Z\)-extensions of \(Y_o\) by \(g_o^*K\) are the isomorphisms of the algebra extensions, and an isomorphism of diagrams is compatible with \(f\) exactly when its ring map is compatible with \(\phi\). \(\square\)

Theorem 1.4. In 1.1, suppose that \(g_o\) is flat and that \(X\) is a flat deformation of \(X_o\) over \(Z\).

1. There is a class \(\omega\in\operatorname{Ext}^2(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\) that vanishes if and only if the flat problem of Definition 1.2 has a solution.
2. If \(\omega=0\), the isomorphism classes of solutions form a torsor under \(\operatorname{Ext}^1(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\).
3. The automorphism group of a solution is isomorphic to \(\operatorname{Ext}^0(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\), the group of derivations of \(\mathcal O_{Y_o}\) over \(g_o^{-1}\mathcal O_{Z_o}\) into \(g_o^*K\), on \(\operatorname{Top}(Y_{o,\mathrm{Zar}})\), whose pullback to \(X_o\) vanishes.

The class \(\omega\) is the obstruction of Theorem 3.2 of the lesson on deforming a morphism together with its target for the ringed topoi of 1.1; so its description as a product (Theorem 4.1 there), its restriction to \(Y_o\) (Proposition 5.1 there) and its description through a given flat deformation of \(Y_o\) (Proposition 5.2 there) apply.

Proof. The morphism of ringed topoi defined by \(g_o\) is flat, and \(\operatorname{Top}(X_{\mathrm{Zar}})\) is a flat deformation of \(\operatorname{Top}(X_{o,\mathrm{Zar}})\) over \(\operatorname{Top}(Z_{\mathrm{Zar}})\) in the sense of Definition 1.1 there, \(h\) being flat and \(w\) an isomorphism. The cotangent complex of the morphism of diagrams \(g_o\) is that of the morphism of ringed topoi (Definition 1.1 of the lesson on deformations of diagrams). Theorem 3.2 there gives the three statements for the solutions of the flat problem for ringed topoi, and Lemma 1.3 identifies those with the solutions of Definition 1.2, with their isomorphisms. \(\square\)

2. Equivariant deformations of the target

Let \(G\) be a category scheme over \(S\), with the notation of the lesson on equivariant extensions and deformations: \(N(G,X)\) is the nerve of the action category of a \(G\)-scheme \(X\), a diagram of type \(\Delta^{\mathrm{op}}\), and a morphism of \(G\)-schemes \(f\) gives the \(\Delta^{\mathrm{op}}\)-morphism \(N(G,f)\). For a module \(M\) of \(B[G,X]^{\mathrm o}\) whose underlying module is quasi-coherent, \(\underline M\) is the termwise quasi-coherent module of \(\operatorname{Top}(N(G,X)_{\mathrm{Zar}})\) of Proposition 3.2 (1) of the lesson on equivariant cotangent complexes, and for a morphism of \(G\)-schemes \(g\colon Y\to Z\) the module belonging to \(g_B^*M\) (Lemma 3.1 there) is \(N(G,g)^*\underline M\).

2.1. The data. Let \(X\xrightarrow{f}Y\xrightarrow{g}Z\to T\) be morphisms of \(G\)-schemes, \(h=gf\), \((Z',k,z')\) an equivariant \(T\)-extension of \(Z\) by a module \(K\) of \(B[G,Z]^{\mathrm o}\) whose underlying module is quasi-coherent (Definition 2.2 of that lesson), and \((X',i,h')\) an equivariant deformation of \(h\) over \(Z'\) (Definition 3.1 there). An equivariant solution is a triple of an equivariant \(Z'\)-extension \((Y',j,g')\) of \(Y\) by \(g_B^*K\), for \(kg\), and a morphism of \(G\)-schemes \(f'\colon X'\to Y'\) with \(f'\circ i=j\circ f\) and \(g'\circ f'=h'\). An isomorphism of equivariant solutions is an isomorphism of equivariant \(Z'\)-extensions compatible with the morphisms from \(X'\).

Theorem 2.2. In 2.1, suppose that \(g\) and \(h'\) are flat. Put \(\Phi=N(G,f)\colon N(G,X)\to N(G,Y)\) and \[ \begin{aligned}E^i&=\operatorname{Ext}^i\bigl(N(G,Y)/N(G,X);\\&\qquad L_{N(G,Y)/N(G,Z)},N(G,g)^*\underline K\bigr).\end{aligned} \]

1. There is a class \(\omega\in E^2\) that vanishes if and only if an equivariant solution exists.
2. If \(\omega=0\), the isomorphism classes of equivariant solutions form a torsor under \(E^1\), and the automorphism group of an equivariant solution is isomorphic to \(E^0\).
3. In every equivariant solution \(g'\) is flat, \(f'\) is an equivariant deformation of \(f\) over \(Y'\), and \(g'\) is a deformation of \(g\) over \(Z'\).

Proof. By Proposition 2.3 (2) of the lesson on equivariant extensions, \(N(G,Z')\), with \(N(G,k)\), is an extension of the diagram \(N(G,Z)\) by \(\underline K\). By Proposition 3.2 (1) there, \((N(G,X'),N(G,i),N(G,h'))\) is a deformation of \(N(G,h)\) over \(N(G,Z')\): an \(N(G,Z')\)-extension of \(N(G,X)\) for \(N(G,kh)\) whose ideal map \(w\colon N(G,h)^*\underline K\to\mathcal I\) is an isomorphism. By Lemma 1.3 of the lesson on equivariant cotangent complexes, \(N_n(G,X')=N_n(G)\times_{G_0}X'\), which is \(N_n(G,Z')\times_{Z'}X'\), this fibre product being formed with the first vertex of \(N_n(G,Z')\); so the level \(n\) of \(N(G,h')\) is a base change of \(h'\), and \(N(G,h')\) is flat. In the same way \(N(G,g)\) is flat. So the data \(\Phi\), \(N(G,g)\), \(N(G,k)\), \(N(G,X')\) satisfy the hypotheses of Theorem 1.4, which gives a class \(\omega\) and the torsor and automorphism statements for the solutions of Definition 1.2.

It remains to identify these solutions with the equivariant ones. The nerve of an equivariant solution \((Y',j,g',f')\) is a solution of Definition 1.2: \(N(G,Y')\) is an \(N(G,Z')\)-extension of \(N(G,Y)\) by \(N(G,g)^*\underline K\) (Proposition 2.3 (2) of the lesson on equivariant extensions, for \(kg\)), each \(N_n(G,g')\) is flat, being a base change of \(g'\) as above once \(g'\) is flat, and \(N(G,f')\) satisfies the two equalities. Conversely, let \((\hat Y,\hat j,\hat g,\hat f)\) be a solution of Definition 1.2 for the nerves. By that Proposition 2.3 (2), \((\hat Y,\hat j,\hat g)\) is isomorphic to \(N(G,-)\) of an equivariant \(Z'\)-extension \((Y',j,g')\) of \(Y\) by \(g_B^*K\), unique up to unique isomorphism; \(g'\), the level \(0\) of \(\hat g\), is flat. Then \(\hat f\) is a morphism of simplicial schemes \(N(G,X')\to N(G,Y')\) over \(N(G,Z')\), hence over \(N(G)\), so \(\hat f=N(G,f')\) for a unique morphism of \(G\)-schemes \(f'\colon X'\to Y'\) (Lemma 2.1 (1) of the lesson on equivariant extensions); the equalities \(f'\circ i=j\circ f\) and \(g'\circ f'=h'\) are those of the level \(0\). Isomorphisms correspond in the same way. This proves (1) and (2), and (3) follows from Definition 1.2: the ideal maps of \(N(G,f')\) and \(N(G,g')\) are isomorphisms, and so are their levels \(0\). \(\square\)

3. Group schemes

With the conventions of the lesson on deformations of flat group schemes: \(S\to T\) is a morphism of schemes and \((S',i,s')\) a \(T\)-extension of \(S\) by a quasi-coherent \(\mathcal O_S\)-module \(I\); group schemes are flat and locally of finite presentation over their base; \(N(G)\) is the nerve of a group scheme \(G\) over \(S\), with \(f_G\colon N(G)\to S\), and \(f_G^*I\) is the cartesian module with the levels \(\pi_n^*I\).

Theorem 3.1. Let \(F'\) be a group scheme over \(S'\), \((F=F'\times_{S'}S,\iota_F)\) its pullback, \(G\) a group scheme over \(S\) and \(u\colon F\to G\) a homomorphism. Call a solution a pair of a deformation \((G',\iota_G)\) of \(G\) over \(S'\) and a homomorphism \(u'\colon F'\to G'\) over \(S'\) with \(u'\iota_F=\iota_Gu\); an isomorphism of solutions is an isomorphism \(\theta\) of deformations of \(G\) with \(\theta u'_1=u'_2\).

Put \(E^i=\operatorname{Ext}^i(N(G)/N(F);L_{N(G)/S},f_G^*I)\).

1. There is a class \(\omega(u,F')\in E^2\) that vanishes if and only if a solution exists.
2. If \(\omega(u,F')=0\), the isomorphism classes of solutions form a torsor under \(E^1\), and the automorphism group of a solution is isomorphic to \(E^0\).

Proof. Apply Theorem 1.4 to the one-object diagram \(Z_o=S\), its extension \(Z=S'\), whose ideal \(I\) is cartesian, \(X_o=N(F)\), \(Y_o=N(G)\), \(f_o=N(u)\), \(g_o=f_G\) and \(X=N(F')\). The levels \(G^n\) are flat over \(S\), so \(f_G\) is flat. By Proposition 1.2 (1) of the lesson on deformations of group schemes, \(N(F')\), with the \(\iota_F^n\) and \(f_{F'}\), is a deformation of \(f_F\) over \(S'\): an \(S'\)-extension of \(N(F)\) whose ideal map \(f_F^*I\to\mathcal I\) is an isomorphism; and \(f_{F'}\) is flat, the \(F'^n\) being flat over \(S'\). So \(N(F')\) is a flat deformation of \(N(F)\) over \(S'\), and Theorem 1.4 gives a class and the torsor and automorphism statements for the solutions of Definition 1.2.

For a solution \((G',\iota_G,u')\), \(N(G')\) is a deformation of \(f_G\) over \(S'\) (Proposition 1.2 (1) there), flat over \(S'\), and \(N(u')\) is a morphism of diagrams with \(N(u')N(\iota_F)=N(\iota_G)N(u)\) and \(f_{G'}N(u')=f_{F'}\); so it is a solution of Definition 1.2. Conversely, in a solution \((Y,j,g,f)\) of Definition 1.2, \(Y\) is a deformation of \(f_G\) over \(S'\), its ideal map being an isomorphism; by Proposition 1.2 (2) and (3) there, it is isomorphic, compatibly with \(j\) and \(g\), to \(N(G')\) for a deformation \((G',\iota_G)\) of \(G\), unique up to unique isomorphism, and by Proposition 1.3 (1) there \(f\) is \(N(u')\) for a unique extension \(u'\colon F'\to G'\) of \(u\). Isomorphisms correspond by the full faithfulness of the nerve functor over \(S'\) (proof of Proposition 1.2 (3) there). \(\square\)

Corollary 3.2. In Theorem 3.1, if \(u\) is faithfully flat, so is \(u'\) for every solution \((G',u')\).

Proof. The morphism \(u'\colon F'\to G'\) of \(S'\)-schemes, with \(F'\) flat over \(S'\), has the base change \(u\) along \(S\to S'\), through \(\iota_F\) and \(\iota_G\). By [Tag 06AG], for the first-order thickening \(S\subseteq S'\), \(u'\) is flat because \(u\) is, and surjective because \(u\) is. \(\square\)

Remark 3.3. By Theorem 4.1 of the lesson on deforming a morphism together with its target, applied to the ringed topoi of the proof, \(\omega(u,F')\) is the product of the Kodaira–Spencer class of \(N(F)N(G)\to N(F)S\to S'\), where \(N(F)S\) is the arrow topos of \(f_F\), with the pullback of the class of its thickening defined by \(f_{F'}\colon N(F')\to S'\). The first depends only on \(u\) and on \(S\subseteq S'\), the second only on \(F'\); in particular \(\omega(u,F')\) is functorial in the data.

4. Module schemes

With the conventions of the lesson on deformations of module schemes: \(A\) is a ring scheme over \(T\) satisfying the Tor independence condition (TI) there, \(N(A,G)\) is the spectral nerve of an \(A_S\)-module scheme \(G\), with the projection \(q_G\colon N(A,G)\to N(A,S)\) and the structure morphism \(p_G\colon N(A,G)\to S\), and module schemes are flat and locally of finite presentation over their base.

Theorem 4.1. Let \(F'\) be an \(A_{S'}\)-module scheme, \((F,\iota_F)\) its pullback to \(S\), \(G\) an \(A_S\)-module scheme and \(u\colon F\to G\) a homomorphism. Call a solution a pair of a deformation \((G',\iota_G)\) of \(G\) over \(S'\) and a homomorphism \(u'\colon F'\to G'\) of \(A_{S'}\)-module schemes with \(u'\iota_F=\iota_Gu\), with isomorphisms as in Theorem 3.1.

Put \[ \begin{aligned}E^i&=\operatorname{Ext}^i\bigl(N(A,G)/N(A,F);\\&\qquad L_{N(A,G)/N(A,S)},p_G^*I\bigr).\end{aligned} \]

1. There is a class \(\omega(u,F')\in E^2\) that vanishes if and only if a solution exists.
2. If \(\omega(u,F')=0\), the isomorphism classes of solutions form a torsor under \(E^1\), and the automorphism group of a solution is isomorphic to \(E^0\).

Proof. Apply Theorem 1.4 to \(Z_o=N(A,S)\), its extension \(Z=N(A,S')\), whose ideal is the cartesian module \(p_S^*I\) (Lemma 1.1 of the lesson on deformations of module schemes), \(X_o=N(A,F)\), \(Y_o=N(A,G)\), \(f_o=N(A,u)\), \(g_o=q_G\) and \(X=N(A,F')\); here \(q_G^*p_S^*I=p_G^*I\). The level \(\lambda\) of \(N(A,G)\) is \(N(A,S)_\lambda\times_SG^P\) for a finite set \(P\), flat over \(N(A,S)_\lambda\) (proof of Proposition 1.2 (1) there), so \(q_G\) is flat; in the same way \(q_{F'}\) is flat, and by Proposition 1.2 (1) there \(N(A,F')\) is a deformation of \(q_F\) over \(N(A,S')\), with an isomorphism as ideal map. So Theorem 1.4 applies. Its solutions are the deformations of \(q_G\) over \(N(A,S')\), flat over \(N(A,S')\), with a morphism from \(N(A,F')\) over \(N(A,S')\) extending \(N(A,u)\). By Proposition 1.2 (2)–(4) there, these are, up to unique isomorphism, the \(N(A,G')\) for deformations \(G'\) of \(G\), with the morphisms \(N(A,u')\) for the extensions \(u'\colon F'\to G'\) of \(u\), and isomorphisms correspond. \(\square\)

Proposition 4.2. In Theorem 4.1, suppose that \(G\) has a deformation \(G'_1\) over \(S'\). Let \(o(G'_1)\) be the class of Proposition 5.2 of the lesson on deforming a morphism together with its target, for the flat deformation \(N(A,G'_1)\) of \(N(A,G)\) over \(N(A,S')\): the obstruction to extending \(N(A,u)\) to a morphism \(N(A,F')\to N(A,G'_1)\) over \(N(A,S')\), that is, by Proposition 1.2 (4) of the lesson on deformations of module schemes, to extending \(u\) to a homomorphism \(F'\to G'_1\).

Write \(L=L_{N(A,G)/N(A,S)}\), \(\Phi=N(A,u)\), and \(w\colon\operatorname{Ext}^1(L,p_G^*I)\to\operatorname{Ext}^1(L\Phi^*L,p_F^*I)\) for the restriction map of the relative Ext sequence (Proposition 3.2 of the arrow topos lesson), \(b\mapsto\) the composite of \(L\Phi^*b\) with \(L\Phi^*p_G^*I\to p_F^*I\).

1. \(\omega(u,F')\) is the image of \(o(G'_1)\) under the connecting map \(\operatorname{Ext}^1(L\Phi^*L,p_F^*I)\to E^2\) of that sequence.
2. For another deformation \(G'_2\) of \(G\), let \(d\in\operatorname{Ext}^1(L,p_G^*I)\) be the difference of the classes of \(N(A,G'_2)\) and \(N(A,G'_1)\) in the torsor of Theorem 10.4 of The cotangent complex. Then \(o(G'_2)-o(G'_1)=-w(d)\), and every \(d\) arises from some \(G'_2\).
3. \(\omega(u,F')=0\) if and only if \(o(G'_1)\) lies in the image of \(w\), and then \(o(G'_2)=0\) for the deformation \(G'_2\) with difference \(d\), where \(w(d)=o(G'_1)\): \(u\) extends to a homomorphism \(F'\to G'_2\).

Proof. The deformations of \(q_G\) over \(N(A,S')\) are the flat deformations of \(N(A,G)\) over \(N(A,S')\) (proof of Theorem 4.1), and by Proposition 1.2 (3) of the lesson on deformations of module schemes they are, up to unique isomorphism, the \(N(A,G')\) for the deformations \(G'\) of \(G\). (1) and the formula of (2) are Proposition 5.2 (3) and (2) of the lesson on deforming a morphism together with its target, through Lemma 1.3. The flat deformations of \(N(A,G)\) over \(N(A,S')\) are the extensions of Theorem 10.4 of The cotangent complex for the canonical map \(q_G^{-1}p_S^*I\to p_G^*I\) (proof of Proposition 5.1 (1) of the lesson on deforming a morphism together with its target), which form a torsor under \(\operatorname{Ext}^1(L,p_G^*I)\); so every \(d\) arises. (3) By (1) and the exactness of the relative Ext sequence, \(\omega(u,F')=0\) if and only if \(o(G'_1)=w(d)\) for some \(d\); then (2) gives \(o(G'_2)=0\), and by Proposition 5.2 (1) there and Proposition 1.2 (4) of the lesson on deformations of module schemes, \(u\) extends to \(F'\to G'_2\). \(\square\)

So the obstruction of Theorem 4.1 is the obstruction to extending \(u\) to a given deformation of \(G\), modulo the change of that deformation, which acts through the restriction map \(w\).

Remark 4.3. Proposition 4.2 and its proof hold for the group schemes of Theorem 3.1, with the nerves \(N(G'_1)\), \(N(G'_2)\), \(L=L_{N(G)/S}\), the coefficients \(f_G^*I\) and \(f_F^*I\), and Propositions 1.2 (3) and 1.3 (1) of the lesson on deformations of group schemes in place of Proposition 1.2 (3) and (4) of the lesson on deformations of module schemes.

Corollary 4.4 (Ext groups over \(A_S\)). In Theorem 4.1, use the notation of Lie complexes of module schemes for the ring scheme \(A_S\) over \(S\): \({}_{A_S}\ell^\vee_G\) is the Lie complex of \(G\) with its action, and \((-)|_{A_S}\) the underlying complex of \(A_S\)-modules. Let \(\operatorname{cone}(u)\) be the complex \(F\xrightarrow{u}G\) of \(A_S\)-modules of the big fppf topos of \(S\), with \(G\) in degree \(0\), and \(M'=({}_{A_S}\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I)|_{A_S}\). Then \(E^i\cong\operatorname{Ext}^i_{A_S}(\operatorname{cone}(u),M')\) for every \(i\). Through these isomorphisms the obstruction \(\omega(u,F')\) is a class in \(\operatorname{Ext}^2_{A_S}(\operatorname{cone}(u),M')\), the isomorphism classes of solutions form a torsor under \(\operatorname{Ext}^1_{A_S}(\operatorname{cone}(u),M')\), and the automorphism group of a solution is isomorphic to \(\operatorname{Ext}^0_{A_S}(\operatorname{cone}(u),M')\).

Proof. Write \(L=L_{N(A,G)/N(A,S)}\) and \(\Phi=N(A,u)\). By Proposition 3.2 of the arrow topos lesson, \(E^i\) is the cohomology of a cocone of the map \(R\operatorname{Hom}(L,p_G^*I)\to R\operatorname{Hom}(L\Phi^*L,p_F^*I)\) given by the functor \(L\Phi^*\), formed on the Zariski topoi, where \(L\Phi^*p_G^*I=p_F^*I\) (Conventions). By Lemma 4.2 (2) of Quasi-coherent and perfect complexes on big fppf sites, with the truncation index \(0\), applying \(L\varepsilon^*\) and composing with \(L\varepsilon^*p_G^*I\to\varepsilon^*p_G^*I\) gives an isomorphism \(R\operatorname{Hom}(L,p_G^*I)\cong R\operatorname{Hom}(L\varepsilon^*L,\varepsilon^*p_G^*I)\), and likewise over \(N(A,F)\). Here \(\varepsilon^*p_G^*I=(\varepsilon^*I)_{N(A,G)}\) and \(\varepsilon^*p_F^*I=(\varepsilon^*I)_{N(A,F)}\), the pullbacks of quasi-coherent modules composing (proof of Corollary 4.4 of the lesson on Lie complexes of module schemes). These isomorphisms commute with the maps given by \(L\Phi^*\) and by the inverse image \(\Phi_{\mathrm{big}}^*\) of the fppf topoi, as \(L\varepsilon^*L\Phi^*\cong\Phi_{\mathrm{big}}^*L\varepsilon^*\) [Tag 0D6D]. So \(E^i\) is the cohomology of a cocone of the map \[ \begin{aligned}&R\operatorname{Hom}\bigl(L\varepsilon^*L,(\varepsilon^*I)_{N(A,G)}\bigr)\\&\quad\to R\operatorname{Hom}\bigl(\Phi_{\mathrm{big}}^*L\varepsilon^*L,(\varepsilon^*I)_{N(A,F)}\bigr)\end{aligned} \] given by \(\Phi_{\mathrm{big}}^*\) [Tag 014A]. At every level \(\Phi_{\mathrm{big}}^*\) is the restriction to a localization, and the complexes in the definition of the relative \(R\operatorname{Hom}(G/F;-,-)\) in Section 4 of the lesson on Lie complexes of module schemes compute these two derived Hom complexes and this map. So that cocone is \(R\operatorname{Hom}(G/F;L\varepsilon^*L,\varepsilon^*I)\), and Theorem 4.3 there, with \(M=\varepsilon^*I\in D^b(\mathcal O)\), gives the isomorphism. Theorem 4.1 gives the rest. \(\square\)

5. Exercises

Exercise 5.1. In Theorem 3.1, suppose that \(G\) is étale over \(S\). Show that there is a solution, unique up to unique isomorphism.

Exercise 5.2. In Theorem 3.1, take \(G=F\) and \(u=\mathrm{id}_F\). Show that \((F',\mathrm{id}_{F'})\) is a solution, and that every solution is isomorphic to it by a unique isomorphism.

Exercise 5.3. In Theorem 3.1, suppose that \(u\) is a closed immersion. Show that \(u'\) is a closed immersion in every solution \((G',u')\).

6. Complete solutions

Solution 5.1. The levels \(G^n\) of \(N(G)\) are étale over \(S\), so \(L_{G^n/S}=0\) (Proposition 5.1 of The cotangent complex, on affine opens), and \(L_{N(G)/S}\), whose levels are the \(L_{G^n/S}\) (Proposition 1.2 of the lesson on deformations of diagrams), is acyclic. Then \((0\leftarrow L_{N(G)/S})\) is zero in the derived category of the arrow topos, and all relative Ext groups \(\operatorname{Ext}^i(N(G)/N(F);L_{N(G)/S},f_G^*I)\) vanish (Definition 3.1 of the arrow topos lesson). By Theorem 3.1, \(\omega(u,F')=0\), the solutions form a torsor under the zero group, so there is exactly one isomorphism class, and its automorphism group is trivial.

Solution 5.2. The pair \((F',\mathrm{id}_{F'})\), with \(\iota_G=\iota_F\), is a solution. For the identity \(N(\mathrm{id}_F)\) of \(N(F)\), the relative derived Hom vanishes (Exercise 5.3 of the arrow topos lesson), so all groups of Theorem 3.1 are zero: the solutions form a single isomorphism class, and a solution has only the identity automorphism. So every solution is isomorphic to \((F',\mathrm{id}_{F'})\), and the isomorphism is unique, two isomorphisms differing by an automorphism.

Solution 5.3. A morphism of schemes is a closed immersion if and only if it is a proper monomorphism [Tag 04XV]. As in the proof of Corollary 3.2, \(u\) is the base change of \(u'\) along \(S\subseteq S'\), with \(F'\) flat over \(S'\), so by [Tag 06AG] \(u'\) is proper and a monomorphism because \(u\) is. Hence \(u'\) is a closed immersion.

Sources

The deformation of a morphism together with its target, for diagrams of schemes and in its equivariant form, and its application to group schemes and to module schemes through nerves and spectral nerves are due to Illusie, in his work on the cotangent complex.