Deformations of module schemes

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 over a base \(T\), with a quasi-coherent ideal \(I\), and \(A\) a ring scheme over \(T\). An \(A\)-module scheme \(G\) over \(S\), flat and locally of finite presentation, is determined by its spectral nerve \(N(A,G)\), a diagram of products of copies of \(A\) and \(G\) over the diagram \(N(A,S)\) of products of copies of \(A\). A deformation of \(G\) to an \(A\)-module scheme over \(S'\) is the same as a deformation of the morphism \(N(A,G)\to N(A,S)\) over \(N(A,S')\): the conditions that characterize spectral nerves are isomorphisms and cartesian squares, and they survive deformation. The deformation theory of diagrams of schemes (Deformations of diagrams of schemes) therefore gives an obstruction, a classification and automorphisms in Ext groups of the cotangent complex of the spectral nerve, and the Ext formula of Lie complexes of module schemes turns these into Ext groups of \(A\)-modules:

For \(A\) the constant ring scheme \(\mathbb Z\) these are statements about commutative group schemes, and for \(A=\mathcal O\) and vector groups they recover the deformation theory of locally free modules.

Prerequisites. Lie complexes of module schemes (Conventions, Lemma 1.1, Propositions 3.2 and 3.4, Theorems 4.1 and 4.2, Corollary 4.4, Exercise 5.1); Deformations of flat group schemes and of homomorphisms (Lemma 1.1, Propositions 1.2 and 3.2, Theorem 3.1 and its proof, Proposition 3.5 and its proof, Exercise 4.2); Lemma 5.1 of Equivariant extensions and deformations; Deformations of diagrams of schemes (Proposition 1.4, Definitions 2.1, 3.1 and 4.1, Construction 2.2, Theorem 2.4, Lemmas 3.2–3.4, Construction 4.2, Theorem 4.3, Proposition 4.4); Graded monoids and the nerves of their actions (Lemma 2.5, Proposition 2.6); Graded rings and the spectra of modules (Proposition 3.5 and the remark after it); Section 3 of Multisimplicial objects and spectral diagrams (Propositions 3.1 and 3.3); Lie complexes of commutative group schemes (Proposition 1.3); The co-Lie complex of a group scheme (Proposition 1.5, Corollary 2.3); Quasi-coherent and perfect complexes on big fppf sites (Propositions 1.1 and 2.1, Corollary 2.2, Lemma 4.2); Theorem 2.2 of the programme lesson The cotangent complex (functoriality maps compose). From the AI-integrated Stacks project: closed immersions and fibre products [Stacks, Tag 01JU]; Tor independence [Tag 08IA]; the flatness criterion over a nilpotent ideal [Tag 06A5]; duals of perfect complexes on ringed sites [Tag 08JJ]; vector bundles [Tag 01M1]; thickenings of affine schemes [Tag 06AD]; finitely presented flat modules are finite projective [Tag 00NX]; maps between truncated objects [Tag 06XS]; composition of derived inverse images [Tag 0D6D].

Conventions

\(S\to T\) is a morphism of schemes, and \((S',i,s')\) a \(T\)-extension of \(S\) by a quasi-coherent \(\mathcal O_S\)-module \(I\) (Definition 2.1 of the lesson on deformations of diagrams), with class \(e(i)\in\operatorname{Ext}^1(L_{S/T},I)\) (Theorem 2.4 there). \(A\) is a ring scheme over \(T\), and we assume:

(TI) for every finite set \(P\), the fibre product \(A^P\) of copies of \(A\) over \(T\) is Tor independent of \(S\) and of \(S'\) over \(T\).

Example 1.3 shows that it does not follow from the Tor independence of \(A\) itself. It holds if \(A\) is flat over \(T\), for instance for the constant ring schemes \(\mathbb Z\) and \(\mathbb Z/n\mathbb Z\), for \(\mathcal O=\mathbf A^1_T\), and for monoid rings \(\mathbb Z[\Gamma]_T\). For a \(T\)-scheme \(X\), \(A_X=A\times_TX\), a ring scheme over \(X\). Over \(S\) and over \(S'\) the notation of Lie complexes of module schemes is used for the rings \(A_S\) and \(A_{S'}\); in particular \(N(A,G)\) is the spectral nerve of an \(A_S\)-module scheme \(G\), \(q_G\colon N(A,G)\to N(A,S)\) its projection, and \(p_G\colon N(A,G)\to S\) its structure morphism, and \(N(A,S')\), \(N(A,G')\) are formed over \(S'\). These are diagrams of schemes indexed by the same category \(\mathfrak L\). For an \(\mathcal O_S\)-module \(J\), \(p_G^*J\) is the module with the levels \(p_{G,\lambda}^*J\).

Definition 0.1. Let \(G\) be an \(A_S\)-module scheme. A deformation of \(G\) over \(S'\) is a pair \((G',\iota)\) of an \(A_{S'}\)-module 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 additions and the actions of \(A_S\to A_{S'}\), inducing an isomorphism of \(A_S\)-module schemes \(G\cong G'\times_{S'}S\). Isomorphisms of deformations and extensions of homomorphisms to given deformations are defined as in Definition 0.1 of the lesson on deformations of group schemes, with \(A\)-linear maps.

1. Spectral nerves over a thickening

Lemma 1.1. \(N(A,S')\), with the closed immersions \(N(A,S)_\lambda\to N(A,S')_\lambda\), is a \(T\)-extension of \(N(A,S)\) whose ideal is the cartesian module \(p_S^*I\); so it is a deformation of \(N(A,S)\). Let \(e(N(A,S'))\in\operatorname{Ext}^1(L_{N(A,S)/T},p_S^*I)\) be its class (Theorem 2.4 of the lesson on deformations of diagrams).

Proof. The level \(\lambda\) of \(N(A,S')\) is \(A_{S'}^P=A^P\times_TS'\) for a finite set \(P\) (Lemma 1.1 of the lesson on Lie complexes of module schemes), and that of \(N(A,S)\) is \(A^P\times_TS=A_{S'}^P\times_{S'}S\). So \(N(A,S)_\lambda\to N(A,S')_\lambda\) is the base change of \(i\), a first-order thickening whose ideal is the image of \(I\otimes\mathcal O_{A^P_{S'}}\) [Tag 01JU]. On affine opens \(\operatorname{Spec}R\) of \(T\), \(\operatorname{Spec}C'\) of \(S'\) over it, \(C=C'/I\), and \(\operatorname{Spec}B\) of \(A^P\) over \(\operatorname{Spec}R\), (TI) says \(B\otimes^{\mathbf L}_RC'=B\otimes_RC'\) and \(B\otimes^{\mathbf L}_RC=B\otimes_RC\) [Tag 08IA]. By the associativity of derived tensor products, \((B\otimes_RC')\otimes^{\mathbf L}_{C'}C=B\otimes^{\mathbf L}_RC=B\otimes_RC\), so \(\operatorname{Tor}_1^{C'}(B\otimes_RC',C)=0\), and by Lemma 3.2 (2) of the lesson on deformations of diagrams the ideal is \(p_{S,\lambda}^*I\). The maps of \(N(A,S')\) restrict to those of \(N(A,S)\) and carry these identifications into each other, so the ideal is the cartesian module \(p_S^*I\). The morphism \(N(A,S')\to S'\to T\) makes it a \(T\)-extension. \(\square\)

Proposition 1.2 (deformations of module schemes and of their spectral nerves). Let \(G\) be an \(A_S\)-module scheme.

  1. For a deformation \((G',\iota)\) of \(G\), the diagram \(N(A,G')\), with the closed immersions \(N(A,\iota)_\lambda\) and the projection \(q_{G'}\), is a deformation of \(q_G\colon N(A,G)\to N(A,S)\) over \(N(A,S')\) (Definition 4.1 of the lesson on deformations of diagrams).
  2. Every deformation of \(q_G\) over \(N(A,S')\) is isomorphic, compatibly with the maps from \(N(A,G)\) and to \(N(A,S')\), to \(N(A,G')\) for a deformation \((G',\iota)\) of \(G\).
  3. \((G',\iota)\mapsto N(A,G')\) is an equivalence from the groupoid of deformations of \(G\) onto the groupoid of deformations of \(q_G\) over \(N(A,S')\).
  4. For deformations \(F'\) of \(F\) and \(G'\) of \(G\) and a homomorphism \(u\colon F\to G\), the extensions \(u'\) of \(u\) correspond to the morphisms \(N(A,F')\to N(A,G')\) over \(N(A,S')\) whose composite with \(N(A,\iota_F)\) is \(N(A,\iota_G)N(A,u)\). For a deformation \(G'\) of \(G\), the pairs of a deformation \(F'\) of \(F\) and an extension \(u'\colon F'\to G'\) of \(u\), with their isomorphisms, correspond to the deformations of \(N(A,u)\) over \(N(A,G')\).

Proof. (1) The level \(\lambda\) of \(N(A,G')\) is \(N(A,S')_\lambda\times_{S'}G'^P\), flat over \(N(A,S')_\lambda\), and its base change to \(N(A,S)_\lambda\) is \(N(A,S)_\lambda\times_SG^P\), because \(G'^P\times_{S'}S\cong G^P\) through \(\iota^P\). So \(N(A,\iota)_\lambda\) is a first-order thickening [Tag 01JU], and by Lemma 1.1 of the lesson on deformations of group schemes, applied to \(N(A,G')_\lambda\to N(A,S')_\lambda\), it is a deformation of \(q_{G,\lambda}\). The maps \(N(A,\iota)_\lambda\) form a morphism of diagrams, \(\iota\) being additive and compatible with the actions. So the ideal map \(q_G^*p_S^*I\to\mathcal I\) is an isomorphism at every level.

(2) Let \(X'\) be a deformation of \(q_G\) over \(N(A,S')\), with \(\kappa\colon N(A,G)\to X'\). Regard \(X'\), through its morphism to \(N(A,S')\), as an object of the category \(\mathcal Y\) of Proposition 2.6 of the lesson on graded monoids for the graded monoid \(A_{S'}(1)^\times\) of the topos of sheaves on \((\mathrm{Sch}/S')_{\mathrm{fppf}}\). By Proposition 3.5 of the lesson on graded rings, \(X'\) is isomorphic to the spectral nerve of an \(A_{S'}\)-module if it satisfies conditions 1 and 2 of Proposition 2.6 there and conditions 3 and 4 of Proposition 3.5. Each of these conditions says that certain maps formed from the maps of the diagram and from its morphism to \(N(A,S')\) are isomorphisms, or that certain commutative squares are cartesian: condition 1 says that the squares formed by the vertex \(0\) in the direction of the action and the morphism to \(N(A,S')\) are cartesian, \(N(A,S')\) having the final object at the vertex \(0\); conditions 2 and 3 say that certain maps of the diagram are isomorphisms; for \(n\geq2\), condition 4 says, by Proposition 3.3 of the lesson on multisimplicial objects, that the lines of \(X'_{0,0}\) satisfy the conditions (i)–(iii) of Proposition 3.1 there, which say that a map to the final object and the maps to products over it, given by faces, are isomorphisms; and for \(n=1\) condition 4 follows from condition 3 and the case \(n=2\) (remark after Proposition 3.5 of the lesson on graded rings). All these conditions hold for \(N(A,G)\) over \(N(A,S)\). By Proposition 4.4 of the lesson on deformations of diagrams, applied to the cylinder of \(q_G\), the cartesian squares of \(N(A,G)\) and of \(q_G\) have cartesian images in \(X'\) and its morphism to \(N(A,S')\). A map of \(X'\) or of the cylinder whose image in \(N(A,G)\) is an isomorphism is a deformation of that isomorphism, the ideal being cartesian, hence an isomorphism (Lemma 3.3 there). A map to a product over \(S'\) is an isomorphism exactly when the corresponding squares over the final object are cartesian. So \(X'\) satisfies the four conditions, and \(X'\cong N(A_{S'},G')\) over \(N(A,S')\) for an \(A_{S'}\)-module \(G'\), whose underlying object is the value of \(X'\) at the object \(o\) of the level \(0\). That value is a deformation of \(G\to S\), so \(G'\) is a scheme, flat and locally of finite presentation over \(S'\) (Lemma 1.1 of the lesson on deformations of group schemes). The square formed by \(\kappa_o\), \(i\) and the structure morphisms is cartesian (Lemma 3.2 (1) of the lesson on deformations of diagrams), so \(\iota=\kappa_o\) induces an isomorphism of schemes \(G\cong G'\times_{S'}S\). The morphism \(\kappa\) lies over the morphism \(N(A,S)\to N(A,S')\) induced by the homomorphism \(A_S\to A_{S'}\) of graded monoids; by Lemma 2.5 of the lesson on graded monoids, applied at every level of the second simplicial direction, \(\kappa\) is induced by a morphism of graded objects \(G(1)^\times\to G'(1)^\times\) compatible with the actions, which is \(\iota(1)\) by the full faithfulness of the spectrum (as in the proof of Proposition 3.4 of the lesson on graded rings). So \(\iota\) is additive and compatible with the actions. So \((G',\iota)\) is a deformation of \(G\), and \(X'\cong N(A,G')\) compatibly with \(\kappa\) and \(N(A,\iota)\).

(3) By (1) and (2) the functor is essentially surjective, and it is fully faithful by the full faithfulness of \(M\mapsto N(A_{S'},M)\) on \(A_{S'}\)-modules (Proposition 3.5 of the lesson on graded rings), an isomorphism of deformations of \(q_G\) being in particular a morphism over \(N(A,S')\).

(4) The first statement is again full faithfulness, for morphisms over \(N(A,S')\). For the second, a deformation \(X'\) of \(N(A,u)\) over \(N(A,G')\), composed with \(q_{G'}\), is a deformation of \(q_F\) over \(N(A,S')\), by the composition rule of Lemma 3.4 (1) of the lesson on deformations of diagrams and \(N(A,u)^*q_G^*=q_F^*\). By (2) it is \(N(A,F')\) for a deformation \(F'\) of \(F\), and its morphism to \(N(A,G')\) is \(N(A,u')\) for an extension \(u'\) of \(u\), by the first statement. Conversely a pair \((F',u')\) gives the deformation \(N(A,F')\to N(A,G')\) of \(N(A,u)\): its ideal map is the ideal map of \(N(A,F')\) over \(N(A,S')\), an isomorphism by (1). \(\square\)

Example 1.3 (Tor independence of \(A\) alone). Condition (TI) cannot be weakened to the Tor independence of \(A\) itself: then \(N(A,S')\) need not be a deformation of \(N(A,S)\). Let \(k\) be a field, \(T=\operatorname{Spec}\Lambda\) with \(\Lambda=k[x,y]\), \(S=V(x)\) and \(S'=V(x^2)\), a \(T\)-extension of \(S\) by \(I=x\Lambda/x^2\Lambda\cong\mathcal O_S\). Let \(G\subseteq\mathbf G_a^2\) be the kernel of the \(\mathcal O\)-linear map \((t_1,t_2)\mapsto xt_1-yt_2\), an \(\mathcal O\)-module scheme over \(T\), and \(A=\mathcal O\oplus G\) the ring scheme with the product \((a,g)(a',g')=(aa',ag'+a'g)\). Then \(A\) is Tor independent of \(S\) and of \(S'\) over \(T\), but \(A\times_TA\) is not Tor independent of \(S\), and at the level of \(N(A,S')\) given by a set \(P\) with two elements the map from \(p_S^*I\) onto the ideal of \(N(A,S)_\lambda\) is not injective.

Proof. \(A=\operatorname{Spec}C\) with \(C=\Lambda[a,t_1,t_2]/(xt_1-yt_2)\). The quadratic form \(xt_1-yt_2\) has rank \(4\), while a product of two linear forms has rank at most \(2\), so it is irreducible and \(C\) is a domain. As \(\Lambda/x^e\) has the free resolution \(\Lambda\xrightarrow{x^e}\Lambda\), \(\operatorname{Tor}_1^\Lambda(C,\Lambda/x^e)\) is the \(x^e\)-torsion of \(C\), which vanishes, and \(\operatorname{Tor}_i^\Lambda(C,\Lambda/x^e)=0\) for \(i\geq2\). Next, \[ \begin{aligned}C_2&=C\otimes_\Lambda C\\&=\Lambda[a,t_1,t_2,b,u_1,u_2]/\\&\qquad(xt_1-yt_2,\,xu_1-yu_2)\end{aligned} \] is the ring of \(A\times_TA\), and \(z=t_2u_1-t_1u_2\) satisfies \(xz=t_2(xu_1)-u_2(xt_1)=\) \(t_2yu_2-u_2yt_2=0\) in \(C_2\). Modulo \(x\) and \(y\), \(C_2\) becomes the polynomial ring \(k[a,t_1,t_2,b,u_1,u_2]\), in which \(z\neq0\), while \(xC_2\) goes to \(0\). So the class of \(z\) in \(C_2/xC_2\) is not zero and is killed by \(x\): \(\operatorname{Tor}_1^\Lambda(C_2,\Lambda/x)\neq0\). For the same reason the map \((C_2/xC_2)\otimes I\to xC_2/x^2C_2\), \(c\otimes\bar x\mapsto xc\), from \(p_S^*I\) to the ideal of \(N(A,S)_\lambda=\operatorname{Spec}C_2/xC_2\) in \(N(A,S')_\lambda=\operatorname{Spec}C_2/x^2C_2\), sends the class of \(z\) to \(0\). \(\square\)

Lemma 1.4 (the class of the spectral nerve over the thickening). Let \(N(A,T)\) be the spectral nerve over \(T\), with the levels \(A^P\), so that \(N(A,S)=N(A,T)\times_TS\) levelwise, and let \(\gamma\colon L_{N(A,S)/T}\to Lp_S^*L_{S/T}\) be the functoriality map \(L_{N(A,S)/T}\to L_{N(A,S)/N(A,T)}\) followed by the inverse of the functoriality map \(Lp_S^*L_{S/T}\to L_{N(A,S)/N(A,T)}\). The latter is an isomorphism by Proposition 1.4 of the lesson on deformations of diagrams, the squares being cartesian and Tor independent by (TI). Then \[ e(N(A,S'))=p_S^*e(i)\circ\gamma, \] where \(p_S^*e(i)\) is \(Lp_S^*(e(i))\) followed by the map \(Lp_S^*I[1]\to p_S^*I[1]\) inducing the identity on \(H^{-1}\).

Proof. Write \(X\), \(X'\), \(Y\) for \(N(A,S)\), \(N(A,S')\), \(N(A,T)\), and work on \(\operatorname{Top}(X_{\mathrm{Zar}})\) with the ring \(\mathcal O_{X'}\) of Construction 2.2 of the lesson on deformations of diagrams. By Theorem 2.4 there, \(e(X')=\tau_X\circ c_X\), where \(c_X\colon L_{X/T}\to L_{\mathcal O_X/\mathcal O_{X'}}\) is the functoriality map and \(\tau_X\) the conormal truncation; likewise \(e(i)=\tau_S\circ c_S\). The morphisms \(X\to X'\to Y\to T\) give maps of ring pairs, from the pair of \(T\) to the pair of \(Y\) and from that to the pair \(\mathcal O_{X'}\to\mathcal O_X\). Functoriality maps compose (Theorem 2.2 of The cotangent complex), so \(c_X=d\circ\ell\), with \(\ell\colon L_{X/T}\to L_{X/Y}\) and \(d\colon L_{X/Y}\to L_{\mathcal O_X/\mathcal O_{X'}}\) the functoriality maps. Let \(\beta\colon Lp_S^*L_{S/T}\to L_{X/Y}\) be the isomorphism of the statement and \(b\colon Lp_S^*L_{\mathcal O_S/\mathcal O_{S'}}\to L_{\mathcal O_X/\mathcal O_{X'}}\) the functoriality map for the square \(X\to X'\) over \(S\to S'\). The composites \(d\circ\beta\) and \(b\circ Lp_S^*(c_S)\) are both the functoriality map of the map of ring pairs from the inverse image of the pair of \(S\) over \(T\) to the pair \(\mathcal O_{X'}\to\mathcal O_X\), factored through \(Y\) and through \(S'\); so they are equal. The complex \(Lp_S^*L_{\mathcal O_S/\mathcal O_{S'}}\) lies in \(D^{\leq-1}\), with \(H^{-1}=p_S^*I\), and \(b\) induces on \(H^{-1}\) the identification of \(p_S^*I\) with the ideal of \(X\) in \(X'\) (Lemma 1.1). Maps from it to \(p_S^*I[1]\) are determined by their effect on \(H^{-1}\) [Tag 06XS], so \(\tau_X\circ b=p_S^*(\tau_S)\), with the same convention for \(p_S^*\). Hence \[ \begin{aligned}e(X')&=\tau_X\circ d\circ\beta\circ\beta^{-1}\circ\ell\\&=\tau_X\circ b\circ Lp_S^*(c_S)\circ\gamma\\&=p_S^*(\tau_S\circ c_S)\circ\gamma=p_S^*e(i)\circ\gamma.\end{aligned} \] \(\square\)

2. The deformation theorems

For an \(A_S\)-module scheme \(G\) write \(M_G={}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I\), an object of the derived category of \(A_S\otimes^{\mathbf L}\mathcal O\) (Proposition 3.2 of the lesson on Lie complexes of module schemes), and \(M_G|_A\) its underlying complex of \(A_S\)-modules. Let \(\omega(G,i)\in\operatorname{Ext}^2(L_{N(A,G)/N(A,S)},p_G^*I)\) be the obstruction \(\omega(q_G,N(A,S'))\) of Construction 4.2 of the lesson on deformations of diagrams, \(q_G^*p_S^*I=p_G^*I\) being the pullback of the ideal of \(N(A,S')\).

Theorem 2.1 (deformations of module schemes). Let \(G\) be an \(A_S\)-module scheme.

  1. \(G\) has a deformation over \(S'\) if and only if \(\omega(G,i)=0\).
  2. If \(\omega(G,i)=0\), the isomorphism classes of deformations of \(G\) form a torsor under \(\operatorname{Ext}^1(L_{N(A,G)/N(A,S)},p_G^*I)\), and the automorphism group of every deformation is \(\operatorname{Ext}^0(L_{N(A,G)/N(A,S)},p_G^*I)\).
  3. \(\omega(G,i)=\) \(q_G^*e(N(A,S'))[1]\circ\) \(K(N(A,G)/N(A,S)/T)\), with the class of Lemma 1.1 and the Kodaira–Spencer class of \(N(A,G)\to N(A,S)\to T\).
  4. For every \(j\), \(\operatorname{Ext}^j(L_{N(A,G)/N(A,S)},p_G^*I)\cong\) \(\operatorname{Ext}^j_{A_S}(G,M_G|_A)\). So the obstruction lies in \(\operatorname{Ext}^2_{A_S}(G,M_G|_A)\), the deformations form a torsor under \(\operatorname{Ext}^1_{A_S}(G,M_G|_A)\), and the automorphisms of a deformation form \(\operatorname{Ext}^0_{A_S}(G,M_G|_A)\).

Proof. The ideal \(p_S^*I\) of \(N(A,S')\) is termwise quasi-coherent (Lemma 1.1), so Theorem 4.3 of the lesson on deformations of diagrams applies to \(q_G\), of identity type, and to the extension \(N(A,S')\) of \(N(A,S)\). By Proposition 1.2 (3), its deformations of \(q_G\) 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 4.4 of the lesson on Lie complexes of module schemes, for \(u=\mathrm{id}_G\). \(\square\)

Theorem 2.2 (extending homomorphisms). Let \((F',\iota_F)\) and \((G',\iota_G)\) be deformations of \(A_S\)-module schemes \(F\) and \(G\), \(u\colon F\to G\) a homomorphism, and \(L_u=LN(A,u)^*L_{N(A,G)/N(A,S)}\).

  1. There is a class \(\omega(u,F',G')\in\operatorname{Ext}^1(L_u,p_F^*I)\), isomorphic to \(\operatorname{Ext}^1_{A_S}(F,M_G|_A)\), whose vanishing is necessary and sufficient for the existence of an extension \(u'\colon F'\to G'\) of \(u\).
  2. If \(\omega(u,F',G')=0\), the extensions of \(u\) form a torsor under \(\operatorname{Ext}^0(L_u,p_F^*I)\cong\operatorname{Ext}^0_{A_S}(F,M_G|_A)\).
  3. For \(F=G\) and \(u=\mathrm{id}_G\), \(\omega(\mathrm{id}_G,G',G'')\) is the element of the torsor of Theorem 2.1 (2) carrying the class of \(G''\) to that of \(G'\).

Proof. By Proposition 1.2 (4), the extensions of \(u\) are the morphisms \(N(A,F')\to N(A,G')\) over \(N(A,S')\) extending \(N(A,u)\). The proof of Theorem 3.1 (1), (2) of the lesson on deformations of group schemes applies word for word with \(N(A,F)\), \(N(A,G)\), \(N(A,S)\), \(N(A,S')\), \(N(A,u)\), \(q_G\) and the ideals \(p_F^*I\), \(p_G^*I\) in place of \(N(G)\), \(N(H)\), \(S\), \(S'\), \(N(u)\), \(f_H\) and \(f_G^*I\), \(f_H^*I\): the rings on the Zariski topos of \(N(A,F)\) are the inverse images of the structure rings, the lifts are given by Lemma 5.1 of Equivariant extensions and deformations, and the passage from \(N(A,S')\) to \(N(A,S)\) uses the transitivity triangle of \(N(A,G)\to N(A,S)\to N(A,S')\), with \(L_{N(A,S)/N(A,S')}\) in \(D^{\leq-1}\), the levels being first-order thickenings. The identifications of the Ext groups are Corollary 4.4 of the lesson on Lie complexes of module schemes. (3) is the proof of Proposition 3.2 of the lesson on deformations of group schemes, with Proposition 1.2 (3) in place of Proposition 1.2 there. \(\square\)

In particular an extension of the identity exists exactly when the two deformations are isomorphic, and every extension of the identity is an isomorphism of deformations (Lemma 3.3 of the lesson on deformations of diagrams).

Theorem 2.3 (deforming a homomorphism with a given target). Let \((G',\iota_G)\) be a deformation of \(G\), \(u\colon F\to G\) a homomorphism of \(A_S\)-module schemes, and \(K\) the first vertex of a distinguished triangle \(K\to{}_A\ell^\vee_F\to{}_A\ell^\vee_G\to K[1]\) on \({}_A\ell^\vee(u)\). Let \(\omega(u,G')=\) \(\omega(N(A,u),N(A,G'))\in\) \(\operatorname{Ext}^2(L_{N(A,F)/N(A,G)},p_F^*I)\) be the obstruction of Construction 4.2 of the lesson on deformations of diagrams.

  1. There is a pair of a deformation \((F',\iota_F)\) of \(F\) and an extension \(u'\colon F'\to G'\) of \(u\) if and only if \(\omega(u,G')=0\).
  2. If \(\omega(u,G')=0\), the isomorphism classes of such pairs form a torsor under \(\operatorname{Ext}^1(L_{N(A,F)/N(A,G)},p_F^*I)\), and the automorphism group of each pair is \(\operatorname{Ext}^0(L_{N(A,F)/N(A,G)},p_F^*I)\).
  3. \(\omega(u,G')=\) \(N(A,u)^*e(N(A,G'))[1]\circ\) \(K(N(A,F)/N(A,G)/T)\), where \(e(N(A,G'))\) is the class of \(N(A,G')\) as a \(T\)-extension of \(N(A,G)\).
  4. For every \(j\), \(\operatorname{Ext}^j(L_{N(A,F)/N(A,G)},p_F^*I)\cong\) \(\operatorname{Ext}^j_{A_S}(F,(K\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I)|_A)\); if \(u\) is faithfully flat with kernel \(E\), \(K\cong{}_A\ell^\vee_E\).
  5. For \(G=0\), \(G'=0\) over \(S'\), the pairs are the deformations of \(F\), and (1)–(4) are Theorem 2.1.
  6. If \(u\) is a closed immersion, respectively faithfully flat, so is \(u'\) for every pair \((F',u')\).

Proof. The ideal \(p_G^*I\) of \(N(A,G')\) is termwise quasi-coherent, and \(N(A,u)^*p_G^*I=p_F^*I\). By Proposition 1.2 (4), the deformations of \(N(A,u)\) over \(N(A,G')\), 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). (4) is the Zariski form of Theorem 4.2 of the lesson on Lie complexes of module schemes (Corollary 4.4 there). For (5), \(N(A,0)=N(A,S)\) and \(N(A,u)\) is \(q_F\). For (6), the solution of Exercise 4.2 of the lesson on deformations of group schemes shows that \(u'\) is a closed immersion if \(u\) is. If \(u\) is faithfully flat, then \(u'\) is flat by the flatness criterion over the nilpotent ideal \(I\) [Tag 06A5], applied to \(\mathcal O_{F'}\) over \(\mathcal O_{G'}\) and \(\mathcal O_{S'}\): \(F'\) is flat over \(S'\) and \(F=F'\times_{S'}S\) is flat over \(G=G'\times_{S'}S\). It is surjective, having the underlying map of \(u\). \(\square\)

Corollary 2.4 (the obstruction as a product). Let \(G\) be an \(A_S\)-module scheme. Put \(M=t_{[-1}L\varepsilon^*L_{S/T}\in D^b(\mathcal O)\), and let \(e_\varepsilon\colon M\to\varepsilon^*I[1]\) be the map induced by \(e(i)\), through \(t_{[-1}L\varepsilon^*(I[1])=\varepsilon^*I[1]\). There is a class \[ \begin{gathered}c(A,G/S/T)\in\\ \operatorname{Ext}^1_A\bigl(G,({}_A\ell^\vee_G\otimes^{\mathbf L}_{\mathcal O}M)|_A\bigr)\end{gathered} \] such that, under the isomorphism of Theorem 2.1 (4), \[ \begin{gathered}\omega(G,i)=\bigl(({}_A\ell^\vee_G\otimes^{\mathbf L}e_\varepsilon)|_A\bigr)[1]\\ \circ c(A,G/S/T).\end{gathered} \] It is the image of the Kodaira–Spencer class \(K(N(A,G)/N(A,S)/T)\), followed by \(q_G^*\gamma[1]\) and by the truncation of \(Lp_G^*L_{S/T}[1]\) to degrees \(\geq-2\).

Proof. The morphism \(q_G\) is flat at every level, \(G\) being flat over \(S\), so \(Lq_G^*=q_G^*\), and \(q_G^*Lp_S^*=Lp_G^*\). By Theorem 2.1 (3) and Lemma 1.4, \[ \begin{aligned}\omega(G,i)&=q_G^*\bigl(p_S^*e(i)\circ\gamma\bigr)[1]\circ K\\&=p_G^*e(i)[1]\circ q_G^*\gamma[1]\circ K,\end{aligned} \] with \(K=K(N(A,G)/N(A,S)/T)\) and \(p_G^*e(i)\) as in Lemma 1.4. The target \(p_G^*I[2]\) lies in degrees \(\geq-2\), so \(p_G^*e(i)[1]\) factors through the truncation \(F[1]\) of \(Lp_G^*L_{S/T}[1]\), where \(F=t_{[-1}Lp_G^*L_{S/T}\) [Tag 06XS]. Let \(\kappa\in\operatorname{Ext}^1(L_{N(A,G)/N(A,S)},F)\) be the truncation of \(q_G^*\gamma[1]\circ K\), and \(e_F\colon F\to p_G^*I[1]\) the induced map, so that \(\omega(G,i)=e_F[1]\circ\kappa\). The complex \(F\) has termwise quasi-coherent cohomology in degrees \(-1\) and \(0\). By Lemma 4.2 (2) of the lesson on big fppf sites, with \(n=-1\), \(\operatorname{RHom}(L_{N(A,G)/N(A,S)},F)\cong\) \(\operatorname{RHom}(L\varepsilon^*L_{N(A,G)/N(A,S)},t_{[-1}L\varepsilon^*F)\), naturally in \(F\). Derived inverse images compose [Tag 0D6D], and the inverse images of the big fppf topoi along the levels of \(p_G\) are restrictions, which are exact. As \(L\varepsilon^*\) is right t-exact, \(t_{[-1}L\varepsilon^*F=M_{N(A,G)}\), the restriction of \(M\) to the levels. Theorem 4.1 of the lesson on Lie complexes of module schemes, for \(u=\mathrm{id}_G\), gives \(\operatorname{RHom}(L\varepsilon^*L_{N(A,G)/N(A,S)},M_{N(A,G)})\cong\) \(\operatorname{RHom}_A(G,({}_A\ell^\vee_G\otimes^{\mathbf L}M)|_A)\), naturally in \(M\). Let \(c(A,G/S/T)\) be the image of \(\kappa\). The isomorphism of Theorem 2.1 (4) is the same chain for the coefficients \(p_G^*I\) and \(\varepsilon^*I\) (Corollary 4.4 there). The map \(e_F\colon F\to p_G^*I[1]\) induces \(e_\varepsilon\) on the restrictions to the levels, after \(t_{[-1}L\varepsilon^*\). By the naturality of both isomorphisms in the coefficients, composition with \(e_F[1]\) corresponds to composition with \((({}_A\ell^\vee_G\otimes^{\mathbf L}e_\varepsilon)|_A)[1]\), which gives the formula. \(\square\)

The class \(c(A,G/S/T)\) depends on \(G\), \(A\) and \(S\to T\) only, not on the extension \(S'\): the obstruction is the product of an \(A\)-linear Kodaira–Spencer class of \(G\) with the class \(e(i)\) of the thickening.

Proposition 2.5 (two deformations of the target). Let \((G',\iota')\) and \((G'',\iota'')\) be deformations of \(G\), and let \(d\in\operatorname{Ext}^1(L_{N(A,G)/N(A,S)},p_G^*I)\) be the element that carries the class of \(G''\) to that of \(G'\) in the torsor of Theorem 2.1 (2); by Theorem 2.2 (3), \(d=\omega(\mathrm{id}_G,G',G'')\). Let \(u\colon F\to G\) be a homomorphism, and put \(N(A,u)^*d=\) \(\tau\circ LN(A,u)^*(d)\colon\) \(L_u\to\) \(p_F^*I[1]\), where \(\tau\colon LN(A,u)^*p_G^*I[1]\to p_F^*I[1]\) is the map inducing the identity on \(H^{-1}\).

  1. For every deformation \((F',\iota_F)\) of \(F\), \(\omega(u,F',G')-\omega(u,F',G'')=\) \(-N(A,u)^*d\) in \(\operatorname{Ext}^1(L_u,p_F^*I)\).
  2. Let \(K\) be the Kodaira–Spencer class \(K(N(A,F)/N(A,G)/N(A,S))\), the third map of the transitivity triangle of \(N(A,F)\to N(A,G)\to N(A,S)\). The difference of the obstructions of Theorem 2.3 is the image of \(N(A,u)^*d\) under the connecting map of that triangle: in \(\operatorname{Ext}^2(L_{N(A,F)/N(A,G)},p_F^*I)\), \[ \begin{aligned}&\omega(u,G')-\omega(u,G'')\\&\quad=(N(A,u)^*d)[1]\circ K.\end{aligned} \]

Proof. The proof of Proposition 3.5 of the lesson on deformations of flat group schemes applies word for word, with \(N(A,F)\), \(N(A,G)\), \(N(A,S)\), \(N(A,S')\), \(N(A,u)\) and the ideals \(p_F^*I\), \(p_G^*I\) in place of \(N(G)\), \(N(H)\), \(S\), \(S'\), \(N(u)\) and \(f_G^*I\), \(f_H^*I\), and with Theorem 2.2 and Theorem 2.3 (3) in place of Theorems 3.1 and 3.3 (3) there. The two facts it uses hold here: the class \(\kappa(G')\in\operatorname{Ext}^1(L_{N(A,G)/N(A,S')},p_G^*I)\) of \(N(A,G')\) as an \(N(A,S')\)-extension of \(N(A,G)\) satisfies \(\kappa(G')-\kappa(G'')=d\circ w\), with the functoriality map \(w\colon L_{N(A,G)/N(A,S')}\to L_{N(A,G)/N(A,S)}\) (proof of Theorem 2.2 (3)); and the class of \(N(A,G')\) as a \(T\)-extension is \(e(N(A,G'))=\kappa(G')\circ\rho\), with the functoriality map \(\rho\colon L_{N(A,G)/T}\to L_{N(A,G)/N(A,S')}\), functoriality maps being compatible with composition. \(\square\)

3. Examples

Proposition 3.1 (étale module schemes). Let \(G\) be an \(A_S\)-module scheme, étale over \(S\). Then \(G\) has a deformation over \(S'\), unique up to unique isomorphism, and for every deformation \(F'\) of an \(A_S\)-module scheme \(F\) and every homomorphism \(u\colon F\to G\), \(u\) has a unique extension \(F'\to G'\).

Proof. \(\ell_G=0\) (Corollary 2.3 of the lesson on the co-Lie complex), so \(\ell^!_G=0\), and \({}_A\ell^\vee_G=0\), its restriction of scalars to \(\mathcal O\) being \(\ell^!_G\) (Proposition 3.2 (2) of the lesson on Lie complexes of module schemes). So \(M_G=0\), and all the Ext groups of Theorems 2.1 and 2.2 vanish. \(\square\)

Proposition 3.2 (vector groups). Let \(T=S'\), so that \(\mathcal O_{S'}\) acts on \(S'\)-schemes, \(A=\mathcal O\) the ring scheme \(\mathbf A^1_{S'}\), and \(G=\mathbf V(E)\) the vector group of a finite locally free \(\mathcal O_S\)-module \(E\), with the points \(\mathbf V(E)(X)=\Gamma(X,E_X)\), an \(A_S\)-module scheme. Then \({}_A\ell^\vee_G\cong\bar\varepsilon^*(\varepsilon^*E)\), with \(A\otimes\mathcal O\) acting on \(\varepsilon^*E\) through the product, and for every \(j\) \[ \begin{aligned}&\operatorname{Ext}^j_{A_S}(G,M_G|_A)\\&\quad\cong H^j\bigl(S,\mathcal Hom(E,E\otimes I)\bigr),\end{aligned} \] the cohomology of the Zariski site. So the obstruction to deforming \(\mathbf V(E)\) as a module over \(\mathcal O\) lies in \(H^2(S,\mathcal End(E)\otimes I)\), the deformations form a torsor under \(H^1\), and the automorphisms form \(H^0\).

Proof. The Lie complex of \(\mathbf V(E)\) is \(\varepsilon^*E\) in degree \(0\), with \(a\in\mathcal O\) acting by multiplication, and the identification of \({}_A\ell^\vee_G\) is that of Exercise 5.1 of the lesson on Lie complexes of module schemes, whose argument applies to a finite locally free \(E\) in place of \(\mathcal O^r\). Then \(M_G=\bar\varepsilon^*(\varepsilon^*E\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I)\), and \(M_G|_A\) is \(\varepsilon^*E\otimes_{\mathcal O}\varepsilon^*I=\varepsilon^*(E\otimes I)\), a module in degree \(0\), \(E\) being flat, with \(A=\mathcal O\) acting through \(\mathcal O\). The \(A_S\)-modules are the \(\mathcal O\)-modules of the big fppf topos of \(S\), and \(G=\varepsilon^*E\) as an \(\mathcal O\)-module. Since \(\varepsilon^*E\) is finite locally free, \(\operatorname{RHom}_{\mathcal O}(\varepsilon^*E,\varepsilon^*(E\otimes I))\cong\) \(R\Gamma(S_{\mathrm{fppf}},\varepsilon^*(E^\vee\otimes E\otimes I))\) [Tag 08JJ]. By Proposition 1.1 (3) of the lesson on big fppf sites, \(R\varepsilon_*\varepsilon^*N=N\) for the quasi-coherent module \(N=E^\vee\otimes E\otimes I=\mathcal Hom(E,E\otimes I)\), so this is \(R\Gamma(S,\mathcal Hom(E,E\otimes I))\). \(\square\)

For \(S\) affine the groups \(H^j\) vanish for \(j\geq1\): a vector group over an affine base deforms, and all its deformations are isomorphic, as for a finite locally free module.

Proposition 3.3 (deformations of vector groups). In the situation of Proposition 3.2:

  1. For every finite locally free \(\mathcal O_{S'}\)-module \(E'\), \(\mathbf V(E')\) is an \(A_{S'}\)-module scheme, flat and of finite presentation over \(S'\). The \(A_{S'}\)-linear homomorphisms \(\mathbf V(E')\to\mathbf V(F')\) are the maps \(\mathbf V(a)\) for the \(\mathcal O_{S'}\)-linear maps \(a\colon E'\to F'\). The same holds over \(S\).
  2. Every deformation \((G',\iota)\) of \(\mathbf V(E)\) is isomorphic to \((\mathbf V(E'),\mathbf V(\varphi))\) for a finite locally free \(\mathcal O_{S'}\)-module \(E'\) and an isomorphism \(\varphi\colon E\to E'\otimes_{\mathcal O_{S'}}\mathcal O_S\), and the isomorphisms of such deformations are the maps \(\mathbf V(a)\) for the isomorphisms \(a\colon E'\to E''\) with \((a\otimes\mathcal O_S)\circ\varphi'=\varphi''\).

So the deformations of \(\mathbf V(E)\) as an \(A_S\)-module scheme, with their isomorphisms, are the deformations of \(E\) to finite locally free \(\mathcal O_{S'}\)-modules, and the homomorphisms between them are the module maps.

Proof. (1) For a finite locally free \(E'\), \(\mathbf V(E')\) is the relative spectrum of \(\operatorname{Sym}(E'^\vee)\) [Tag 01M1], locally a polynomial ring over \(\mathcal O_{S'}\), so it is flat and of finite presentation. The addition and the action of \(\mathcal O\) correspond to the maps of algebras that send a linear form \(x\in E'^\vee\) to \(x\otimes1+1\otimes x\) and to \(x\otimes t\), \(t\) the coordinate of \(\mathbf A^1\). A homomorphism \(\mathbf V(E')\to\mathbf V(F')\) corresponds to a map of algebras \(\operatorname{Sym}(F'^\vee)\to\operatorname{Sym}(E'^\vee)\). Compatibility with the actions of \(\mathcal O\) means that it preserves the gradings, so it is determined by its part of degree one, a linear map \(F'^\vee\to E'^\vee\), the transpose of a linear map \(a\colon E'\to F'\). Conversely, \(\mathbf V(a)\) sends a point \(e\in\Gamma(X,E'_X)\) to \(a(e)\), which is linear. The same argument applies over \(S\).

(2) The scheme \(G'\times_{S'}S\cong\mathbf V(E)\) is affine over \(S\), and over an affine open \(U'\) of \(S'\) it is a thickening of the restriction of \(G'\). So \(G'\) is affine over \(S'\) [Tag 06AD]: \(G'\) is the relative spectrum of a quasi-coherent \(\mathcal O_{S'}\)-algebra \(R'\), flat over \(\mathcal O_{S'}\). The action of the multiplicative monoid of \(\mathcal O\) is a coaction \(c\colon R'\to R'[t]\). Writing \(c(r)=\sum_nr_nt^n\), associativity gives \(c(r_n)=r_nt^n\) and the unit gives \(r=\sum_nr_n\), so \(R'=\bigoplus_{n\geq0}R'_n\) with \(R'_n=\{r\colon c(r)=rt^n\}\), a grading by quasi-coherent submodules. The action of \(0\) is the composite of the structure morphism and the zero section, and its comorphism \(r\mapsto r_0\) factors through \(\mathcal O_{S'}\); so \(R'_0=\mathcal O_{S'}\). Through \(\iota\), \(R'/IR'\) is \(\operatorname{Sym}(E^\vee)\) with its grading by degree.

Each \(R'_n\) is a direct summand of \(R'\), hence flat, and \(R'_n/IR'_n=\operatorname{Sym}^n(E^\vee)\) is finite locally free. Over an affine open, \(R'_n\) is generated by lifts of generators of \(\operatorname{Sym}^n(E^\vee)\), \(I\) being nilpotent. If \(K\) is the kernel of a surjection \(\mathcal O^k\to R'_n\), flatness of \(R'_n\) makes \(0\to K/IK\to\mathcal O_S^k\to\operatorname{Sym}^n(E^\vee)\to0\) exact, so \(K/IK\), hence \(K\), is finitely generated. So \(R'_n\) is finitely presented and flat, hence finite locally free [Tag 00NX]. Put \(E'=(R'_1)^\vee\), finite locally free, with \(\varphi\colon E\cong E'\otimes\mathcal O_S\) given by \(R'_1/IR'_1=E^\vee\). The multiplication \(\operatorname{Sym}^n(R'_1)\to R'_n\) is surjective, as it is modulo \(I\). Its kernel \(N\) satisfies \(N/IN=0\), by flatness of \(R'_n\) and since the map is bijective modulo \(I\); so \(N=0\). Hence \(R'=\operatorname{Sym}(R'_1)\) as graded algebras, and \(G'=\mathbf V(E')\) with its action of \(\mathcal O\).

The addition is compatible with the actions of \(\mathcal O\), so its comorphism \(\Delta\colon R'\to R'\otimes R'\) preserves the gradings. For \(x\in R'_1\), \(\Delta(x)=x_1\otimes1+1\otimes x_2\) with \(x_1,x_2\in R'_1\), as \(R'_0=\mathcal O_{S'}\). Since the zero section is neutral, \(x_1=x_2=x\). So the addition of \(G'\) is that of \(\mathbf V(E')\). By (1) over \(S\), \(\iota\) is \(\mathbf V(\varphi)\) up to these identifications, and by (1) over \(S'\) the isomorphisms of deformations are as stated. \(\square\)

4. Exercises

4.1. Let \(A=\mathbb Z\), the constant ring scheme over \(T\). Show that an \(A_S\)-module scheme is a commutative group scheme over \(S\), that Definition 0.1 is the definition of deformations of commutative group schemes with commutative deformations, and that Theorem 2.1 (4) puts the obstruction to deforming \(G\) as a commutative group scheme in \(\operatorname{Ext}^2_{\mathbb Z}(G,\ell^!_G\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I)\), Ext groups of abelian sheaves of the big fppf topos of \(S\).

4.2. Let \(k\) be a field, \(S=\operatorname{Spec}k\), \(S'=T=\operatorname{Spec}k[\epsilon]\) with \(\epsilon^2=0\), \(A=\mathbb Z\), and \(G=(\mathbb Z/n\mathbb Z)_S\), the constant group, with \(n\) invertible or not in \(k\). Show that \(G\) has, up to unique isomorphism, exactly one deformation as a commutative group scheme over \(S'\), and find it.

4.3. In Theorem 2.3, let \(u\colon F\to G\) be faithfully flat with kernel \(E\), and suppose \(E\) is étale over \(S\). Show that the pairs \((F',u')\) are unique up to unique isomorphism.

5. Complete solutions

4.1. A module over the constant ring \(\mathbb Z\) of a topos is an abelian group object, so an \(A_S\)-module scheme is a commutative group scheme, flat and locally of finite presentation, and \(A_S\)-linear maps are homomorphisms. A deformation \(G'\) is then a commutative group scheme over \(S'\) with \(\iota\) additive, as in Definition 0.1 of the lesson on deformations of group schemes with the extra condition that \(G'\) be commutative. By Proposition 3.4 of the lesson on Lie complexes of module schemes, \({}_{\mathbb Z}\ell^\vee_G\) is \(\ell^!_G\), so \(M_G=\ell^!_G\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I\), and Theorem 2.1 (4) gives the obstruction in \(\operatorname{Ext}^2_{\mathbb Z}(G,M_G)\). The hypothesis (TI) holds, \(\mathbb Z_T\) being flat over \(T\).

4.2. \(G\) is étale over \(S\), so by Proposition 3.1 it has a unique deformation. The constant group \((\mathbb Z/n\mathbb Z)_{S'}\), with the closed immersion induced by \(i\), is a deformation: it is flat and of finite presentation over \(S'\), commutative, and its base change to \(S\) is \(G\). So it is the deformation, whichever \(n\) is.

4.3. By Theorem 2.3 (4), the groups \(\operatorname{Ext}^j(L_{N(A,F)/N(A,G)},p_F^*I)\) are \(\operatorname{Ext}^j_{A_S}(F,({}_A\ell^\vee_E\otimes^{\mathbf L}_{\mathcal O}\varepsilon^*I)|_A)\), and \({}_A\ell^\vee_E=0\) as in the proof of Proposition 3.1. So the obstruction vanishes, the pairs form a torsor under the zero group, and the automorphism group of a pair is trivial: the pair exists and is unique up to unique isomorphism.

Sources

The deformation theory of module schemes through their spectral nerves, with the obstruction, classification and automorphisms in Ext groups of modules into the Lie complex, the extension of homomorphisms and the deformation of a homomorphism with a given target, is due to Illusie, in his work on the cotangent complex; he applies the case of the constant ring schemes \(\mathbb Z\) and \(\mathbb Z/n\mathbb Z\) to deformations of Barsotti–Tate groups.