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Deforming a morphism together with its target

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

Let \(Z\) be a square-zero thickening of a ringed topos \(Z_o\) by a module \(K\), and let \(f_o\colon X_o\to Y_o\) and \(g_o\colon Y_o\to Z_o\) be morphisms of ringed topoi, 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 as a morphism \(X\to Y\) over \(Z\)? The problem is a deformation problem for the arrow topos \(X_oY_o\) of \(f_o\) over the arrow topos of \(X\to Z\), and the cotangent complex of that morphism of arrow topoi is \(L_{Y_o/Z_o}\) placed over \(Y_o\). So the answer lies in the relative Ext groups of the arrow topos. With \(h_o=g_of_o\), the lesson proves:

Prerequisites. The arrow topos and relative Ext groups (Definition 1.1, Proposition 1.2, 1.3, Definitions 2.6 and 3.1, Propositions 2.7, 2.8, 3.2, 4.1 and 4.2, Exercise 5.3); Deformations of diagrams of schemes (the conormal truncation and the isomorphism (0.1) of its Conventions); Equivariant extensions and deformations (Lemma 5.1); Cotangent complexes of simplicial rings (Corollary 3.7). From the programme lesson The cotangent complex: Theorem 2.2, the transitivity triangle and its naturality (Theorem 4.1), Theorem 6.1, Lemma 7.1, the classification of extensions and the obstruction theorem, Theorems 10.3 and 10.4, and the sheaf construction of its Section 10, with its argument that square-zero thickenings of schemes by quasi-coherent ideals are schemes. From the AI-integrated Stacks project: flatness over a square-zero thickening [Stacks, Tag 08M4]; K-flat resolutions [Tag 06YS], bounded above flat complexes [Tag 06YQ], their pullbacks [Tag 0G7E] and derived pullback [Tag 06YY]; the cotangent complex of a morphism of ringed topoi [Tag 08SU] and its transitivity triangle [Tag 08V1]; Ext groups between bounded objects [Tag 06XS] and long exact Hom sequences [Tag 0149], triangulated categories [Tag 0145] and the triangle of a short exact sequence [Tag 0152]; inverse images of cotangent complexes [Tag 08SV]; locally ringed sites and topoi and their morphisms [Tags 04ES, 04EU, 04H8, 04HA]; closed immersions of schemes [Tag 01LD]; cohomology of quasi-coherent modules on affine schemes [Tag 01XB].

Conventions

Rings of a topos are commutative. For a morphism of ringed topoi \(\varphi\), \(\varphi^{-1}\) is the inverse image of sheaves, \(\varphi^*\) the inverse image of modules, \(L\varphi^*\) its derived functor [Tag 06YY] and \(\varphi^\sharp\) the ring map. The cotangent complex of a morphism of ringed topoi \(\varphi\colon P\to Q\) is \(L_{P/Q}=L_{\mathcal O_P/\varphi^{-1}\mathcal O_Q}\) [Tag 08SU], made by the sheaf construction of Section 10 of The cotangent complex. It is a bounded above complex of flat modules, hence K-flat [Tag 06YQ], so that \(L\psi^*L_{P/Q}=\psi^*L_{P/Q}\) for every morphism \(\psi\) into \(P\). Inverse images under morphisms of topoi are exact and preserve the polynomial constructions (Section 10 there): for a morphism of topoi \(\psi\), \(\psi^{-1}L_{B/A}=L_{\psi^{-1}B/\psi^{-1}A}\), compatibly with the functoriality maps and the transitivity triangles. The cotangent complex is concentrated in degrees \(\leq0\) with \(H^0(L_{B/A})=\Omega_{B/A}\) (Section 10 there).

For \(P\in D^{\leq a}\) and \(Q\in D^{\geq b}\), \(\operatorname{Ext}^n(P,Q)=0\) for \(n<b-a\) and \(\operatorname{Ext}^{b-a}(P,Q)=\) \(\operatorname{Hom}(H^aP,H^bQ)\) [Tag 06XS]. For a distinguished triangle \(P_1\xrightarrow{p}P_2\to P_3\xrightarrow{w}P_1[1]\) and an object \(Q\), the sequence \[ \begin{aligned}&\operatorname{Hom}(P_2,Q[1])\xrightarrow{\ p^*\ }\operatorname{Hom}(P_1,Q[1])\\&\quad\xrightarrow{\ \varphi\mapsto\varphi[1]\circ w\ }\operatorname{Hom}(P_3,Q[2])\end{aligned} \] is exact [Tag 0149, applied to the rotated triangle]; these are the connecting maps of the long exact sequences below, and the product of Ext classes is the composite of shifted morphisms. For a surjection \(\mathcal A'\to\mathcal A\) of rings of a topos whose kernel \(\mathcal N\) has square zero, \(L_{\mathcal A/\mathcal A'}\) lies in \(D^{\leq-1}\) with \(H^{-1}=\mathcal N\), and its conormal truncation \(t\colon L_{\mathcal A/\mathcal A'}\to\mathcal N[1]\) is the map inducing the identity on \(H^{-1}\); for \(P\in D^{\leq-1}(\mathcal A)\) and a ring map \(\mathcal A\to\mathcal B\), \(H^{-1}(P\otimes^{\mathbf L}_{\mathcal A}\mathcal B)=H^{-1}(P)\otimes_{\mathcal A}\mathcal B\) (Conventions of Deformations of diagrams of schemes, (0.1)).

The arrow topos of a morphism of ringed topoi \(\varphi\colon P\to Q\) is written \(PQ\), ringed by \((\mathcal O_P\leftarrow\mathcal O_Q)\), with its restrictions \(i_P\), \(i_Q\) and its projection \(\pi\colon PQ\to Q\) (Definition 1.1 and Proposition 1.2 of The arrow topos and relative Ext groups).

1. Extensions and the deformation problem

Definition 1.1. Let \(q\colon Y_o\to Z\) be a morphism of ringed topoi and \(J\) an \(\mathcal O_{Y_o}\)-module. A \(Z\)-extension of \(Y_o\) by \(J\) is a ringed topos \(Y\) with the underlying topos of \(Y_o\), together with a surjection of rings \(j^\sharp\colon\mathcal O_Y\to\mathcal O_{Y_o}\) whose kernel has square zero and is identified with \(J\) as an \(\mathcal O_{Y_o}\)-module, and a ring map \(g^\sharp\colon q^{-1}\mathcal O_Z\to\mathcal O_Y\) with \(j^\sharp\circ g^\sharp=q^\sharp\). We write \(j\colon Y_o\to Y\) and \(g\colon Y\to Z\) for the morphisms of ringed topoi with these ring maps, whose underlying morphisms of topoi are the identity and that of \(q\); so \(g\circ j=q\). Thus a \(Z\)-extension of \(Y_o\) by \(J\) is a \(q^{-1}\mathcal O_Z\)-algebra extension of \(\mathcal O_{Y_o}\) by \(J\) in the sense of Theorem 10.3 of The cotangent complex. An isomorphism of \(Z\)-extensions of \(Y_o\) by \(J\) is a ring isomorphism of the structure sheaves compatible with the maps from \(q^{-1}\mathcal O_Z\), the maps to \(\mathcal O_{Y_o}\) and the identifications of the kernels with \(J\).

Let \(k\colon Z_o\to Z\) be given by a surjection \(\mathcal O_Z\to\mathcal O_{Z_o}\) whose kernel \(K\) has square zero, and \(q=k\circ g_o\) for a morphism \(g_o\colon Y_o\to Z_o\). Then \(q^{-1}=g_o^{-1}\), and \(g^\sharp\) carries \(g_o^{-1}K\) into \(J\), its image in \(\mathcal O_{Y_o}\) being \(g_o^\sharp\) of the image of \(K\) in \(\mathcal O_{Z_o}\), which is zero. The \(\mathcal O_{Y_o}\)-linear extension of this map is denoted \[ v_Y\colon g_o^*K\longrightarrow J. \] The extension \(Y\) is a flat deformation of \(Y_o\) over \(Z\) if \(g\) is flat, that is, \(\mathcal O_Y\) is a flat \(g_o^{-1}\mathcal O_Z\)-module, and \(v_Y\) is an isomorphism.

Lemma 1.2. In Definition 1.1, suppose that \(g_o\) is flat. Then \(v_Y\) is an isomorphism if and only if \(g\) is flat and \(J\) is the ideal \((g_o^{-1}K)\mathcal O_Y\) generated by \(g_o^{-1}K\). In particular \(Y\) is a flat deformation of \(Y_o\) over \(Z\) if and only if \(g\) is flat and \(\mathcal O_Y/(g_o^{-1}K)\mathcal O_Y=\mathcal O_{Y_o}\).

Proof. The ideal \((g_o^{-1}K)\mathcal O_Y\) lies in \(J\), and since \(J\cdot J=0\) it is the \(\mathcal O_{Y_o}\)-submodule of \(J\) generated by the image of \(g_o^{-1}K\), that is, the image of \(v_Y\). Apply [Tag 08M4] to the surjection \(\mathcal O'=g_o^{-1}\mathcal O_Z\to\mathcal O=g_o^{-1}\mathcal O_{Z_o}\), whose kernel \(g_o^{-1}K\) has square zero, and to \(\mathcal F'=\mathcal O_Y\): \(\mathcal O_Y\) is \(\mathcal O'\)-flat if and only if \(\mathcal F=\mathcal O_Y/(g_o^{-1}K)\mathcal O_Y\) is \(\mathcal O\)-flat and \(g_o^{-1}K\otimes_{\mathcal O}\mathcal F\to\mathcal O_Y\) is injective. If \(v_Y\) is an isomorphism, its image \(J\) equals \((g_o^{-1}K)\mathcal O_Y\), so \(\mathcal F=\mathcal O_{Y_o}\), which is \(\mathcal O\)-flat because \(g_o\) is flat, and \(g_o^{-1}K\otimes_{\mathcal O}\mathcal O_{Y_o}=g_o^*K\to\mathcal O_Y\) is \(v_Y\) followed by the inclusion of \(J\), which is injective; so \(g\) is flat. Conversely, if \(g\) is flat and \(J=(g_o^{-1}K)\mathcal O_Y\), then \(\mathcal F=\mathcal O_{Y_o}\), and the injectivity says that \(v_Y\) is injective; its image is \(J\). \(\square\)

1.3. The data. Throughout, \(S\) is a ringed topos and \(Z\to S\) a morphism of ringed topoi; \(k\colon Z_o\to Z\) is given by a surjection \(\mathcal O_Z\to\mathcal O_{Z_o}\) whose kernel \(K\) has square zero, and \(Z_o\) maps to \(S\) through \(Z\). Further \(f_o\colon X_o\to Y_o\) and \(g_o\colon Y_o\to Z_o\) are morphisms of ringed topoi, \(h_o=g_o\circ f_o\), and \(X\) is a \(Z\)-extension of \(X_o\) by an \(\mathcal O_{X_o}\)-module \(I\) for \(k\circ h_o\), with \(i\colon X_o\to X\), \(h\colon X\to Z\) and \[ w=v_X\colon h_o^*K\longrightarrow I. \] The general problem takes, in addition, an \(\mathcal O_{Y_o}\)-module \(J\) and maps \[ \begin{gathered}v\colon g_o^*K\to J,\qquad u\colon f_o^*J\to I,\\ u\circ f_o^*(v)=w,\end{gathered} \] where \(f_o^*g_o^*K=h_o^*K\).

Definition 1.4. A solution for \((J,v,u)\) is a \(Z\)-extension \((Y,j,g)\) of \(Y_o\) by \(J\) for \(k\circ g_o\) with \(v_Y=v\), together with a morphism of ringed topoi \(f\colon X\to Y\) whose underlying morphism of topoi is that of \(f_o\), such that \(f\circ i=j\circ f_o\), \(g\circ f=h\), and the map \(f_o^*J\to I\) induced by \(f^\sharp\) is \(u\). (Since \(f\circ i=j\circ f_o\), \(f^\sharp\) carries \(f_o^{-1}J\) into \(I\).) An isomorphism of solutions \((Y,f)\to(Y',f')\) is an isomorphism of \(Z\)-extensions \(\theta\colon Y\to Y'\) with \(\theta\circ f=f'\).

If \(g_o\) is flat and \(X\) is a flat deformation of \(X_o\) over \(Z\), the flat problem asks for a flat deformation \(Y\) of \(Y_o\) over \(Z\) together with such an \(f\). These are the solutions for \((g_o^*K,\mathrm{id},w)\): the condition \(u\circ f_o^*(\mathrm{id})=w\) forces \(u=w\).

2. The problem on the arrow topos

2.1. The rings. Let \(T=X_oY_o\) be the arrow topos of \(f_o\), with \(\mathcal B=\mathcal O_T=(\mathcal O_{X_o}\leftarrow\mathcal O_{Y_o})\). The square \(g_o\circ f_o=h_o\circ\mathrm{id}\) gives the morphism \(\gamma=(\mathrm{id},g_o)\colon X_oY_o\to X_oZ_o\) of Proposition 4.1 of the arrow topos lesson, with \(\gamma^{-1}(E\leftarrow F)=\) \((E\leftarrow g_o^{-1}F)\). The morphisms \(i\) and \(k\) are the identity on topoi, so \(h\) has the underlying morphism of \(h_o\), and the arrow topos \(XZ\) of \(h\) has the underlying topos of \(X_oZ_o\), ringed by \((\mathcal O_X\leftarrow\mathcal O_Z)\). The surjection \((i^\sharp,k^\sharp)\colon\mathcal O_{XZ}\to\mathcal O_{X_oZ_o}\) has the kernel \[ \mathcal N=(I\leftarrow K), \] of square zero, whose structure map, as an \(\mathcal O_{X_oZ_o}\)-module, is \(w\). Put \[ \begin{aligned}\mathcal A'&=\gamma^{-1}\mathcal O_{XZ}=(\mathcal O_X\leftarrow g_o^{-1}\mathcal O_Z),\\ \mathcal A&=\gamma^{-1}\mathcal O_{X_oZ_o}\\&=(\mathcal O_{X_o}\leftarrow g_o^{-1}\mathcal O_{Z_o}).\end{aligned} \] These are rings of \(T\); \(\mathcal A'\to\mathcal A\) is surjective with kernel \(\gamma^{-1}\mathcal N=(I\leftarrow g_o^{-1}K)\) of square zero, \(\gamma^{-1}\) being exact, and \(\gamma^\sharp\colon\mathcal A\to\mathcal B\). For data \((J,v,u)\) as in 1.3, \(M=(I\xleftarrow{u}J)\) is a \(\mathcal B\)-module (1.3 of the arrow topos lesson), and the map \[ c\colon\gamma^{-1}\mathcal N\longrightarrow M, \] which is the identity of \(I\) over \(X_o\) and \(g_o^{-1}K\to g_o^*K\xrightarrow{v}J\) over \(Y_o\), is \(\mathcal A\)-linear. It is compatible with the structure maps because both composites \(h_o^{-1}K\to I\) are \(w\) composed with \(h_o^{-1}K\to h_o^*K\), by \(u\circ f_o^*(v)=w\).

Proposition 2.2. The solutions for \((J,v,u)\) and their isomorphisms correspond to the \(\mathcal A'\)-algebra extensions of \(\mathcal B\) by \(M\) whose structure map carries \(\gamma^{-1}\mathcal N\) into \(M\) by \(c\), in the sense of Theorem 10.4 of The cotangent complex, and their isomorphisms.

Proof. A ring of \(T\) is a pair of rings \((E_X\leftarrow E_Y)\) with a ring map \(f_o^{-1}E_Y\to E_X\) (1.3 of the arrow topos lesson).

Given a solution \((Y,f)\), put \(\mathcal E=(\mathcal O_X\leftarrow\mathcal O_Y)\) with the structure map \(f^\sharp\colon f_o^{-1}\mathcal O_Y\to\mathcal O_X\); this is a ring of \(T\) since \(f\) has the underlying morphism of \(f_o\). The pair \((\mathrm{id}_{\mathcal O_X},g^\sharp)\colon\mathcal A'\to\mathcal E\) is a ring map of \(T\) because \(f^\sharp\circ f_o^{-1}(g^\sharp)=h^\sharp\), which is \(g\circ f=h\); and \((i^\sharp,j^\sharp)\colon\mathcal E\to\mathcal B\) is one because \(f\circ i=j\circ f_o\). The second is surjective with kernel \((I\leftarrow J)\), whose structure map as a \(\mathcal B\)-module is the map \(f_o^*J\to I\) induced by \(f^\sharp\), that is \(u\); so the kernel is \(M\). The structure map carries \(\gamma^{-1}\mathcal N\) into \(M\) by the identity of \(I\) over \(X_o\) and by \(v_Y=v\) over \(Y_o\), that is by \(c\).

Conversely, let \(\mathcal E=(E_X\xleftarrow{\phi}E_Y)\) be such an extension. Over \(X_o\), \(E_X\) is an extension of \(\mathcal O_{X_o}\) by \(I\) and an \(\mathcal O_X\)-algebra, and \(I\subseteq\mathcal O_X\) maps to \(I\subseteq E_X\) by the identity. The algebra map \(\mathcal O_X\to E_X\) is compatible with the surjections onto \(\mathcal O_{X_o}\), since \(\mathcal A'\to\mathcal A\to\mathcal B\) is \(i^\sharp\) over \(X_o\), and with the identity on \(I\). By the five lemma it is an isomorphism; we identify \(E_X=\mathcal O_X\). Over \(Y_o\), \(Y=(Y_o,E_Y)\), with \(E_Y\to\mathcal O_{Y_o}\) and \(g_o^{-1}\mathcal O_Z\to E_Y\), is a \(Z\)-extension of \(Y_o\) by \(J\), and \(v_Y=v\) because \(\mathcal E\) carries \(g_o^{-1}K\) into \(J\) by \(c\). The structure map \(\phi\colon f_o^{-1}E_Y\to E_X=\mathcal O_X\) is a ring map and defines \(f\colon X\to Y\) with the underlying morphism of \(f_o\). That \(\mathcal E\to\mathcal B\) is a map of rings of \(T\) says \(f\circ i=j\circ f_o\); that \(\mathcal A'\to\mathcal E\) is one says \(g\circ f=h\); and the map \(f_o^*J\to I\) induced by \(\phi\) is the structure map \(u\) of the kernel \(M\).

The two constructions are inverse to each other. An isomorphism of solutions \(\theta\colon(Y,f)\to(Y',f')\) gives the isomorphism \((\mathrm{id}_{\mathcal O_X},\theta^\sharp)\colon\mathcal E'\to\mathcal E\) of extensions, which is a map of rings of \(T\) because \(\theta\circ f=f'\). Conversely, an isomorphism of extensions \(\mathcal E'\to\mathcal E\) is an \(\mathcal O_X\)-algebra map over \(X_o\), hence the identity under the identifications \(E'_X=\mathcal O_X=E_X\), and its component over \(Y_o\) is an isomorphism of solutions. \(\square\)

Lemma 2.3. \(L_{X_oY_o/X_oZ_o}=L_{\mathcal B/\mathcal A}\), and the map \((0,\mathrm{id})\colon L_{\mathcal B/\mathcal A}\to(0\leftarrow L_{Y_o/Z_o})\) is an isomorphism of \(D(\mathcal B)\).

Proof. The first equality is [Tag 08SU], since \(\gamma^{-1}\mathcal O_{X_oZ_o}=\mathcal A\). Inverse images commute with the construction of the cotangent complex, so \(i_X^{-1}L_{\mathcal B/\mathcal A}=L_{\mathcal O_{X_o}/\mathcal O_{X_o}}\) and \(i_Y^{-1}L_{\mathcal B/\mathcal A}=L_{\mathcal O_{Y_o}/g_o^{-1}\mathcal O_{Z_o}}=L_{Y_o/Z_o}\), in agreement with Proposition 4.2 of the arrow topos lesson. The first is acyclic, the cotangent complex of a polynomial algebra in no variables (Theorem 2.2 of The cotangent complex). Represent \(L_{\mathcal B/\mathcal A}\) by its complex \((L_X\leftarrow L_Y)\) of \(\mathcal B\)-modules. The pair \((0,\mathrm{id})\) is a map of complexes of \(\mathcal B\)-modules onto \((0\leftarrow L_Y)\), a quasi-isomorphism over \(X_o\), where \(L_X\) is acyclic, and the identity over \(Y_o\); cohomology sheaves are computed on the two levels. \(\square\)

3. Obstructions, solutions and automorphisms

Theorem 3.1. In 1.3, let \((J,v,u)\) be data for the general problem and put \[ E^i(J)=\operatorname{Ext}^i_T\bigl((0\leftarrow L_{Y_o/Z_o}),(I\xleftarrow{u}J)\bigr). \]

  1. There is a class \(\omega(J,v,u)\in E^2(J)\) that vanishes if and only if a solution for \((J,v,u)\) exists.
  2. If \(\omega(J,v,u)=0\), the isomorphism classes of solutions form a torsor under \(E^1(J)\), and the automorphism group of each solution is isomorphic to \(E^0(J)\).
  3. There is a long exact sequence \[ \begin{aligned}&\cdots\to\operatorname{Ext}^{i-1}(Lf_o^*L_{Y_o/Z_o},I)\\&\quad\to E^i(J)\\&\quad\to\operatorname{Ext}^i(L_{Y_o/Z_o},J)\\&\quad\to\operatorname{Ext}^i(Lf_o^*L_{Y_o/Z_o},I)\to\cdots,\end{aligned} \] whose second map is restriction to \(Y_o\) and whose third map sends \(b\) to the composite of \(Lf_o^*b\) with \(Lf_o^*J\to f_o^*J\xrightarrow{u}I\).

Explicitly, let \(D=L_{\mathcal A/\mathcal A'}\otimes^{\mathbf L}_{\mathcal A}\mathcal B\in D^{\leq-1}(\mathcal B)\), with \(H^{-1}D=\mathcal B\otimes_{\mathcal A}\gamma^{-1}\mathcal N=\gamma^*\mathcal N\); let \(\tilde c\colon D\to M[1]\) be the map inducing on \(H^{-1}\) the \(\mathcal B\)-linear extension \(c_{\mathcal B}\colon\gamma^*\mathcal N\to M\) of \(c\) [Tag 06XS]; and let \[ D\to L_{\mathcal B/\mathcal A'}\to L_{\mathcal B/\mathcal A}\xrightarrow{\ \delta\ }D[1] \] be the transitivity triangle of \(\mathcal A'\to\mathcal A\to\mathcal B\). Then \(\omega(J,v,u)=\tilde c[1]\circ\delta\), read in \(E^2(J)\) through Lemma 2.3.

Proof. (1), (2) By Proposition 2.2 the solutions and their isomorphisms are the extensions of Theorem 10.4 of The cotangent complex, for the rings \(\mathcal A'\to\mathcal A\to\mathcal B\) of the topos \(T\), the module \(M\) and the map \(c\). That theorem gives an obstruction \(\partial c\in\operatorname{Ext}^2_{\mathcal B}(L_{\mathcal B/\mathcal A},M)\) whose vanishing is equivalent to existence, the torsor under \(\operatorname{Ext}^1_{\mathcal B}(L_{\mathcal B/\mathcal A},M)\) and the automorphisms \(\operatorname{Ext}^0_{\mathcal B}(L_{\mathcal B/\mathcal A},M)\). In its proof, \(\operatorname{Ext}^1(D,M)\) is identified with \(\operatorname{Hom}_{\mathcal A}(\gamma^{-1}\mathcal N,M)\) by composing \(D\to\gamma^*\mathcal N[1]\) with the \(\mathcal B\)-linear extension of a map; so \(c\) corresponds to \(\tilde c\). The connecting map \(\partial\) of the exact sequence (10.9) there is composition with \(\delta\), by the exact sequence of the Conventions; hence \(\partial c=\tilde c[1]\circ\delta\). Lemma 2.3 identifies the groups with \(E^i(J)\).

(3) This is Proposition 2.8 (3) of the arrow topos lesson for \(L_{Y_o/Z_o}\in D^-(Y_o)\) and the module \(C=(I\xleftarrow{u}J)\), for which the map \(u_C\) of Definition 2.6 there is \(Lf_o^*J\to f_o^*J\xrightarrow{u}I\). \(\square\)

Theorem 3.2. In 1.3, suppose that \(g_o\) is flat and that \(X\) is a flat deformation of \(X_o\) over \(Z\).

  1. The morphism \(h_o\) is flat, and the canonical map \(L\pi^*(g_o^*K)\to\pi^*(g_o^*K)\), where \(\pi^*(g_o^*K)=(h_o^*K\leftarrow g_o^*K)\), and the map \((w,\mathrm{id})\colon\pi^*(g_o^*K)\to M\), where \(M=(I\xleftarrow{w}g_o^*K)\), are isomorphisms. Hence \[ \begin{aligned}&E^i(g_o^*K)\\&\quad\cong\operatorname{Ext}^i(Y_o/X_o;L_{Y_o/Z_o},g_o^*K).\end{aligned} \]
  2. 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 has a solution. 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 kernel of \[ \begin{aligned}&\operatorname{Hom}(\Omega_{Y_o/Z_o},g_o^*K)\\&\quad\to\operatorname{Hom}(f_o^*\Omega_{Y_o/Z_o},h_o^*K),\\&\quad\alpha\mapsto f_o^*\alpha,\end{aligned} \] that is, the group of derivations of \(\mathcal O_{Y_o}\) over \(g_o^{-1}\mathcal O_{Z_o}\) with values in \(g_o^*K\) whose pullback to \(X_o\) vanishes.

Proof. (1) Since \(w\) is surjective, \((h_o^{-1}K)\mathcal O_X=I\), so \(\mathcal O_X/(h_o^{-1}K)\mathcal O_X=\mathcal O_{X_o}\); as \(h\) is flat, [Tag 08M4] shows that \(\mathcal O_{X_o}\) is flat over \(h_o^{-1}\mathcal O_{Z_o}\), that is, \(h_o\) is flat. Let \(P\to K\) be a quasi-isomorphism from a K-flat complex with flat terms [Tag 06YS]. As \(g_o\) is flat, \(g_o^*\) is exact, so \(g_o^*P\to g_o^*K\) is a quasi-isomorphism, and \(g_o^*P\) is K-flat with flat terms [Tag 0G7E]. Hence \(Lf_o^*(g_o^*K)=f_o^*g_o^*P=h_o^*P\) [Tag 06YY], and the canonical map \(Lf_o^*(g_o^*K)\to f_o^*g_o^*K\) is \(h_o^*P\to h_o^*K\), a quasi-isomorphism because \(h_o^*\) is exact. So \(L\pi^*(g_o^*K)=\) \((Lf_o^*g_o^*K\leftarrow g_o^*K)\to\pi^*(g_o^*K)\) is an isomorphism on both levels. The pair \((w,\mathrm{id})\) is a map of \(\mathcal B\)-modules, since \(w\circ\mathrm{id}=u\circ f_o^*(\mathrm{id})\) with \(u=w\), and an isomorphism, since \(w\) is. Definition 3.1 of the arrow topos lesson applies to \(L_{Y_o/Z_o}\in D^-(Y_o)\) and \(g_o^*K\), with \(Lf_o^*(g_o^*K)\cong h_o^*K\) bounded below, and gives the last isomorphism.

(2) By Lemma 1.2 and Definition 1.4, the solutions of the flat problem are those for \((g_o^*K,\mathrm{id},w)\); apply Theorem 3.1 and (1).

(3) By Theorem 3.1 (2) and (1), the automorphism group is \(\operatorname{Ext}^0(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\). In the long exact sequence of Proposition 3.2 of the arrow topos lesson, \[ \begin{aligned}&\operatorname{Ext}^{-1}(Lf_o^*L_{Y_o/Z_o},h_o^*K)\\&\quad\to\operatorname{Ext}^0(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\\&\quad\to\operatorname{Ext}^0(L_{Y_o/Z_o},g_o^*K)\\&\quad\to\operatorname{Ext}^0(Lf_o^*L_{Y_o/Z_o},h_o^*K),\end{aligned} \] the last map is \(Lf_o^*\). The complexes \(L_{Y_o/Z_o}\) and \(Lf_o^*L_{Y_o/Z_o}\) lie in \(D^{\leq0}\), with \(H^0\) equal to \(\Omega_{Y_o/Z_o}\) and \(f_o^*\Omega_{Y_o/Z_o}\) by (0.1). By [Tag 06XS] the first group vanishes, the third is \(\operatorname{Hom}(\Omega_{Y_o/Z_o},g_o^*K)\), the fourth \(\operatorname{Hom}(f_o^*\Omega_{Y_o/Z_o},h_o^*K)\), and the last map is \(\alpha\mapsto f_o^*\alpha\). Finally \(\operatorname{Hom}(\Omega_{Y_o/Z_o},N)\) is the group of derivations of \(\mathcal O_{Y_o}\) over \(g_o^{-1}\mathcal O_{Z_o}\) into \(N\). \(\square\)

4. The obstruction as a product

Let \(\mathcal R_0\) be the inverse image of \(\mathcal O_S\) on \(X_oZ_o\), through \(X_oZ_o\xrightarrow{\pi}Z_o\to S\). The ring \(\mathcal O_{XZ}\) is an \(\mathcal R_0\)-algebra extension of \(\mathcal O_{X_oZ_o}\) by \(\mathcal N\), through \(XZ\xrightarrow{\pi}Z\to S\). Its class (Theorem 10.3 of The cotangent complex) is \[ \begin{aligned}&e(i,k)=t\circ\psi\colon L_{X_oZ_o/S}\\&\quad\to L_{\mathcal O_{X_oZ_o}/\mathcal O_{XZ}}\to\mathcal N[1],\end{aligned} \] where \(\psi\) is the functoriality map and \(t\) the conormal truncation. Let \[ \begin{aligned}&K(X_oY_o/X_oZ_o/S)\colon L_{X_oY_o/X_oZ_o}\\&\quad\longrightarrow L\gamma^*L_{X_oZ_o/S}[1]\end{aligned} \] be the Kodaira–Spencer class, the third map of the transitivity triangle of \(X_oY_o\xrightarrow{\gamma}X_oZ_o\to S\) [Tag 08V1].

Theorem 4.1. In 1.3, for data \((J,v,u)\), \[ \begin{aligned}&\omega(J,v,u)\\&\quad=\bigl(c_{\mathcal B}[1]\circ\tau[1]\circ L\gamma^*e(i,k)\bigr)[1]\\&\qquad\circ K(X_oY_o/X_oZ_o/S),\end{aligned} \] where \(\tau\colon L\gamma^*\mathcal N\to\gamma^*\mathcal N\) is the canonical map and \(c_{\mathcal B}\colon\gamma^*\mathcal N\to M\) the \(\mathcal B\)-linear extension of \(c\). In the flat problem \(\gamma^*\mathcal N=(I\xleftarrow{w}g_o^*K)=M\) and \(c_{\mathcal B}\) is the identity, so \(\omega\) is the product of \(K(X_oY_o/X_oZ_o/S)\) with the pullback of the class of the thickening \(XZ\) of \(X_oZ_o\).

Proof. Let \(\mathcal R=\gamma^{-1}\mathcal R_0\), the inverse image of \(\mathcal O_S\) on \(T\). The ring maps \(\mathcal R\to\mathcal A'\to\mathcal A\to\mathcal B\) give a map of triples from \((\mathcal R,\mathcal A,\mathcal B)\) to \((\mathcal A',\mathcal A,\mathcal B)\), the identity on \(\mathcal A\) and \(\mathcal B\). Here \(L_{\mathcal A/\mathcal R}=\gamma^{-1}L_{X_oZ_o/S}\), so \(L_{\mathcal A/\mathcal R}\otimes^{\mathbf L}_{\mathcal A}\mathcal B=L\gamma^*L_{X_oZ_o/S}\), and \(L_{\mathcal B/\mathcal R}=L_{X_oY_o/S}\); so the transitivity triangle of the first triple is that of \(X_oY_o\to X_oZ_o\to S\), with third map \(K(X_oY_o/X_oZ_o/S)\). By the naturality of the transitivity triangle (Theorem 4.1 of The cotangent complex, in the sheaf form of its Section 10), it maps to the triangle of the second triple by \(\varphi\otimes1\), the functoriality map \(L_{\mathcal B/\mathcal R}\to L_{\mathcal B/\mathcal A'}\), the identity of \(L_{\mathcal B/\mathcal A}\) and \((\varphi\otimes1)[1]\), where \(\varphi\colon L_{\mathcal A/\mathcal R}\to L_{\mathcal A/\mathcal A'}\) is the functoriality map. Hence \(\delta=(\varphi\otimes1)[1]\circ K(X_oY_o/X_oZ_o/S)\), and by Theorem 3.1 \[ \begin{aligned}&\omega(J,v,u)\\&\quad=\bigl(\tilde c\circ(\varphi\otimes1)\bigr)[1]\\&\qquad\circ K(X_oY_o/X_oZ_o/S).\end{aligned} \] It remains to show \(\tilde c\circ(\varphi\otimes1)=c_{\mathcal B}[1]\circ\tau[1]\circ L\gamma^*e(i,k)\). Since \(\gamma^{-1}\) preserves the polynomial constructions and their functoriality, \(\gamma^{-1}\psi=\varphi\), so \(L\gamma^*(\psi)=\varphi\otimes1\), and \(L\gamma^*L_{\mathcal O_{X_oZ_o}/\mathcal O_{XZ}}=D\). Thus \(c_{\mathcal B}[1]\circ\tau[1]\circ L\gamma^*e(i,k)=\) \(c_{\mathcal B}[1]\circ\tau[1]\circ L\gamma^*(t)\circ(\varphi\otimes1)\). The maps \(\tilde c\) and \(c_{\mathcal B}[1]\circ\tau[1]\circ L\gamma^*(t)\) go from \(D\in D^{\leq-1}\) to \(M[1]\), so by [Tag 06XS] they are equal if they induce the same map on \(H^{-1}\). For \(\tilde c\) this is \(c_{\mathcal B}\) by definition; for the other it is \(c_{\mathcal B}\circ\gamma^*H^{-1}(t)=c_{\mathcal B}\), under the identification \(H^{-1}D=\gamma^*\mathcal N\) of (0.1). \(\square\)

5. Deforming the target first

Let \(\omega_Y\in\operatorname{Ext}^2(L_{Y_o/Z_o},g_o^*K)\) be the obstruction of Theorem 10.4 of The cotangent complex for the rings \(g_o^{-1}\mathcal O_Z\to g_o^{-1}\mathcal O_{Z_o}\to\mathcal O_{Y_o}\) of the topos of \(Y_o\), the module \(g_o^*K\) and the canonical map \(c_Y\colon g_o^{-1}K\to g_o^*K\).

Proposition 5.1. Suppose that \(g_o\) is flat and that \(X\) is a flat deformation of \(X_o\) over \(Z\).

  1. \(\omega_Y=0\) if and only if \(Y_o\) has a flat deformation over \(Z\).
  2. The restriction map \[ \begin{aligned}&\operatorname{Ext}^2(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\\&\quad\to\operatorname{Ext}^2(L_{Y_o/Z_o},g_o^*K)\end{aligned} \] of Proposition 3.2 of the arrow topos lesson carries \(\omega\) to \(\omega_Y\).
  3. If \(Y_o\) has a flat deformation over \(Z\), then \(\omega\) is the image under the connecting map of an element of \(\operatorname{Ext}^1(Lf_o^*L_{Y_o/Z_o},h_o^*K)\). The image of this connecting map is isomorphic to the cokernel of \(Lf_o^*\colon\operatorname{Ext}^1(L_{Y_o/Z_o},g_o^*K)\to\) \(\operatorname{Ext}^1(Lf_o^*L_{Y_o/Z_o},h_o^*K)\).

Proof. (1) The \(g_o^{-1}\mathcal O_Z\)-algebra extensions of \(\mathcal O_{Y_o}\) by \(g_o^*K\) whose structure map carries \(g_o^{-1}K\) into \(g_o^*K\) by \(c_Y\) are the \(Z\)-extensions \(Y\) of \(Y_o\) by \(g_o^*K\) with \(v_Y=\mathrm{id}\); by Lemma 1.2 these are the flat deformations of \(Y_o\) over \(Z\). Theorem 10.4 of The cotangent complex says that one exists if and only if \(\omega_Y=0\).

(2) The inverse image \(i_Y^{-1}\) is exact and carries the data \((\mathcal A',\mathcal A,\mathcal B,\gamma^{-1}\mathcal N,M,c)\) of the flat problem to \((g_o^{-1}\mathcal O_Z,g_o^{-1}\mathcal O_{Z_o},\mathcal O_{Y_o},g_o^{-1}K,g_o^*K,c_Y)\). It commutes with the polynomial constructions, so it carries \(D\), the transitivity triangle of \(\mathcal A'\to\mathcal A\to\mathcal B\) and \(\tilde c\) to the corresponding objects for the data over \(Y_o\). Restriction \(\operatorname{Ext}_T(P,Q)\to\operatorname{Ext}_{Y_o}(i_Y^*P,i_Y^*Q)\) is compatible with composition and shifts, so it carries \(\omega=\tilde c[1]\circ\delta\) to \(\tilde c_Y[1]\circ\delta_Y\), which is \(\omega_Y\) by the proof of Theorem 3.1 applied to the data over \(Y_o\). The isomorphisms of Lemma 2.3 and Theorem 3.2 (1) restrict to identities over \(Y_o\), so this restriction is the first map of Proposition 3.2 of the arrow topos lesson.

(3) If \(\omega_Y=0\), exactness of the sequence of Proposition 3.2 there, \[ \begin{aligned}&\operatorname{Ext}^1(L_{Y_o/Z_o},g_o^*K)\\&\quad\to\operatorname{Ext}^1(Lf_o^*L_{Y_o/Z_o},h_o^*K)\\&\quad\to\operatorname{Ext}^2(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\\&\quad\to\operatorname{Ext}^2(L_{Y_o/Z_o},g_o^*K),\end{aligned} \] places \(\omega\) in the image of the connecting map, which is isomorphic to the cokernel of the first map. \(\square\)

5.2. A given deformation of the target. Let \(Y\) be a flat deformation of \(Y_o\) over \(Z\), with \(j\colon Y_o\to Y\) and \(g\colon Y\to Z\). Put \(\mathcal C_X=h_o^{-1}\mathcal O_Z\), \(L=L_{Y_o/Z_o}\) and \(L'=L_{\mathcal O_{Y_o}/g_o^{-1}\mathcal O_Z}\), the cotangent complex of \(Y_o\to Z\). As \(g_o\) is flat, the transitivity triangle of \(g_o^{-1}\mathcal O_Z\to g_o^{-1}\mathcal O_{Z_o}\to\mathcal O_{Y_o}\) reads \[ \begin{aligned}&g_o^*L_{Z_o/Z}\xrightarrow{\ \iota\ }L'\xrightarrow{\ \psi\ }L\\&\quad\xrightarrow{\ K_Y\ }g_o^*L_{Z_o/Z}[1].\end{aligned} \] Let \(e_Z\colon L_{Z_o/Z}\to K[1]\) be the conormal truncation, which is the class (10.6) of The cotangent complex of \(\mathcal O_Z\) as an extension of \(\mathcal O_{Z_o}\) by \(K\) over \(\mathcal O_Z\); let \([Y]\in\operatorname{Ext}^1(L',g_o^*K)\) be the class (10.6) of \(\mathcal O_Y\) as a \(g_o^{-1}\mathcal O_Z\)-algebra extension of \(\mathcal O_{Y_o}\) by \(g_o^*K\), and \(e_X\in\operatorname{Ext}^1(L_{\mathcal O_{X_o}/\mathcal C_X},I)\) that of \(\mathcal O_X\) as a \(\mathcal C_X\)-algebra extension of \(\mathcal O_{X_o}\) by \(I\). Let \(F_f\colon Lf_o^*L'\to L_{\mathcal O_{X_o}/\mathcal C_X}\) and \(F_h\colon h_o^*L_{Z_o/Z}\to L_{\mathcal O_{X_o}/\mathcal C_X}\) be the functoriality maps. For \(b\in\operatorname{Ext}^i(L',g_o^*K)\) let \(w(b)\in\operatorname{Ext}^i(Lf_o^*L',I)\) be the composite of \(Lf_o^*b\) with \(Lf_o^*g_o^*K\to h_o^*K\xrightarrow{w}I\), and in the same way for \(L\) in place of \(L'\); this is the third map of the sequence of Theorem 3.1 (3).

Proposition 5.2. Suppose that \(g_o\) is flat, that \(X\) is a flat deformation of \(X_o\) over \(Z\), and that \(Y\) is a flat deformation of \(Y_o\) over \(Z\).

  1. There is a unique \(o(Y)\in\operatorname{Ext}^1(Lf_o^*L,I)\) with \(o(Y)\circ Lf_o^*\psi=e_X\circ F_f-w([Y])\). It vanishes if and only if there is a morphism of ringed topoi \(f\colon X\to Y\) with the underlying morphism of \(f_o\) such that \(f\circ i=j\circ f_o\) and \(g\circ f=h\).
  2. Let \(Y'\) be another flat deformation of \(Y_o\) over \(Z\), and \(d\in\operatorname{Ext}^1(L,g_o^*K)\) the difference of the classes of \(Y'\) and \(Y\) in the torsor of Theorem 10.4 of The cotangent complex, that is \([Y']-[Y]=d\circ\psi\). Then \(o(Y')-o(Y)=-w(d)\).
  3. The connecting map \(\partial\colon\operatorname{Ext}^1(Lf_o^*L,I)\to E^2(g_o^*K)\) of Theorem 3.1 (3) sends \(o(Y)\) to the obstruction \(\omega=\omega(g_o^*K,\mathrm{id},w)\) of the flat problem.

Through \(w\colon h_o^*K\cong I\), \(o(Y)\) is a class in \(\operatorname{Ext}^1(Lf_o^*L_{Y_o/Z_o},h_o^*K)\), and (3) identifies the element of Proposition 5.1 (3).

Proof. The lifting problem. A morphism \(f\) as in (1) is a ring map \(\phi\colon f_o^{-1}\mathcal O_Y\to\mathcal O_X\) that is a map of \(\mathcal C_X\)-algebras, through \(f_o^{-1}(g^\sharp)\) and \(h^\sharp\), and lifts \(f_o^\sharp\circ f_o^{-1}(j^\sharp)\colon f_o^{-1}\mathcal O_Y\to\mathcal O_{X_o}\). Such a \(\phi\) carries the kernel \(f_o^{-1}(g_o^*K)\) of \(f_o^{-1}\mathcal O_Y\to f_o^{-1}\mathcal O_{Y_o}\) into \(I\) by the map \(c\) induced by \(w\): that kernel is generated by the image of \(f_o^{-1}g_o^{-1}K\) (Lemma 1.2), and \(\phi(g^\sharp(k))=h^\sharp(k)\). So these maps \(\phi\) are the lifts of \(f_o^\sharp\colon f_o^{-1}\mathcal O_{Y_o}\to\mathcal O_{X_o}\), in the sense of Lemma 5.1 of Equivariant extensions and deformations, for \(\mathcal C_X\), the extensions \(f_o^{-1}\mathcal O_Y\) and \(\mathcal O_X\), and \(c\). By that lemma they exist if and only if \(\mathrm{Ob}(Y)=e_X\circ F_f-w([Y])\) vanishes, read in \(\operatorname{Ext}^1(Lf_o^*L',I)\): inverse images commute with the cotangent complex [Tag 08SV], so that \(L_{f_o^{-1}\mathcal O_{Y_o}/\mathcal C_X}\otimes^{\mathbf L}\mathcal O_{X_o}=Lf_o^*L'\), and \(\kappa(f_o^{-1}\mathcal O_Y)=f_o^{-1}[Y]\).

(1) The map \(h^\sharp\colon h_o^{-1}\mathcal O_Z\to\mathcal O_X\) is a lift of \(h_o^\sharp\) inducing \(w\) on the kernels, and the class of \(h_o^{-1}\mathcal O_Z\) as an extension of \(h_o^{-1}\mathcal O_{Z_o}\) by \(h_o^{-1}K\) is \(h_o^{-1}e_Z\). By Lemma 5.1 (3) of the lesson on equivariant extensions, \(e_X\circ F_h=w\circ h_o^*(e_Z)\). In the same way \(g^\sharp\), which induces \(v=\mathrm{id}\), gives \([Y]\circ\iota=g_o^*(e_Z)\). Functoriality maps compose, so \(F_f\circ Lf_o^*\iota=F_h\). Hence \[ \begin{aligned}&\mathrm{Ob}(Y)\circ Lf_o^*\iota\\&\quad=w\circ h_o^*(e_Z)-w(g_o^*(e_Z))=0.\end{aligned} \] The image under \(Lf_o^*\) of the triangle above, with \(Lf_o^*g_o^*L_{Z_o/Z}=h_o^*L_{Z_o/Z}\), gives the exact sequence [Tag 0149] \[ \begin{aligned}&\operatorname{Hom}(h_o^*L_{Z_o/Z},I)\to\operatorname{Ext}^1(Lf_o^*L,I)\\&\quad\xrightarrow{\ \circ Lf_o^*\psi\ }\operatorname{Ext}^1(Lf_o^*L',I)\\&\quad\xrightarrow{\ \circ Lf_o^*\iota\ }\operatorname{Ext}^1(h_o^*L_{Z_o/Z},I),\end{aligned} \] whose first term vanishes because \(h_o^*L_{Z_o/Z}\) lies in \(D^{\leq-1}\) [Tag 06XS]. So \(o(Y)\) exists and is unique, and it vanishes exactly when \(\mathrm{Ob}(Y)\) does.

(2) The flat deformations of \(Y_o\) over \(Z\) are the extensions of Theorem 10.4 of The cotangent complex for the canonical map \(g_o^{-1}K\to g_o^*K\) (proof of Proposition 5.1 (1)), which form a torsor under \(\operatorname{Ext}^1(L,g_o^*K)\) acting through \(\psi\); this defines \(d\). Then \(\mathrm{Ob}(Y')-\mathrm{Ob}(Y)=-w(d\circ\psi)\), which is \(-w(d)\circ Lf_o^*\psi\), and the uniqueness in (1) gives \(o(Y')-o(Y)=-w(d)\).

(3) First, \(K_Y=0\). The ring \(\mathcal O_Y\) is flat over \(g_o^{-1}\mathcal O_Z\), and \(\mathcal O_Y\otimes_{g_o^{-1}\mathcal O_Z}g_o^{-1}\mathcal O_{Z_o}=\mathcal O_{Y_o}\) (Lemma 1.2). By Corollary 3.7 of Cotangent complexes of simplicial rings the functoriality map \(\beta\colon L_{\mathcal O_Y/g_o^{-1}\mathcal O_Z}\otimes^{\mathbf L}\mathcal O_{Y_o}\to L\) is an isomorphism. It is \(\psi\circ\psi_1\), with \(\psi_1\) the first map of the transitivity triangle of \(g_o^{-1}\mathcal O_Z\to\mathcal O_Y\to\mathcal O_{Y_o}\). So \(\psi\) has the right inverse \(\psi_1\beta^{-1}\), and \(K_Y=K_Y\circ\psi\circ\psi_1\beta^{-1}=0\).

The obstruction as a Kodaira–Spencer class. By Theorem 4.1 with \(S=Z\), \(\omega=e'[1]\circ K\), where \(U\to V\to W\xrightarrow{K}U[1]\) is the transitivity triangle of \(X_oY_o\to X_oZ_o\to Z\), with \(U=L\gamma^*L_{X_oZ_o/Z}\), \(V=L_{X_oY_o/Z}\), \(W=L_{X_oY_o/X_oZ_o}\), and \(e'=\tau[1]\circ L\gamma^*e(i,k)\colon U\to M[1]\). Inverse images under \(i_X\) and \(i_Y\) commute with the cotangent complex and its transitivity triangles, as in the proof of Lemma 2.3. Over \(X_o\), \(U\to V\) is the identity of \(L_{\mathcal O_{X_o}/\mathcal C_X}\) and \(e'\) is \(e_X\); over \(Y_o\) the triangle is the one above. Represent \(V\) by its complex \((V_X\leftarrow L')\), whose structure map \(f_o^*L'\to V_X\) is \(F_f\), the polynomial constructions being functorial; \(L'\) is a bounded above complex of flat modules, so \(Lf_o^*L'=f_o^*L'\). Replace \(W\) by \((0\leftarrow L)\) through Lemma 2.3, and \(K\) accordingly.

Comparison. For \(N=L'\) or \(L\), let \(\theta_N\colon(0\leftarrow N)\to(f_o^*N\leftarrow0)[1]\) be the third map of the triangle [Tag 0152] of the exact sequence \(0\to(f_o^*N\leftarrow0)\to\pi^*N\) \(\to(0\leftarrow N)\to0\), \(N\) represented by a bounded above complex of flat modules. By the Conventions, the connecting map is \(\partial_N(\varphi)=\tilde\varphi[1]\circ\theta_N\), where \(\tilde\varphi\colon(f_o^*N\leftarrow0)\to M[1]\) is the morphism with component \(\varphi\) over \(X_o\) (adjunction of \(i_{X!}=(-\leftarrow0)\) and \(i_X^*\)); \(\partial=\partial_L\). These triangles are natural in \(N\), so \(\theta_L\circ(0\leftarrow\psi)=(f_o^*\psi\leftarrow0)[1]\circ\theta_{L'}\) and \(\partial(\varphi)\circ(0\leftarrow\psi)=\partial_{L'}(\varphi\circ Lf_o^*\psi)\).

The morphism of complexes \(\sigma=(F_f,\mathrm{id})\colon\pi^*L'\to V\) of the proof of Proposition 2.7 of the arrow topos lesson, followed by \(V\to(0\leftarrow L)\), is the map of complexes \((0,\psi)\), which is also the composite of \(\pi^*L'\to(0\leftarrow L')\) with \((0\leftarrow\psi)\). So \(\sigma\) and \((0\leftarrow\psi)\) form the middle square of a morphism from the triangle of \(L'\) to \(U\to V\to(0\leftarrow L)\xrightarrow{K}U[1]\), which a morphism \(\tilde\rho\colon(f_o^*L'\leftarrow0)\to U\) completes (axioms TR2 and TR3 of triangulated categories [Tag 0145], applied to the rotated triangles). Its first square says that \(\tilde\rho\) followed by \(U\to V\) is \(\sigma\) restricted to \((f_o^*L'\leftarrow0)\); over \(X_o\) this reads \(\tilde\rho_X=F_f\), which determines \(\tilde\rho\) by adjunction. Its third square says \(K\circ(0\leftarrow\psi)=\tilde\rho[1]\circ\theta_{L'}\). Hence \[ \begin{aligned}\omega\circ(0\leftarrow\psi)&=(e'\circ\tilde\rho)[1]\circ\theta_{L'}\\&=\partial_{L'}(e_X\circ F_f).\end{aligned} \] On the other hand \(\partial(o(Y))\circ(0\leftarrow\psi)\) equals \(\partial_{L'}(e_X\circ F_f)-\partial_{L'}(w([Y]))\), and \(\partial_{L'}\circ w=0\): \(w\) and \(\partial_{L'}\) are consecutive maps of the long exact sequence of Proposition 2.8 (3) of the arrow topos lesson for \(L'\), the first being restriction along \((f_o^*L'\leftarrow0)\to\pi^*L'\). So \((\omega-\partial(o(Y)))\circ(0\leftarrow\psi)=0\). The functor \((0\leftarrow-)\) is exact, so \((0\leftarrow L')\to(0\leftarrow L)\) \(\xrightarrow{(0\leftarrow K_Y)}(0\leftarrow g_o^*L_{Z_o/Z})[1]\) is part of a distinguished triangle, and by [Tag 0149] \(\omega-\partial(o(Y))\) factors through \((0\leftarrow K_Y)=0\). So \(\omega=\partial(o(Y))\). \(\square\)

6. Locally ringed topoi, schemes and closed immersions

Lemma 6.1. Let \(E\to E_o\) be a surjection of rings of a topos whose kernel has square zero.

  1. A section of \(E\) over an object \(U\) is invertible if and only if its image in \(E_o\) is.
  2. \(E\) is a local ring, that is, the ringed topos it defines is locally ringed [Tags 04EU, 04H8], if and only if \(E_o\) is.

Proof. (1) Let \(x\in E(U)\) have an invertible image with inverse \(\bar y\). Since \(E\to E_o\) is an epimorphism of sheaves, \(\bar y\) lifts on a covering \(U_\alpha\to U\) to \(y_\alpha\in E(U_\alpha)\). Then \(xy_\alpha=1+n_\alpha\) with \(n_\alpha\) in the kernel, \(n_\alpha^2=0\), so \(xy_\alpha(1-n_\alpha)=1\). Inverses are unique, so these local inverses agree on overlaps and glue to an inverse of \(x\). (2) Over an object where \(0=1\) in \(E_o\), \(1\) lies in the kernel of \(E\to E_o\), so \(1=1^2=0\) in \(E\); hence the subsheaves where \(0=1\) in \(E\) and in \(E_o\) coincide, and one is the initial sheaf if and only if the other is. For the condition of [Tag 04ES] (1): if \(E_o\) satisfies it, then for \(x\in E(U)\) there is a covering on which the image of \(x\) or of \(1-x\) is invertible, and by (1) so is \(x\) or \(1-x\). If \(E\) satisfies it, lift a section of \(E_o\) on a covering and apply the condition to the lifts. \(\square\)

Proposition 6.2. In 1.3, let \((Y,f)\) be a solution for data \((J,v,u)\).

  1. If \(X_o\), \(Y_o\), \(Z_o\), \(X\) and \(Z\) are locally ringed and \(f_o\), \(g_o\) and \(k\) are morphisms of locally ringed topoi [Tag 04HA], then \(Y\) is locally ringed and \(f\), \(g\) and \(j\) are morphisms of locally ringed topoi.
  2. If the topoi in 1.3 are the Zariski topoi of schemes and the morphisms come from morphisms of schemes, and \(J\) is quasi-coherent, then \(Y\) is a scheme and \(f\) and \(g\) are morphisms of schemes. In the flat problem this holds when \(K\) is quasi-coherent.
  3. If \(f_o^\sharp\colon f_o^{-1}\mathcal O_{Y_o}\to\mathcal O_{X_o}\) and \(u\) are surjective, then so is \(f^\sharp\). In particular, in the flat problem for schemes, if \(f_o\) is a closed immersion, so is \(f\).

Proof. (1) By Lemma 6.1 (2) for \(\mathcal O_Y\to\mathcal O_{Y_o}\), \(Y\) is locally ringed. By [Tag 04HA], a morphism of ringed topoi \(\varphi\colon P\to Q\) between locally ringed topoi is a morphism of locally ringed topoi when every section \(s\) of \(\varphi^{-1}\mathcal O_Q\) with \(\varphi^\sharp(s)\) invertible is invertible; \(\varphi^{-1}\) preserves units, commuting with finite limits. Let \(s\) be a section of \(f^{-1}\mathcal O_Y\) with \(f^\sharp(s)\) invertible. The image of \(f^\sharp(s)\) in \(\mathcal O_{X_o}\) is \(f_o^\sharp(\bar s)\), where \(\bar s\) is the image of \(s\) in \(f_o^{-1}\mathcal O_{Y_o}\), since \(f\circ i=j\circ f_o\). It is invertible, so \(\bar s\) is invertible, \(f_o\) being a morphism of locally ringed topoi, and \(s\) is invertible by Lemma 6.1 (1) for the surjection \(f^{-1}\mathcal O_Y\to f^{-1}\mathcal O_{Y_o}\), whose kernel \(f^{-1}J\) has square zero. The same argument with \(g\circ j=k\circ g_o\) and the surjection \(g_o^{-1}\mathcal O_Z\to g_o^{-1}\mathcal O_{Z_o}\) applies to \(g\), and Lemma 6.1 (1) itself to \(j\).

(2) The ringed space \(Y\) is the underlying space of \(Y_o\) with a square-zero thickening \(\mathcal O_Y\) of \(\mathcal O_{Y_o}\) by the quasi-coherent ideal \(J\). The argument following Theorem 10.4 of The cotangent complex shows that such a ringed space is a scheme: it uses only the vanishing of \(H^1\) of quasi-coherent modules on affine schemes [Tag 01XB] and the description of principal opens. By (1), \(f\) and \(g\) are morphisms of locally ringed spaces, hence of schemes. In the flat problem \(J=g_o^*K\), which is quasi-coherent when \(K\) is.

(3) The rows \(0\to f^{-1}J\to f^{-1}\mathcal O_Y\to f^{-1}\mathcal O_{Y_o}\to0\) and \(0\to I\to\mathcal O_X\to\mathcal O_{X_o}\to0\) are exact, \(f^{-1}\) being exact, and \(f^\sharp\) maps the first to the second. On the right it is \(f_o^\sharp\), surjective; on the left it is \(f^{-1}J\to f_o^*J\xrightarrow{u}I\), surjective because \(f_o^*J=\mathcal O_{X_o}\otimes_{f^{-1}\mathcal O_{Y_o}}f^{-1}J\) with \(\mathcal O_{X_o}\) a quotient of \(f^{-1}\mathcal O_{Y_o}\). By the snake lemma \(f^\sharp\) is surjective. In the flat problem \(u=w\) is an isomorphism. For schemes, \(f\) has the underlying map of \(f_o\), a homeomorphism onto a closed subset, and \(\mathcal O_Y\to f_*\mathcal O_X\) is surjective, \(f_*\) being exact for this closed embedding of spaces and \(\mathcal O_Y\to f_*f^{-1}\mathcal O_Y\) surjective; so \(f\) is a closed immersion [Tag 01LD]. \(\square\)

7. Exercises

Exercise 7.1. In the flat problem, let \(Y_o=X_o\) and \(f_o=\mathrm{id}\). Show that the flat problem has a solution, unique up to a unique isomorphism, namely \(Y=X\) and \(f=\mathrm{id}\).

Exercise 7.2. Let \(k\) be a field, \(S=Z_o=\operatorname{Spec}k\), \(Z=\operatorname{Spec}k[\varepsilon]\) with \(\varepsilon^2=0\), \(Y_o=\operatorname{Spec}k[x,y]/(xy)\), \(X_o=\operatorname{Spec}k[x]\) embedded by \(y=0\), and \(X=\operatorname{Spec}k[x,\varepsilon]\); all on Zariski topoi.

  1. For \(a\in k\), let \(Y_a=\operatorname{Spec}R_a\) with \(R_a=k[x,y,\varepsilon]/(\varepsilon^2,xy-a\varepsilon)\). Show that \(Y_a\) is a flat deformation of \(Y_o\) over \(Z\), and that \(f_o\) extends to a morphism \(X\to Y_a\) over \(Z\) if and only if \(a=0\).
  2. Compute the groups \(\operatorname{Ext}^i(Y_o/X_o;L_{Y_o/k},\mathcal O_{Y_o})\) for \(i=0,1,2\) with Theorem 3.1 (3), using \(L_{Y_o/k}\cong[\mathcal O_{Y_o}e\to\mathcal O_{Y_o}dx\oplus\mathcal O_{Y_o}dy]\) in degrees \(-1,0\), \(e\mapsto y\,dx+x\,dy\).
  3. Show that the solutions \((Y_0,f_p)\), where \(f_p^\sharp\colon x\mapsto x+\varepsilon p(x)\), \(y\mapsto0\) for \(p\in k[x]\), fall into isomorphism classes indexed by \(p(0)\in k\), and that the automorphisms of each are the maps \(x\mapsto x\), \(y\mapsto y+\varepsilon\,y\,b(y)\) with \(b\in k[y]\). Compare with Theorem 3.2.

Exercise 7.3. Show that Theorem 3.2 fails if the hypothesis that \(w\) is an isomorphism is dropped. Take \(k\) a field, \(S=Z=Z_o=\operatorname{Spec}k\) (so \(K=0\)), \(Y_o=X_o=\operatorname{Spec}k[t]/(t^2)\), \(f_o=\mathrm{id}\), and \(X=\operatorname{Spec}k[s]/(s^4)\) with \(i\) given by \(s\mapsto t\), so that \(I=(s^2)\). Show that \(h\) is flat, that the groups \(\operatorname{Ext}^i(Y_o/X_o;L_{Y_o/Z_o},g_o^*K)\) vanish, and that there is no solution. Which group of Theorem 3.1 contains the obstruction?

8. Complete solutions

Solution 7.1. Here \(h_o=g_o\), and \(X\) is a flat deformation of \(Y_o\) over \(Z\) whose kernel is identified with \(g_o^*K\) through \(w\); with \(f=\mathrm{id}_X\) it is a solution, \(u=w\). For \(f_o=\mathrm{id}\) the relative Ext groups vanish (Exercise 5.3 of the arrow topos lesson), so by Theorem 3.2 the solutions form a torsor under the zero group, that is, there is exactly one isomorphism class, and the automorphism group of a solution is trivial. So every solution is isomorphic to \((X,\mathrm{id})\) by a unique isomorphism.

Solution 7.2. (1) With the lexicographic order \(x>y>\varepsilon\), the leading monomials \(\varepsilon^2\) and \(xy\) of the generators are coprime, so the generators form a Gröbner basis, and the monomials \(x^i\varepsilon^e\), \(y^j\varepsilon^e\) with \(e\in\{0,1\}\) form a \(k\)-basis of \(R_a\). Hence \(\varepsilon R_a\) has the basis \(\varepsilon x^i\), \(\varepsilon y^j\), and \(r\mapsto\varepsilon r\) is an isomorphism from \(k[x,y]/(xy)\) onto \(\varepsilon R_a\), which is the kernel \(J\) of \(R_a\to k[x,y]/(xy)\) because \(xy=a\varepsilon\). This map is \(v_{Y_a}\colon g_o^*K\to J\), with \(K=k\varepsilon\). By Lemma 1.2, \(Y_a\) is a flat deformation; \(g_o\) is flat over the field \(k\). A morphism \(X\to Y_a\) over \(Z\) extending \(f_o\) is a \(k[\varepsilon]\)-algebra map \(\phi\colon R_a\to k[x,\varepsilon]\) with \(\phi(x)=x+\varepsilon p(x)\) and \(\phi(y)=\varepsilon q(x)\). It must kill \(xy-a\varepsilon\): \[ \begin{aligned}&(x+\varepsilon p)\,\varepsilon q-a\varepsilon\\&\quad=\varepsilon\bigl(xq(x)-a\bigr)=0,\end{aligned} \] so \(xq(x)=a\) in \(k[x]\), which forces \(a=0\) and \(q=0\). For \(a=0\) every \(p\in k[x]\) gives the extension \(f_p\).

(2) For an affine scheme \(\operatorname{Spec}R\), a bounded complex \(L\) of finite free modules and a quasi-coherent module \(N\), \(R\operatorname{Hom}(L,N)\) is computed by \(\operatorname{Hom}^\bullet_R(L,N)\), the sheaf Hom complex having quasi-coherent terms without higher cohomology [Tag 01XB]. The complex \(L=L_{Y_o/k}\) is the cotangent complex of a hypersurface (Lemma 7.1 and Theorem 4.1 of The cotangent complex), and \(Lf_o^*L=f_o^*L\) is \([k[x]e\to k[x]dx\oplus k[x]dy]\), \(e\mapsto x\,dy\). With \(N\) in degree \(0\), \(\operatorname{Hom}^\bullet(L,N)\) is \(N^2\to N\), \((\alpha,\beta)\mapsto y\alpha+x\beta\), and \(\operatorname{Hom}^\bullet(f_o^*L,k[x])\) is \(k[x]^2\to k[x]\), \((\alpha,\beta)\mapsto x\beta\). In \(k[x,y]/(xy)\), whose elements are \(c+\sum_{i\geq1}a_ix^i+\sum_{j\geq1}b_jy^j\), the equation \(y\alpha+x\beta=0\) holds exactly for \(\alpha\in xk[x]\), \(\beta\in yk[y]\). Hence \[ \begin{aligned}\operatorname{Ext}^0(L,\mathcal O_{Y_o})&=xk[x]\oplus yk[y],\\ \operatorname{Ext}^1(L,\mathcal O_{Y_o})&=k[x,y]/(x,y)=k,\\ \operatorname{Ext}^0(f_o^*L,\mathcal O_{X_o})&=k[x]\oplus0,\\ \operatorname{Ext}^1(f_o^*L,\mathcal O_{X_o})&=k[x]/(x)=k,\end{aligned} \] and \(\operatorname{Ext}^2\) of both vanish. The map \(Lf_o^*\) is restriction to \(y=0\): in degree \(0\) it sends \((xa(x),yb(y))\) to \((xa(x),0)\), with cokernel \(k[x]/xk[x]=k\) and kernel \(0\oplus yk[y]\); in degree \(1\) it sends the class of \(1\) to the class of \(1\), an isomorphism. Here \(g_o^*K=\mathcal O_{Y_o}\) and \(h_o^*K=\mathcal O_{X_o}\). The long exact sequence gives \[ \begin{aligned}\operatorname{Ext}^0(Y_o/X_o;L,\mathcal O_{Y_o})&\cong yk[y],\\ \operatorname{Ext}^1(Y_o/X_o;L,\mathcal O_{Y_o})&\cong k,\\ \operatorname{Ext}^2(Y_o/X_o;L,\mathcal O_{Y_o})&=0.\end{aligned} \] (3) An isomorphism of solutions \(\theta\colon(Y_0,f_p)\to(Y_0,f_{p'})\) is given by an automorphism \(\theta^\sharp\) of \(R_0\) over \(k[\varepsilon]\) inducing the identity on \(k[x,y]/(xy)\) and on \(J\), so \(\theta^\sharp(x)=x+\varepsilon P\), \(\theta^\sharp(y)=y+\varepsilon Q\) with \(P,Q\in k[x,y]/(xy)\), and \(\theta^\sharp(xy)=\varepsilon(xQ+yP)\) must vanish: \(P\in xk[x]\) and \(Q\in yk[y]\), as in (2). The condition \(\theta\circ f_p=f_{p'}\) reads \(f_p^\sharp\circ\theta^\sharp=f_{p'}^\sharp\), and \(f_p^\sharp(\theta^\sharp(x))=x+\varepsilon(p+P(x,0))\), \(f_p^\sharp(\theta^\sharp(y))=\varepsilon f_p^\sharp(Q)=0\), as \(Q\in yk[y]\) and \(f_p^\sharp(y)=0\). So \((Y_0,f_p)\cong(Y_0,f_{p'})\) if and only if \(p'-p\in xk[x]\), that is \(p(0)=p'(0)\); and the automorphisms of \((Y_0,f_p)\) are the \(\theta\) with \(P=0\), that is \(y\mapsto y+\varepsilon\,yb(y)\). These match \(\operatorname{Ext}^1\cong k\) and \(\operatorname{Ext}^0\cong yk[y]\) of (2); \(\operatorname{Ext}^2=0\) matches the existence of solutions, and by (1) the smoothings \(Y_a\), \(a\neq0\), carry none.

Solution 7.3. \(\mathcal O_X=k[s]/(s^4)\) is flat over the field \(k\), so \(h\) is flat, and \(I=(s^2)\) has square zero with \(\mathcal O_X/I=k[t]/(t^2)\); but \(h_o^*K=0\), so \(w\colon0\to I\) is not an isomorphism. As \(g_o^*K=0\), the relative Ext groups with coefficients in \(g_o^*K\) vanish. A flat deformation of \(Y_o\) over \(Z=Z_o\) has kernel \(g_o^*K=0\), so it is \(Y_o\), and a solution would be a morphism \(f\colon X\to Y_o\) with \(f\circ i=\mathrm{id}\), that is, a \(k\)-algebra map \(k[t]/(t^2)\to k[s]/(s^4)\) lifting the identity of \(k[t]/(t^2)\). It sends \(t\) to \(s+\alpha s^2+\beta s^3\), whose square \(s^2+2\alpha s^3\) is not \(0\). So there is no solution. In Theorem 3.1 the data are \((J,v,u)=(0,0,0)\), admissible since \(w=0\); the obstruction lies in \(E^2(0)=\operatorname{Ext}^2((0\leftarrow L),(I\leftarrow0))\), which by the long exact sequence is \(\operatorname{Ext}^1(Lf_o^*L_{Y_o/Z_o},I)\), the obstruction group for extending \(f_o\) to \(X\). It is not zero: \(L_{Y_o/k}=[\mathcal O_{Y_o}e\to\mathcal O_{Y_o}dt]\), \(e\mapsto2t\,dt\), and \(I\cong\mathcal O_{Y_o}\) through \(s^2\), so \(\operatorname{Ext}^1(L,I)=I/2tI\neq0\). The identification of \(E^i(J)\) with relative Ext groups in Theorem 3.2 (1) needs \(M\cong L\pi^*(g_o^*K)\), and here \(M=(I\leftarrow0)\) while \(L\pi^*(g_o^*K)=0\).

Sources

The deformation of a morphism together with its target through the arrow topos, the general problem with arbitrary coefficients, the description of the obstruction as a product with the Kodaira–Spencer class, and its comparison with the obstruction to extending a morphism into a given deformation of the target are due to Illusie, in his work on the cotangent complex.