Deformation Theory

Introduction

The goal of this chapter is to give a (relatively) gentle introduction to deformation theory of modules, morphisms, etc. In this chapter we deal with those results that can be proven using the naive cotangent complex. In the chapter on the cotangent complex we will extend these results a little bit. The advanced reader may wish to consult the treatise by Illusie on this subject, see [cotangent].

Deformations of rings and the naive cotangent complex

In this section we use the naive cotangent complex to do a little bit of deformation theory. We start with a surjective ring map \(A' \to A\) whose kernel is an ideal \(I\) of square zero. Moreover we assume given a ring map \(A \to B\), a \(B\)-module \(N\), and an \(A\)-module map \(c : I \to N\). In this section we ask ourselves whether we can find the question mark fitting into the following diagram [08S4]\[\begin{equation} \vcenter{ \xymatrix{ 0 \ar[r] & N \ar[r] & {?} \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & I \ar[u]^c \ar[r] & A' \ar[u] \ar[r] & A \ar[u] \ar[r] & 0 } } \end{equation}\] and moreover how unique the solution is (if it exists). More precisely, we look for a surjection of \(A'\)-algebras \(B' \to B\) whose kernel is an ideal of square zero and is identified with \(N\) such that \(A' \to B'\) induces the given map \(c\). We will say \(B'\) is a solution to (08S4).

Lemma

Given a commutative diagram \[\xymatrix{ & 0 \ar[r] & N_2 \ar[r] & B'_2 \ar[r] & B_2 \ar[r] & 0 \\ & 0 \ar[r]|\hole & I_2 \ar[u]_{c_2} \ar[r] & A'_2 \ar[u] \ar[r]|\hole & A_2 \ar[u] \ar[r] & 0 \\ 0 \ar[r] & N_1 \ar[ruu] \ar[r] & B'_1 \ar[r] & B_1 \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & I_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & A'_1 \ar[ruu]|\hole \ar[u] \ar[r] & A_1 \ar[ruu]|\hole \ar[u] \ar[r] & 0 }\] with front and back solutions to (08S4) we have

1. There exist a canonical element in \(\Ext^1_{B_1}(\NL_{B_1/A_1}, N_2)\) whose vanishing is a necessary and sufficient condition for the existence of a ring map \(B'_1 \to B'_2\) fitting into the diagram.

2. If there exists a map \(B'_1 \to B'_2\) fitting into the diagram the set of all such maps is a principal homogeneous space under \(\Hom_{B_1}(\Omega_{B_1/A_1}, N_2)\).

Proof

Let \(E = B_1\) viewed as a set. Consider the surjection \(A_1[E] \to B_1\) with kernel \(J\) used to define the naive cotangent complex by the formula \[\NL_{B_1/A_1} = (J/J^2 \to \Omega_{A_1[E]/A_1} \otimes_{A_1[E]} B_1)\] in Algebra, Section 00S0. Since \(\Omega_{A_1[E]/A_1} \otimes B_1\) is a free \(B_1\)-module we have \[\Ext^1_{B_1}(\NL_{B_1/A_1}, N_2) = \frac{\Hom_{B_1}(J/J^2, N_2)} {\Hom_{B_1}(\Omega_{A_1[E]/A_1} \otimes B_1, N_2)}\] We will construct an obstruction in the module on the right. Let \(J' = \Ker(A'_1[E] \to B_1)\). Note that there is a surjection \(J' \to J\) whose kernel is \(I_1A'_1[E]\). For every \(e \in E\) denote \(x_e \in A_1[E]\) the corresponding variable. Choose a lift \(y_e \in B'_1\) of the image of \(x_e\) in \(B_1\) and a lift \(z_e \in B'_2\) of the image of \(x_e\) in \(B_2\). These choices determine \(A'_1\)-algebra maps \[A'_1[E] \to B'_1 \quad\text{and}\quad A'_1[E] \to B'_2\] The first of these gives a map \(J' \to N_1\), \(f' \mapsto f'(y_e)\) and the second gives a map \(J' \to N_2\), \(f' \mapsto f'(z_e)\). A calculation shows that these maps annihilate \((J')^2\). Because the left square of the diagram (involving \(c_1\) and \(c_2\)) commutes we see that these maps agree on \(I_1A'_1[E]\) as maps into \(N_2\). Observe that \(B'_1\) is the pushout of \(J' \to A'_1[E]\) and \(J' \to N_1\). Thus, if the maps \(J' \to N_1 \to N_2\) and \(J' \to N_2\) agree, then we obtain a map \(B'_1 \to B'_2\) fitting into the diagram. Thus we let the obstruction be the class of the map \[J/J^2 \to N_2,\quad f \mapsto f'(z_e) - \nu(f'(y_e))\] where \(\nu : N_1 \to N_2\) is the given map and where \(f' \in J'\) is a lift of \(f\). This is well defined by our remarks above. Note that we have the freedom to modify our choices of \(z_e\) into \(z_e + \delta_{2, e}\) and \(y_e\) into \(y_e + \delta_{1, e}\) for some \(\delta_{i, e} \in N_i\). This will modify the map above into \[f \mapsto f'(z_e + \delta_{2, e}) - \nu(f'(y_e + \delta_{1, e})) = f'(z_e) - \nu(f'(z_e)) + \sum (\delta_{2, e} - \nu(\delta_{1, e}))\frac{\partial f}{\partial x_e}\] This means exactly that we are modifying the map \(J/J^2 \to N_2\) by the composition \(J/J^2 \to \Omega_{A_1[E]/A_1} \otimes B_1 \to N_2\) where the second map sends \(\text{d}x_e\) to \(\delta_{2, e} - \nu(\delta_{1, e})\). Thus our obstruction is well defined and is zero if and only if a lift exists.

Part (2) comes from the observation that given two maps \(\varphi, \psi : B'_1 \to B'_2\) fitting into the diagram, then \(\varphi - \psi\) factors through a map \(D : B_1 \to N_2\) which is an \(A_1\)-derivation: \[\begin{align*} D(fg) & = \varphi(f'g') - \psi(f'g') \\ & = \varphi(f')\varphi(g') - \psi(f')\psi(g') \\ & = (\varphi(f') - \psi(f'))\varphi(g') + \psi(f')(\varphi(g') - \psi(g')) \\ & = gD(f) + fD(g) \end{align*}\] Thus \(D\) corresponds to a unique \(B_1\)-linear map \(\Omega_{B_1/A_1} \to N_2\). Conversely, given such a linear map we get a derivation \(D\) and given a ring map \(\psi : B'_1 \to B'_2\) fitting into the diagram the map \(\psi + D\) is another ring map fitting into the diagram.

Lemma

If there exists a solution to (08S4), then the set of isomorphism classes of solutions is principal homogeneous under \(\Ext^1_B(\NL_{B/A}, N)\).

Proof

We observe right away that given two solutions \(B'_1\) and \(B'_2\) to (08S4) we obtain by Lemma 08S5 an obstruction element \(o(B'_1, B'_2) \in \Ext^1_B(\NL_{B/A}, N)\) to the existence of a map \(B'_1 \to B'_2\). Clearly, this element is the obstruction to the existence of an isomorphism, hence separates the isomorphism classes. To finish the proof it therefore suffices to show that given a solution \(B'\) and an element \(\xi \in \Ext^1_B(\NL_{B/A}, N)\) we can find a second solution \(B'_\xi\) such that \(o(B', B'_\xi) = \xi\).

Let \(E = B\) viewed as a set. Consider the surjection \(A[E] \to B\) with kernel \(J\) used to define the naive cotangent complex by the formula \[\NL_{B/A} = (J/J^2 \to \Omega_{A[E]/A} \otimes_{A[E]} B)\] in Algebra, Section 00S0. Since \(\Omega_{A[E]/A} \otimes B\) is a free \(B\)-module we have \[\Ext^1_B(\NL_{B/A}, N) = \frac{\Hom_B(J/J^2, N)} {\Hom_B(\Omega_{A[E]/A} \otimes B, N)}\] Thus we may represent \(\xi\) as the class of a morphism \(\delta : J/J^2 \to N\).

For every \(e \in E\) denote \(x_e \in A[E]\) the corresponding variable. Choose a lift \(y_e \in B'\) of the image of \(x_e\) in \(B\). These choices determine an \(A'\)-algebra map \(\varphi : A'[E] \to B'\). Let \(J' = \Ker(A'[E] \to B)\). Observe that \(\varphi\) induces a map \(\varphi|_{J'} : J' \to N\) and that \(B'\) is the pushout, as in the following diagram \[\xymatrix{ 0 \ar[r] & N \ar[r] & B' \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & J' \ar[u]^{\varphi|_{J'}} \ar[r] & A'[E] \ar[u] \ar[r] & B \ar[u]_{=} \ar[r] & 0 }\] Let \(\psi : J' \to N\) be the sum of the map \(\varphi|_{J'}\) and the composition \[J' \to J'/(J')^2 \to J/J^2 \xrightarrow{\delta} N.\] Then the pushout along \(\psi\) is an other ring extension \(B'_\xi\) fitting into a diagram as above. A calculation shows that \(o(B', B'_\xi) = \xi\) as desired.

Lemma

Let \(A\) be a ring. Let \(B\) be an \(A\)-algebra. Let \(N\) be a \(B\)-module. The set of isomorphism classes of extensions of \(A\)-algebras \[0 \to N \to B' \to B \to 0\] where \(N\) is an ideal of square zero is canonically bijective to \(\Ext^1_B(\NL_{B/A}, N)\).

Proof

To prove this we apply the previous results to the case where (08S4) is given by the diagram \[\xymatrix{ 0 \ar[r] & N \ar[r] & {?} \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & A \ar[u] \ar[r]^{\text{id}} & A \ar[u] \ar[r] & 0 }\] Thus our lemma follows from Lemma 08S7 and the fact that there exists a solution, namely \(N \oplus B\). (See remark below for a direct construction of the bijection.)

Remark

Let \(A \to B\) and \(N\) be as in Lemma 0GPT. Let \(\alpha : P \to B\) be a presentation of \(B\) over \(A\), see Algebra, Section 00S0. With \(J = \Ker(\alpha)\) the naive cotangent complex \(\NL(\alpha)\) associated to \(\alpha\) is the complex \(J/J^2 \to \Omega_{P/A} \otimes_P B\). We have \[\Ext^1_B(\NL(\alpha), N) = \Coker\left(\Hom_B(\Omega_{P/A} \otimes_P B, N) \to \Hom_B(J/J^2, N)\right)\] because \(\Omega_{P/A}\) is a free module. Consider a extension \(0 \to N \to B' \to B \to 0\) as in the lemma. Since \(P\) is a polynomial algebra over \(A\) we can lift \(\alpha\) to an \(A\)-algebra map \(\alpha' : P' \to B'\). Then \(\alpha'|_J : J \to N\) factors as \(J \to J/J^2 \to N\) as \(N\) has square zero in \(B'\). The lemma sends our extension to the class of this map \(J/J^2 \to N\) in the displayed cokernel.

Lemma

Given ring maps \(A \to B \to C\), a \(B\)-module \(M\), a \(C\)-module \(N\), a \(B\)-linear map \(c : M \to N\), and extensions of \(A\)-algebras with square zero kernels

1. \(0 \to M \to B' \to B \to 0\) corresponding to \(\xi \in \Ext^1_B(\NL_{B/A}, M)\), and

2. \(0 \to N \to C' \to C \to 0\) corresponding to \(\zeta \in \Ext^1_C(\NL_{C/A}, N)\).

See Lemma 0GPT. Then there is an \(A\)-algebra map \(B' \to C'\) compatible with \(B \to C\) and \(c\) if and only if \(\xi\) and \(\zeta\) map to the same element of \(\Ext^1_B(\NL_{B/A}, N)\).

Proof

The stament makes sense as we have the maps \[\Ext^1_B(\NL_{B/A}, M) \to \Ext^1_B(\NL_{B/A}, N)\] using the map \(M \to N\) and \[\Ext^1_C(\NL_{C/A}, N) \to \Ext^1_B(\NL_{C/A}, N) \to \Ext^1_B(\NL_{B/A}, N)\] where the first arrows uses the restriction map \(D(C) \to D(B)\) and the second arrow uses the canonical map of complexes \(\NL_{B/A} \to \NL_{C/A}\). The statement of the lemma can be deduced from Lemma 08S5 applied to the diagram \[\xymatrix{ & 0 \ar[r] & N \ar[r] & C' \ar[r] & C \ar[r] & 0 \\ & 0 \ar[r]|\hole & 0 \ar[u] \ar[r] & A \ar[u] \ar[r]|\hole & A \ar[u] \ar[r] & 0 \\ 0 \ar[r] & M \ar[ruu] \ar[r] & B' \ar[r] & B \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[ruu]|\hole \ar[u] \ar[r] & A \ar[ruu]|\hole \ar[u] \ar[r] & A \ar[ruu]|\hole \ar[u] \ar[r] & 0 }\] and a compatibility between the constructions in the proofs of Lemmas 0GPT and 08S5 whose statement and proof we omit. (See remark below for a direct argument.)

Remark

Let \(A \to B \to C\), \(M\), \(N\), \(c : M \to N\), \(0 \to M \to B' \to B \to 0\), \(\xi \in \Ext^1_B(\NL_{B/A}, M)\), \(0 \to N \to C' \to C \to 0\), and \(\zeta \in \Ext^1_C(\NL_{C/A}, N)\) be as in Lemma 0GPV. Using pushout along \(c : M \to N\) we can construct an extension \[\xymatrix{ 0 \ar[r] & N \ar[r] & B'_1 \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & M \ar[u]^c \ar[r] & B' \ar[u] \ar[r] & B \ar[u] \ar[r] & 0 }\] by setting \(B'_1 = (N \times B')/M\) where \(M\) is antidiagonally embedded. Using pullback along \(B \to C\) we can construct an extension \[\xymatrix{ 0 \ar[r] & N \ar[r] & C' \ar[r] & C \ar[r] & 0 \\ 0 \ar[r] & N \ar[u] \ar[r] & B'_2 \ar[u] \ar[r] & B \ar[u] \ar[r] & 0 }\] by setting \(B'_2 = C' \times_C B\) (fibre product of rings). A simple diagram chase tells us that there exists an \(A\)-algebra map \(B' \to C'\) compatible with \(B \to C\) and \(c\) if and only if \(B'_1\) is isomorphic to \(B'_2\) as \(A\)-algebra extensions of \(B\) by \(N\). Thus to see Lemma 0GPV is true, it suffices to show that \(B'_1\) corresponds via the bijection of Lemma 0GPT to the image of \(\xi\) by the map \(\Ext^1_B(\NL_{B/A}, M) \to \Ext^1_B(\NL_{B/A}, N)\) and that \(B'_2\) correspond to the image of \(\zeta\) by the map \(\Ext^1_C(\NL_{C/A}, N) \to \Ext^1_B(\NL_{B/A}, N)\). The first of these two statements is immediate from the construction of the class in Remark 0GPU. For the second, choose a commutative diagram \[\xymatrix{ Q \ar[r]_\beta & C \\ P \ar[u]^\varphi \ar[r]^\alpha & B \ar[u] }\] of \(A\)-algebras, such that \(\alpha\) is a presentation of \(B\) over \(A\) and \(\beta\) is a presentation of \(C\) over \(A\). See Remark 0GPU and references therein. Set \(J = \Ker(\alpha)\) and \(K = \Ker(\beta)\). The map \(\varphi\) induces a map of complexes \(\NL(\alpha) \to \NL(\beta)\) and in particular \(\bar\varphi : J/J^2 \to K/K^2\). Choose \(A\)-algebra homomorphism \(\beta' : Q \to C'\) which is a lift of \(\beta\). Then \(\alpha' = (\beta' \circ \varphi, \alpha) : P \to B'_2 = C' \times_C B\) is a lift of \(\alpha\). With these choices the composition of the map \(K/K^2 \to N\) induced by \(\beta'\) and the map \(\bar\varphi : J/J^2 \to K/K^2\) is the restriction of \(\alpha'\) to \(J/J^2\). Unwinding the constructions of our classes in Remark 0GPU this indeed shows that \(B'_2\) correspond to the image of \(\zeta\) by the map \(\Ext^1_C(\NL_{C/A}, N) \to \Ext^1_B(\NL_{B/A}, N)\).

Lemma

Let \(0 \to I \to A' \to A \to 0\), \(A \to B\), and \(c : I \to N\) be as in (08S4). Denote \(\xi \in \Ext^1_A(\NL_{A/A'}, I)\) the element corresponding to the extension \(A'\) of \(A\) by \(I\) via Lemma 0GPT. The set of isomorphism classes of solutions is canonically bijective to the fibre of \[\Ext^1_B(\NL_{B/A'}, N) \to \Ext^1_A(\NL_{A/A'}, N)\] over the image of \(\xi\).

Proof

By Lemma 0GPT applied to \(A' \to B\) and the \(B\)-module \(N\) we see that elements \(\zeta\) of \(\Ext^1_B(\NL_{B/A'}, N)\) parametrize extensions \(0 \to N \to B' \to B \to 0\) of \(A'\)-algebras. By Lemma 0GPV applied to \(A' \to A \to B\) and \(c : I \to N\) we see that there is an \(A'\)-algebra map \(A' \to B'\) compatible with \(c\) and \(A \to B\) if and only if \(\zeta\) maps to \(\xi\). Of course this is the same thing as saying \(B'\) is a solution of (08S4).

Remark

Observe that in the situation of Lemma 0GPX we have \[\Ext^1_A(\NL_{A/A'}, N) = \Ext^1_B(\NL_{A/A'} \otimes_A^\mathbf{L} B, N) = \Ext^1_B(\NL_{A/A'} \otimes_A B, N)\] The first equality by More on Algebra, Lemma 0GMT and the second by More on Algebra, Lemma 0FUY. We have maps of complexes \[\NL_{A/A'} \otimes_A B \to \NL_{B/A'} \to \NL_{B/A}\] which is close to being a distinguished triangle, see Algebra, Lemma 00S2. If it were a distinguished triangle we would conclude that the image of \(\xi\) in \(\Ext^2_B(\NL_{B/A}, N)\) would be the obstruction to the existence of a solution to (08S4).

If our ring map \(A \to B\) is a local complete intersection, then there is a solution. This is a kind of lifting result; observe that for syntomic ring maps we have proved a rather strong lifting result in Smoothing Ring Maps, Proposition 07M8.

Lemma

If \(A \to B\) is a local complete intersection ring map, then there exists a solution to (08S4).

Proof

Write \(B = A[x_1, \ldots, x_n]/J\). By More on Algebra, Definition 07D0 the ideal \(J\) is Koszul-regular. This implies \(J\) is \(H_1\)-regular and quasi-regular, see More on Algebra, Section 07CU. Let \(J' \subset A'[x_1, \ldots, x_n]\) be the inverse image of \(J\). Denote \(I[x_1, \ldots, x_n]\) the kernel of \(A'[x_1, \ldots, x_n] \to A[x_1, \ldots, x_n]\). By More on Algebra, Lemma 07CX we have \(I[x_1, \ldots, x_n] \cap (J')^2 = J'I[x_1, \ldots, x_n] = JI[x_1, \ldots, x_n]\). Hence we obtain a short exact sequence \[0 \to I \otimes_A B \to J'/(J')^2 \to J/J^2 \to 0\] Since \(J/J^2\) is projective (More on Algebra, Lemma 08RK) we can choose a splitting of this sequence \[J'/(J')^2 = I \otimes_A B \oplus J/J^2\] Let \((J')^2 \subset J'' \subset J'\) be the elements which map to the second summand in the decomposition above. Then \[0 \to I \otimes_A B \to A'[x_1, \ldots, x_n]/J'' \to B \to 0\] is a solution to (08S4) with \(N = I \otimes_A B\). The general case is obtained by doing a pushout along the given map \(I \otimes_A B \to N\).

Proof

Please read Remark 0GPY before reading this proof. By More on Algebra, Lemma 07D4 the maps \(\NL_{A/A'} \otimes_A B \to \NL_{B/A'} \to \NL_{B/A}\) do form a distinguished triangle in \(D(B)\). Hence it suffices to show that \(\Ext^2_B(\NL_{B/A}, N)\) vanishes. By More on Algebra, Lemma 0FV0 the complex \(\NL_{B/A}\) is perfect of tor-amplitude in \([-1, 0]\). This implies our \(\Ext^2\) vanishes for example by More on Algebra, Lemma 0BCG part (1).

Thickenings of ringed spaces

In the following few sections we will use the following notions:

1. A sheaf of ideals \(\mathcal{I} \subset \mathcal{O}_{X'}\) on a ringed space \((X', \mathcal{O}_{X'})\) is locally nilpotent if any local section of \(\mathcal{I}\) is locally nilpotent. Compare with Algebra, Item 0543.

2. A thickening of ringed spaces is a morphism \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) of ringed spaces such that

   1. \(i\) induces a homeomorphism \(X \to X'\),

   2. the map \(i^\sharp : \mathcal{O}_{X'} \to i_*\mathcal{O}_X\) is surjective, and

   3. the kernel of \(i^\sharp\) is a locally nilpotent sheaf of ideals.

3. A first order thickening of ringed spaces is a thickening \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) of ringed spaces such that \(\Ker(i^\sharp)\) has square zero.

4. It is clear how to define morphisms of thickenings, morphisms of thickenings over a base ringed space, etc.

If \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) is a thickening of ringed spaces then we identify the underlying topological spaces and think of \(\mathcal{O}_X\), \(\mathcal{O}_{X'}\), and \(\mathcal{I} = \Ker(i^\sharp)\) as sheaves on \(X = X'\). We obtain a short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\] of \(\mathcal{O}_{X'}\)-modules. By Modules, Lemma 08KS the category of \(\mathcal{O}_X\)-modules is equivalent to the category of \(\mathcal{O}_{X'}\)-modules annihilated by \(\mathcal{I}\). In particular, if \(i\) is a first order thickening, then \(\mathcal{I}\) is a \(\mathcal{O}_X\)-module.

Situation

A morphism of thickenings \((f, f')\) is given by a commutative diagram [08L0]\[\begin{equation} \vcenter{ \xymatrix{ (X, \mathcal{O}_X) \ar[r]_i \ar[d]_f & (X', \mathcal{O}_{X'}) \ar[d]^{f'} \\ (S, \mathcal{O}_S) \ar[r]^t & (S', \mathcal{O}_{S'}) } } \end{equation}\] of ringed spaces whose horizontal arrows are thickenings. In this situation we set \(\mathcal{I} = \Ker(i^\sharp) \subset \mathcal{O}_{X'}\) and \(\mathcal{J} = \Ker(t^\sharp) \subset \mathcal{O}_{S'}\). As \(f = f'\) on underlying topological spaces we will identify the (topological) pullback functors \(f^{-1}\) and \((f')^{-1}\). Observe that \((f')^\sharp : f^{-1}\mathcal{O}_{S'} \to \mathcal{O}_{X'}\) induces in particular a map \(f^{-1}\mathcal{J} \to \mathcal{I}\) and therefore a map of \(\mathcal{O}_{X'}\)-modules \[(f')^*\mathcal{J} \longrightarrow \mathcal{I}\] If \(i\) and \(t\) are first order thickenings, then \((f')^*\mathcal{J} = f^*\mathcal{J}\) and the map above becomes a map \(f^*\mathcal{J} \to \mathcal{I}\).

Definition

In Situation 08KZ we say that \((f, f')\) is a strict morphism of thickenings if the map \((f')^*\mathcal{J} \longrightarrow \mathcal{I}\) is surjective.

The following lemma in particular shows that a morphism \((f, f') : (X \subset X') \to (S \subset S')\) of thickenings of schemes is strict if and only if \(X = S \times_{S'} X'\).

Lemma

In Situation 08KZ the morphism \((f, f')\) is a strict morphism of thickenings if and only if (08L0) is cartesian in the category of ringed spaces.

Proof

Omitted.

Modules on first order thickenings of ringed spaces

In this section we discuss some preliminaries to the deformation theory of modules. Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. We will freely use the notation introduced in Section 08KY, in particular we will identify the underlying topological spaces. In this section we consider short exact sequences [08L4]\[\begin{equation} 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \end{equation}\] of \(\mathcal{O}_{X'}\)-modules, where \(\mathcal{F}\), \(\mathcal{K}\) are \(\mathcal{O}_X\)-modules and \(\mathcal{F}'\) is an \(\mathcal{O}_{X'}\)-module. In this situation we have a canonical \(\mathcal{O}_X\)-module map \[c_{\mathcal{F}'} : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \longrightarrow \mathcal{K}\] where \(\mathcal{I} = \Ker(i^\sharp)\). Namely, given local sections \(f\) of \(\mathcal{I}\) and \(s\) of \(\mathcal{F}\) we set \(c_{\mathcal{F}'}(f \otimes s) = fs'\) where \(s'\) is a local section of \(\mathcal{F}'\) lifting \(s\).

Lemma

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. Assume given extensions \[0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0\] as in (08L4) and maps \(\varphi : \mathcal{F} \to \mathcal{G}\) and \(\psi : \mathcal{K} \to \mathcal{L}\).

1. If there exists an \(\mathcal{O}_{X'}\)-module map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\), then the diagram \[\xymatrix{ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} }\] is commutative.

2. The set of \(\mathcal{O}_{X'}\)-module maps \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\) is, if nonempty, a principal homogeneous space under \(\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})\).

Proof

Part (1) is immediate from the description of the maps. For (2), if \(\varphi'\) and \(\varphi''\) are two maps \(\mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\), then \(\varphi' - \varphi''\) factors as \[\mathcal{F}' \to \mathcal{F} \to \mathcal{L} \to \mathcal{G}'\] The map in the middle comes from a unique element of \(\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})\) by Modules, Lemma 08KS. Conversely, given an element \(\alpha\) of this group we can add the composition (as displayed above with \(\alpha\) in the middle) to \(\varphi'\). Some details omitted.

Lemma

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. Assume given extensions \[0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0\] as in (08L4) and maps \(\varphi : \mathcal{F} \to \mathcal{G}\) and \(\psi : \mathcal{K} \to \mathcal{L}\). Assume the diagram \[\xymatrix{ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} }\] is commutative. Then there exists an element \[o(\varphi, \psi) \in \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L})\] whose vanishing is a necessary and sufficient condition for the existence of a map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\).

Proof

We can construct explicitly an extension \[0 \to \mathcal{L} \to \mathcal{H} \to \mathcal{F} \to 0\] by taking \(\mathcal{H}\) to be the cohomology of the complex \[\mathcal{K} \xrightarrow{1, - \psi} \mathcal{F}' \oplus \mathcal{G}' \xrightarrow{\varphi, 1} \mathcal{G}\] in the middle (with obvious notation). A calculation with local sections using the assumption that the diagram of the lemma commutes shows that \(\mathcal{H}\) is annihilated by \(\mathcal{I}\). Hence \(\mathcal{H}\) defines a class in \[\Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{L}) \subset \Ext^1_{\mathcal{O}_{X'}}(\mathcal{F}, \mathcal{L})\] Finally, the class of \(\mathcal{H}\) is the difference of the pushout of the extension \(\mathcal{F}'\) via \(\psi\) and the pullback of the extension \(\mathcal{G}'\) via \(\varphi\) (calculations omitted). Thus the vanishing of the class of \(\mathcal{H}\) is equivalent to the existence of a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d]_{\psi} & \mathcal{F}' \ar[r] \ar[d]_{\varphi'} & \mathcal{F} \ar[r] \ar[d]_\varphi & 0\\ 0 \ar[r] & \mathcal{L} \ar[r] & \mathcal{G}' \ar[r] & \mathcal{G} \ar[r] & 0 }\] as desired.

Lemma

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. Assume given \(\mathcal{O}_X\)-modules \(\mathcal{F}\), \(\mathcal{K}\) and an \(\mathcal{O}_X\)-linear map \(c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}\). If there exists a sequence (08L4) with \(c_{\mathcal{F}'} = c\) then the set of isomorphism classes of these extensions is principal homogeneous under \(\Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})\).

Proof

Assume given extensions \[0 \to \mathcal{K} \to \mathcal{F}'_1 \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{K} \to \mathcal{F}'_2 \to \mathcal{F} \to 0\] with \(c_{\mathcal{F}'_1} = c_{\mathcal{F}'_2} = c\). Then the difference (in the extension group, see Homology, Section 010I) is an extension \[0 \to \mathcal{K} \to \mathcal{E} \to \mathcal{F} \to 0\] where \(\mathcal{E}\) is annihilated by \(\mathcal{I}\) (local computation omitted). Hence the sequence is an extension of \(\mathcal{O}_X\)-modules, see Modules, Lemma 08KS. Conversely, given such an extension \(\mathcal{E}\) we can add the extension \(\mathcal{E}\) to the \(\mathcal{O}_{X'}\)-extension \(\mathcal{F}'\) without affecting the map \(c_{\mathcal{F}'}\). Some details omitted.

Lemma

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. Assume given \(\mathcal{O}_X\)-modules \(\mathcal{F}\), \(\mathcal{K}\) and an \(\mathcal{O}_X\)-linear map \(c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}\). Then there exists an element \[o(\mathcal{F}, \mathcal{K}, c) \in \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})\] whose vanishing is a necessary and sufficient condition for the existence of a sequence (08L4) with \(c_{\mathcal{F}'} = c\).

Proof

We first show that if \(\mathcal{K}\) is an injective \(\mathcal{O}_X\)-module, then there does exist a sequence (08L4) with \(c_{\mathcal{F}'} = c\). To do this, choose a flat \(\mathcal{O}_{X'}\)-module \(\mathcal{H}'\) and a surjection \(\mathcal{H}' \to \mathcal{F}\) (Modules, Lemma 05NI). Let \(\mathcal{J} \subset \mathcal{H}'\) be the kernel. Since \(\mathcal{H}'\) is flat we have \[\mathcal{I} \otimes_{\mathcal{O}_{X'}} \mathcal{H}' = \mathcal{I}\mathcal{H}' \subset \mathcal{J} \subset \mathcal{H}'\] Observe that the map \[\mathcal{I}\mathcal{H}' = \mathcal{I} \otimes_{\mathcal{O}_{X'}} \mathcal{H}' \longrightarrow \mathcal{I} \otimes_{\mathcal{O}_{X'}} \mathcal{F} = \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}\] annihilates \(\mathcal{I}\mathcal{J}\). Namely, if \(f\) is a local section of \(\mathcal{I}\) and \(s\) is a local section of \(\mathcal{H}\), then \(fs\) is mapped to \(f \otimes \overline{s}\) where \(\overline{s}\) is the image of \(s\) in \(\mathcal{F}\). Thus we obtain \[\xymatrix{ \mathcal{I}\mathcal{H}'/\mathcal{I}\mathcal{J} \ar@{^{(}->}[r] \ar[d] & \mathcal{J}/\mathcal{I}\mathcal{J} \ar@{..>}[d]_\gamma \\ \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-c & \mathcal{K} }\] a diagram of \(\mathcal{O}_X\)-modules. If \(\mathcal{K}\) is injective as an \(\mathcal{O}_X\)-module, then we obtain the dotted arrow. Denote \(\gamma' : \mathcal{J} \to \mathcal{K}\) the composition of \(\gamma\) with \(\mathcal{J} \to \mathcal{J}/\mathcal{I}\mathcal{J}\). A local calculation shows the pushout \[\xymatrix{ 0 \ar[r] & \mathcal{J} \ar[r] \ar[d]_{\gamma'} & \mathcal{H}' \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{F}' \ar[r] & \mathcal{F} \ar[r] & 0 }\] is a solution to the problem posed by the lemma.

General case. Choose an embedding \(\mathcal{K} \subset \mathcal{K}'\) with \(\mathcal{K}'\) an injective \(\mathcal{O}_X\)-module. Let \(\mathcal{Q}\) be the quotient, so that we have an exact sequence \[0 \to \mathcal{K} \to \mathcal{K}' \to \mathcal{Q} \to 0\] Denote \(c' : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}'\) be the composition. By the paragraph above there exists a sequence \[0 \to \mathcal{K}' \to \mathcal{E}' \to \mathcal{F} \to 0\] as in (08L4) with \(c_{\mathcal{E}'} = c'\). Note that \(c'\) composed with the map \(\mathcal{K}' \to \mathcal{Q}\) is zero, hence the pushout of \(\mathcal{E}'\) by \(\mathcal{K}' \to \mathcal{Q}\) is an extension \[0 \to \mathcal{Q} \to \mathcal{D}' \to \mathcal{F} \to 0\] as in (08L4) with \(c_{\mathcal{D}'} = 0\). This means exactly that \(\mathcal{D}'\) is annihilated by \(\mathcal{I}\), in other words, the \(\mathcal{D}'\) is an extension of \(\mathcal{O}_X\)-modules, i.e., defines an element \[o(\mathcal{F}, \mathcal{K}, c) \in \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{Q}) = \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})\] (the equality holds by the long exact cohomology sequence associated to the exact sequence above and the vanishing of higher ext groups into the injective module \(\mathcal{K}'\)). If \(o(\mathcal{F}, \mathcal{K}, c) = 0\), then we can choose a splitting \(s : \mathcal{F} \to \mathcal{D}'\) and we can set \[\mathcal{F}' = \Ker(\mathcal{E}' \to \mathcal{D}'/s(\mathcal{F}))\] so that we obtain the following diagram \[\xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d] & \mathcal{F}' \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] & \mathcal{E}' \ar[r] & \mathcal{F} \ar[r] & 0 }\] with exact rows which shows that \(c_{\mathcal{F}'} = c\). Conversely, if \(\mathcal{F}'\) exists, then the pushout of \(\mathcal{F}'\) by the map \(\mathcal{K} \to \mathcal{K}'\) is isomorphic to \(\mathcal{E}'\) by Lemma 08L7 and the vanishing of higher ext groups into the injective module \(\mathcal{K}'\). This gives a diagram as above, which implies that \(\mathcal{D}'\) is split as an extension, i.e., the class \(o(\mathcal{F}, \mathcal{K}, c)\) is zero.

Remark

Let \((X, \mathcal{O}_X)\) be a ringed space. A first order thickening \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) is said to be trivial if there exists a morphism of ringed spaces \(\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)\) which is a left inverse to \(i\). The choice of such a morphism \(\pi\) is called a trivialization of the first order thickening. Given \(\pi\) we obtain a splitting [08LA]\[\begin{equation} \mathcal{O}_{X'} = \mathcal{O}_X \oplus \mathcal{I} \end{equation}\] as sheaves of algebras on \(X\) by using \(\pi^\sharp\) to split the surjection \(\mathcal{O}_{X'} \to \mathcal{O}_X\). Conversely, such a splitting determines a morphism \(\pi\). The category of trivialized first order thickenings of \((X, \mathcal{O}_X)\) is equivalent to the category of \(\mathcal{O}_X\)-modules.

Remark

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a trivial first order thickening of ringed spaces and let \(\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_X)\) be a trivialization. Then given any triple \((\mathcal{F}, \mathcal{K}, c)\) consisting of a pair of \(\mathcal{O}_X\)-modules and a map \(c : \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}\) we may set \[\mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K}\] and use the splitting (08LA) associated to \(\pi\) and the map \(c\) to define the \(\mathcal{O}_{X'}\)-module structure and obtain an extension (08L4). We will call \(\mathcal{F}'_{c, triv}\) the trivial extension of \(\mathcal{F}\) by \(\mathcal{K}\) corresponding to \(c\) and the trivialization \(\pi\). Given any extension \(\mathcal{F}'\) as in (08L4) we can use \(\pi^\sharp : \mathcal{O}_X \to \mathcal{O}_{X'}\) to think of \(\mathcal{F}'\) as an \(\mathcal{O}_X\)-module extension, hence a class \(\xi_{\mathcal{F}'}\) in \(\Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})\). Lemma 08L7 assures that \(\mathcal{F}' \mapsto \xi_{\mathcal{F}'}\) induces a bijection \[\left\{ \begin{matrix} \text{isomorphism classes of extensions}\\ \mathcal{F}'\text{ as in (\href{defos.html#defos-equation-extension}{08L4}) with }c = c_{\mathcal{F}'} \end{matrix} \right\} \longrightarrow \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K})\] Moreover, the trivial extension \(\mathcal{F}'_{c, triv}\) maps to the zero class.

Remark

Let \((X, \mathcal{O}_X)\) be a ringed space. Let \((X, \mathcal{O}_X) \to (X'_i, \mathcal{O}_{X'_i})\), \(i = 1, 2\) be first order thickenings with ideal sheaves \(\mathcal{I}_i\). Let \(h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})\) be a morphism of first order thickenings of \((X, \mathcal{O}_X)\). Picture \[\xymatrix{ & (X, \mathcal{O}_X) \ar[ld] \ar[rd] & \\ (X'_1, \mathcal{O}_{X'_1}) \ar[rr]^h & & (X'_2, \mathcal{O}_{X'_2}) }\] Observe that \(h^\sharp : \mathcal{O}_{X'_2} \to \mathcal{O}_{X'_1}\) in particular induces an \(\mathcal{O}_X\)-module map \(\mathcal{I}_2 \to \mathcal{I}_1\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2\) be a pair consisting of an \(\mathcal{O}_X\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}_i\). Assume furthermore given a map of \(\mathcal{O}_X\)-modules \(\mathcal{K}_2 \to \mathcal{K}_1\) such that \[\xymatrix{ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 }\] is commutative. Then there is a canonical functoriality \[\left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (\href{defos.html#defos-equation-extension}{08L4}) with }\\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (\href{defos.html#defos-equation-extension}{08L4}) with }\\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\}\] Namely, thinking of all sheaves \(\mathcal{O}_X\), \(\mathcal{O}_{X'_i}\), \(\mathcal{F}\), \(\mathcal{K}_i\), etc as sheaves on \(X\), we set given \(\mathcal{F}'_2\) the sheaf \(\mathcal{F}'_1\) equal to the pushout, i.e., fitting into the following diagram of extensions \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K}_1 \ar[r] & \mathcal{F}'_1 \ar[r] & \mathcal{F} \ar[r] & 0 }\] We omit the construction of the \(\mathcal{O}_{X'_1}\)-module structure on the pushout (this uses the commutativity of the diagram involving \(c_1\) and \(c_2\)).

Remark

Let \((X, \mathcal{O}_X)\), \((X, \mathcal{O}_X) \to (X'_i, \mathcal{O}_{X'_i})\), \(\mathcal{I}_i\), and \(h : (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2})\) be as in Remark 08LC. Assume that we are given trivializations \(\pi_i : X'_i \to X\) such that \(\pi_1 = h \circ \pi_2\). In other words, assume \(h\) is a morphism of trivialized first order thickening of \((X, \mathcal{O}_X)\). Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2\) be a pair consisting of an \(\mathcal{O}_X\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}_i\). Assume furthermore given a map of \(\mathcal{O}_X\)-modules \(\mathcal{K}_2 \to \mathcal{K}_1\) such that \[\xymatrix{ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 }\] is commutative. In this situation the construction of Remark 08LB induces a commutative diagram \[\xymatrix{ \{\mathcal{F}'_2\text{ as in (\href{defos.html#defos-equation-extension}{08L4}) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{\mathcal{F}'_1\text{ as in (\href{defos.html#defos-equation-extension}{08L4}) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1) }\] where the vertical map on the right is given by functoriality of \(\Ext\) and the map \(\mathcal{K}_2 \to \mathcal{K}_1\) and the vertical map on the left is the one from Remark 08LC.

Remark

Let \((X, \mathcal{O}_X)\) be a ringed space. We define a sequence of morphisms of first order thickenings \[(X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3})\] of \((X, \mathcal{O}_X)\) to be a complex if the corresponding maps between the ideal sheaves \(\mathcal{I}_i\) give a complex of \(\mathcal{O}_X\)-modules \(\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1\) (i.e., the composition is zero). In this case the composition \((X'_1, \mathcal{O}_{X'_1}) \to (X_3', \mathcal{O}_{X'_3})\) factors through \((X, \mathcal{O}_X) \to (X'_3, \mathcal{O}_{X'_3})\), i.e., the first order thickening \((X'_1, \mathcal{O}_{X'_1})\) of \((X, \mathcal{O}_X)\) is trivial and comes with a canonical trivialization \(\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)\).

We say a sequence of morphisms of first order thickenings \[(X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3})\] of \((X, \mathcal{O}_X)\) is a short exact sequence if the corresponding maps between ideal sheaves is a short exact sequence \[0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0\] of \(\mathcal{O}_X\)-modules.

Remark

Let \((X, \mathcal{O}_X)\) be a ringed space. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Let \[(X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3})\] be a complex first order thickenings of \((X, \mathcal{O}_X)\), see Remark 08LE. Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2, 3\) be pairs consisting of an \(\mathcal{O}_X\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{K}_i\). Assume given a short exact sequence of \(\mathcal{O}_X\)-modules \[0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0\] such that \[\vcenter{ \xymatrix{ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } } \quad\text{and}\quad \vcenter{ \xymatrix{ \mathcal{I}_3 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]_-{c_3} \ar[d] & \mathcal{K}_3 \ar[d] \\ \mathcal{I}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \ar[r]^-{c_2} & \mathcal{K}_2 } }\] are commutative. Finally, assume given an extension \[0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0\] as in (08L4) with \(\mathcal{K} = \mathcal{K}_2\) of \(\mathcal{O}_{X'_2}\)-modules with \(c_{\mathcal{F}'_2} = c_2\). In this situation we can apply the functoriality of Remark 08LC to obtain an extension \(\mathcal{F}'_1\) on \(X'_1\) (we’ll describe \(\mathcal{F}'_1\) in this special case below). By Remark 08LB using the canonical splitting \(\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)\) of Remark 08LE we obtain \(\xi_{\mathcal{F}'_1} \in \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1)\). Finally, we have the obstruction \[o(\mathcal{F}, \mathcal{K}_3, c_3) \in \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_3)\] see Lemma 08L8. In this situation we claim that the canonical map \[\partial : \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_1) \longrightarrow \Ext^2_{\mathcal{O}_X}(\mathcal{F}, \mathcal{K}_3)\] coming from the short exact sequence \(0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0\) sends \(\xi_{\mathcal{F}'_1}\) to the obstruction class \(o(\mathcal{F}, \mathcal{K}_3, c_3)\).

To prove this claim choose an embedding \(j : \mathcal{K}_3 \to \mathcal{K}\) where \(\mathcal{K}\) is an injective \(\mathcal{O}_X\)-module. We can lift \(j\) to a map \(j' : \mathcal{K}_2 \to \mathcal{K}\). Set \(\mathcal{E}'_2 = j'_*\mathcal{F}'_2\) equal to the pushout of \(\mathcal{F}'_2\) by \(j'\) so that \(c_{\mathcal{E}'_2} = j' \circ c_2\). Picture: \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d]_{j'} & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{E}'_2 \ar[r] & \mathcal{F} \ar[r] & 0 }\] Set \(\mathcal{E}'_3 = \mathcal{E}'_2\) but viewed as an \(\mathcal{O}_{X'_3}\)-module via \(\mathcal{O}_{X'_3} \to \mathcal{O}_{X'_2}\). Then \(c_{\mathcal{E}'_3} = j \circ c_3\). The proof of Lemma 08L8 constructs \(o(\mathcal{F}, \mathcal{K}_3, c_3)\) as the boundary of the class of the extension of \(\mathcal{O}_X\)-modules \[0 \to \mathcal{K}/\mathcal{K}_3 \to \mathcal{E}'_3/\mathcal{K}_3 \to \mathcal{F} \to 0\] On the other hand, note that \(\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3\) hence the class \(\xi_{\mathcal{F}'_1}\) is the class of the extension \[0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3 \to \mathcal{F} \to 0\] seen as a sequence of \(\mathcal{O}_X\)-modules using \(\pi^\sharp\) where \(\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_X)\) is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}/\mathcal{K}_3 \ar[r] & \mathcal{E}'_3/\mathcal{K}_3 \ar[r] & \mathcal{F} \ar[r] & 0 }\] which is \(\mathcal{O}_X\)-linear (with the \(\mathcal{O}_X\)-module structures given above).

Infinitesimal deformations of modules on ringed spaces

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. We freely use the notation introduced in Section 08KY. Let \(\mathcal{F}'\) be an \(\mathcal{O}_{X'}\)-module and set \(\mathcal{F} = i^*\mathcal{F}'\). In this situation we have a short exact sequence \[0 \to \mathcal{I}\mathcal{F}' \to \mathcal{F}' \to \mathcal{F} \to 0\] of \(\mathcal{O}_{X'}\)-modules. Since \(\mathcal{I}^2 = 0\) the \(\mathcal{O}_{X'}\)-module structure on \(\mathcal{I}\mathcal{F}'\) comes from a unique \(\mathcal{O}_X\)-module structure. Thus the sequence above is an extension as in (08L4). As a special case, if \(\mathcal{F}' = \mathcal{O}_{X'}\) we have \(i^*\mathcal{O}_{X'} = \mathcal{O}_X\) and \(\mathcal{I}\mathcal{O}_{X'} = \mathcal{I}\) and we recover the sequence of structure sheaves \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\]

Lemma

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}_{X'}\)-modules. Set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}_X\)-linear map. The set of lifts of \(\varphi\) to an \(\mathcal{O}_{X'}\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) is, if nonempty, a principal homogeneous space under \(\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{I}\mathcal{G}')\).

Proof

This is a special case of Lemma 08L5 but we also give a direct proof. We have short exact sequences of modules \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0 \quad\text{and}\quad 0 \to \mathcal{I}\mathcal{G}' \to \mathcal{G}' \to \mathcal{G} \to 0\] and similarly for \(\mathcal{F}'\). Since \(\mathcal{I}\) has square zero the \(\mathcal{O}_{X'}\)-module structure on \(\mathcal{I}\) and \(\mathcal{I}\mathcal{G}'\) comes from a unique \(\mathcal{O}_X\)-module structure. It follows that \[\Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{I}\mathcal{G}') \quad\text{and}\quad \Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}) = \Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})\] The lemma now follows from the exact sequence \[0 \to \Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') \to \Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}') \to \Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G})\] see Homology, Lemma 05AA.

Lemma

Let \((f, f')\) be a morphism of first order thickenings of ringed spaces as in Situation 08KZ. Let \(\mathcal{F}'\) be an \(\mathcal{O}_{X'}\)-module and set \(\mathcal{F} = i^*\mathcal{F}'\). Assume that \(\mathcal{F}\) is flat over \(S\) and that \((f, f')\) is a strict morphism of thickenings (Definition 08L1). Then the following are equivalent

1. \(\mathcal{F}'\) is flat over \(S'\), and

2. the canonical map \(f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I}\mathcal{F}'\) is an isomorphism.

Moreover, in this case the maps \[f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I}\mathcal{F}'\] are isomorphisms.

Proof

The map \(f^*\mathcal{J} \to \mathcal{I}\) is surjective as \((f, f')\) is a strict morphism of thickenings. Hence the final statement is a consequence of (2).

Proof of the equivalence of (1) and (2). We may check these conditions at stalks. Let \(x \in X \subset X'\) be a point with image \(s = f(x) \in S \subset S'\). Set \(A' = \mathcal{O}_{S', s}\), \(B' = \mathcal{O}_{X', x}\), \(A = \mathcal{O}_{S, s}\), and \(B = \mathcal{O}_{X, x}\). Then \(A = A'/J\) and \(B = B'/I\) for some square zero ideals. Since \((f, f')\) is a strict morphism of thickenings we have \(I = JB'\). Let \(M' = \mathcal{F}'_x\) and \(M = \mathcal{F}_x\). Then \(M'\) is a \(B'\)-module and \(M\) is a \(B\)-module. Since \(\mathcal{F} = i^*\mathcal{F}'\) we see that the kernel of the surjection \(M' \to M\) is \(IM' = JM'\). Thus we have a short exact sequence \[0 \to JM' \to M' \to M \to 0\] Using Sheaves, Lemma 0098 and Modules, Lemma 01CB to identify stalks of pullbacks and tensor products we see that the stalk at \(x\) of the canonical map of the lemma is the map \[(J \otimes_A B) \otimes_B M = J \otimes_A M = J \otimes_{A'} M' \longrightarrow JM'\] The assumption that \(\mathcal{F}\) is flat over \(S\) signifies that \(M\) is a flat \(A\)-module.

Assume (1). Flatness implies \(\text{Tor}_1^{A'}(M', A) = 0\) by Algebra, Lemma 00M5. This means \(J \otimes_{A'} M' \to M'\) is injective by Algebra, Remark 00M6. Hence \(J \otimes_A M \to JM'\) is an isomorphism.

Assume (2). Then \(J \otimes_{A'} M' \to M'\) is injective. Hence \(\text{Tor}_1^{A'}(M', A) = 0\) by Algebra, Remark 00M6. Hence \(M'\) is flat over \(A'\) by Algebra, Lemma 051C.

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08KZ. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}_{X'}\)-modules and set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}_X\)-linear map. Assume that \(\mathcal{G}'\) is flat over \(S'\) and that \((f, f')\) is a strict morphism of thickenings. The set of lifts of \(\varphi\) to an \(\mathcal{O}_{X'}\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) is, if nonempty, a principal homogeneous space under \[\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J})\]

Proof

Combine Lemmas 08LH and 08LI.

Lemma

Let \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) be a first order thickening of ringed spaces. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}_{X'}\)-modules and set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}_X\)-linear map. There exists an element \[o(\varphi) \in \Ext^1_{\mathcal{O}_X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')\] whose vanishing is a necessary and sufficient condition for the existence of a lift of \(\varphi\) to an \(\mathcal{O}_{X'}\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\).

Proof

It is clear from the proof of Lemma 08LH that the vanishing of the boundary of \(\varphi\) via the map \[\Hom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G}) = \Hom_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}) \longrightarrow \Ext^1_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}')\] is a necessary and sufficient condition for the existence of a lift. We conclude as \[\Ext^1_{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \Ext^1_{\mathcal{O}_X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')\] the adjointness of \(i_* = Ri_*\) and \(Li^*\) on the derived category (Cohomology, Lemma 079W).

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08KZ. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}_{X'}\)-modules and set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}_X\)-linear map. Assume that \(\mathcal{F}'\) and \(\mathcal{G}'\) are flat over \(S'\) and that \((f, f')\) is a strict morphism of thickenings. There exists an element \[o(\varphi) \in \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J})\] whose vanishing is a necessary and sufficient condition for the existence of a lift of \(\varphi\) to an \(\mathcal{O}_{X'}\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\).

Proof

This follows from Lemma 08LK as we claim that under the assumptions of the lemma we have \[\Ext^1_{\mathcal{O}_X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') = \Ext^1_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J})\] Namely, we have \(\mathcal{I}\mathcal{G}' = \mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{J}\) by Lemma 08LI. On the other hand, observe that \[H^{-1}(Li^*\mathcal{F}') = \text{Tor}_1^{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{O}_X)\] (local computation omitted). Using the short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\] we see that this \(\text{Tor}_1\) is computed by the kernel of the map \(\mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I}\mathcal{F}'\) which is zero by the final assertion of Lemma 08LI. Thus \(\tau_{\geq -1}Li^*\mathcal{F}' = \mathcal{F}\). On the other hand, we have \[\Ext^1_{\mathcal{O}_X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') = \Ext^1_{\mathcal{O}_X}(\tau_{\geq -1}Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')\] by the dual of Derived Categories, Lemma 05TC.

Proof

We can apply Lemma 08L6 as follows. Note that \(\mathcal{K} = \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}\) and \(\mathcal{L} = \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{G}\) by Lemma 08LI, that \(c_{\mathcal{F}'} = 1 \otimes 1\) and \(c_{\mathcal{G}'} = 1 \otimes 1\) and taking \(\psi = 1 \otimes \varphi\) the diagram of the lemma commutes. Thus \(o(\varphi) = o(\varphi, 1 \otimes \varphi)\) works.

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08KZ. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Assume \((f, f')\) is a strict morphism of thickenings and \(\mathcal{F}\) flat over \(S\). If there exists a pair \((\mathcal{F}', \alpha)\) consisting of an \(\mathcal{O}_{X'}\)-module \(\mathcal{F}'\) flat over \(S'\) and an isomorphism \(\alpha : i^*\mathcal{F}' \to \mathcal{F}\), then the set of isomorphism classes of such pairs is principal homogeneous under \(\Ext^1_{\mathcal{O}_X}( \mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F})\).

Proof

If we assume there exists one such module, then the canonical map \[f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}\] is an isomorphism by Lemma 08LI. Apply Lemma 08L7 with \(\mathcal{K} = \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}\) and \(c = 1\). By Lemma 08LI the corresponding extensions \(\mathcal{F}'\) are all flat over \(S'\).

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08KZ. Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Assume \((f, f')\) is a strict morphism of thickenings and \(\mathcal{F}\) flat over \(S\). There exists an \(\mathcal{O}_{X'}\)-module \(\mathcal{F}'\) flat over \(S'\) with \(i^*\mathcal{F}' \cong \mathcal{F}\), if and only if

1. the canonical map \(f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}\) is an isomorphism, and

2. the class \(o(\mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F}, 1) \in \Ext^2_{\mathcal{O}_X}( \mathcal{F}, \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F})\) of Lemma 08L8 is zero.

Proof

This follows immediately from the characterization of \(\mathcal{O}_{X'}\)-modules flat over \(S'\) of Lemma 08LI and Lemma 08L8.

Application to flat modules on flat thickenings of ringed spaces

Consider a commutative diagram \[\xymatrix{ (X, \mathcal{O}_X) \ar[r]_i \ar[d]_f & (X', \mathcal{O}_{X'}) \ar[d]^{f'} \\ (S, \mathcal{O}_S) \ar[r]^t & (S', \mathcal{O}_{S'}) }\] of ringed spaces whose horizontal arrows are first order thickenings as in Situation 08KZ. Set \(\mathcal{I} = \Ker(i^\sharp) \subset \mathcal{O}_{X'}\) and \(\mathcal{J} = \Ker(t^\sharp) \subset \mathcal{O}_{S'}\). Let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module. Assume that

1. \((f, f')\) is a strict morphism of thickenings,

2. \(f'\) is flat, and

3. \(\mathcal{F}\) is flat over \(S\).

Note that (1) \(+\) (2) imply that \(\mathcal{I} = f^*\mathcal{J}\) (apply Lemma 08LI to \(\mathcal{O}_{X'}\)). The theory of the preceding section is especially nice under these assumptions. We summarize the results already obtained in the following lemma.

Lemma

In the situation above.

1. There exists an \(\mathcal{O}_{X'}\)-module \(\mathcal{F}'\) flat over \(S'\) with \(i^*\mathcal{F}' \cong \mathcal{F}\), if and only if the class \(o(\mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F}, 1) \in \Ext^2_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})\) of Lemma 08L8 is zero.

2. If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under \(\Ext^1_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})\).

3. Given a lift \(\mathcal{F}'\), the set of automorphisms of \(\mathcal{F}'\) which pull back to \(\text{id}_\mathcal{F}\) is canonically isomorphic to \(\Ext^0_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J} \otimes_{\mathcal{O}_X} \mathcal{F})\).

Proof

Part (1) follows from Lemma 08LN as we have seen above that \(\mathcal{I} = f^*\mathcal{J}\). Part (2) follows from Lemma 08LM. Part (3) follows from Lemma 08LJ.

Situation

Let \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\) be a morphism of ringed spaces. Consider a commutative diagram \[\xymatrix{ (X'_1, \mathcal{O}'_1) \ar[r]_h \ar[d]_{f'_1} & (X'_2, \mathcal{O}'_2) \ar[r] \ar[d]_{f'_2} & (X'_3, \mathcal{O}'_3) \ar[d]_{f'_3} \\ (S'_1, \mathcal{O}_{S'_1}) \ar[r] & (S'_2, \mathcal{O}_{S'_2}) \ar[r] & (S'_3, \mathcal{O}_{S'_3}) }\] where (a) the top row is a short exact sequence of first order thickenings of \(X\), (b) the lower row is a short exact sequence of first order thickenings of \(S\), (c) each \(f'_i\) restricts to \(f\), (d) each pair \((f, f_i')\) is a strict morphism of thickenings, and (e) each \(f'_i\) is flat. Finally, let \(\mathcal{F}'_2\) be an \(\mathcal{O}'_2\)-module flat over \(S'_2\) and set \(\mathcal{F} = \mathcal{F}'_2|_X\). Let \(\pi : X'_1 \to X\) be the canonical splitting (Remark 08LE).

Lemma

In Situation 08VS the modules \(\pi^*\mathcal{F}\) and \(h^*\mathcal{F}'_2\) are \(\mathcal{O}'_1\)-modules flat over \(S'_1\) restricting to \(\mathcal{F}\) on \(X\). Their difference (Lemma 08VR) is an element \(\theta\) of \(\Ext^1_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J}_1 \otimes_{\mathcal{O}_X} \mathcal{F})\) whose boundary in \(\Ext^2_{\mathcal{O}_X}( \mathcal{F}, f^*\mathcal{J}_3 \otimes_{\mathcal{O}_X} \mathcal{F})\) equals the obstruction (Lemma 08VR) to lifting \(\mathcal{F}\) to an \(\mathcal{O}'_3\)-module flat over \(S'_3\).

Proof

Note that both \(\pi^*\mathcal{F}\) and \(h^*\mathcal{F}'_2\) restrict to \(\mathcal{F}\) on \(X\) and that the kernels of \(\pi^*\mathcal{F} \to \mathcal{F}\) and \(h^*\mathcal{F}'_2 \to \mathcal{F}\) are given by \(f^*\mathcal{J}_1 \otimes_{\mathcal{O}_X} \mathcal{F}\). Hence flatness by Lemma 08LI. Taking the boundary makes sense as the sequence of modules \[0 \to f^*\mathcal{J}_3 \otimes_{\mathcal{O}_X} \mathcal{F} \to f^*\mathcal{J}_2 \otimes_{\mathcal{O}_X} \mathcal{F} \to f^*\mathcal{J}_1 \otimes_{\mathcal{O}_X} \mathcal{F} \to 0\] is short exact due to the assumptions in Situation 08VS and the fact that \(\mathcal{F}\) is flat over \(S\). The statement on the obstruction class is a direct translation of the result of Remark 08LF to this particular situation.

Deformations of ringed spaces and the naive cotangent complex

In this section we use the naive cotangent complex to do a little bit of deformation theory. We start with a first order thickening \(t : (S, \mathcal{O}_S) \to (S', \mathcal{O}_{S'})\) of ringed spaces. We denote \(\mathcal{J} = \Ker(t^\sharp)\) and we identify the underlying topological spaces of \(S\) and \(S'\). Moreover we assume given a morphism of ringed spaces \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\), an \(\mathcal{O}_X\)-module \(\mathcal{G}\), and an \(f\)-map \(c : \mathcal{J} \to \mathcal{G}\) of sheaves of modules (Sheaves, Definition 008J and Section 0094). In this section we ask ourselves whether we can find the question mark fitting into the following diagram [08U7]\[\begin{equation} \vcenter{ \xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & \mathcal{J} \ar[u]^c \ar[r] & \mathcal{O}_{S'} \ar[u] \ar[r] & \mathcal{O}_S \ar[u] \ar[r] & 0 } } \end{equation}\] (where the vertical arrows are \(f\)-maps) and moreover how unique the solution is (if it exists). More precisely, we look for a first order thickening \(i : (X, \mathcal{O}_X) \to (X', \mathcal{O}_{X'})\) and a morphism of thickenings \((f, f')\) as in (08L0) where \(\Ker(i^\sharp)\) is identified with \(\mathcal{G}\) such that \((f')^\sharp\) induces the given map \(c\). We will say \(X'\) is a solution to (08U7).

Lemma

Assume given a commutative diagram of morphisms of ringed spaces [08U9]\[\begin{equation} \vcenter{ \xymatrix{ & (X_2, \mathcal{O}_{X_2}) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_g & (X'_2, \mathcal{O}_{X'_2}) \ar[d]^{f'_2} \\ & (S_2, \mathcal{O}_{S_2}) \ar[r]^{t_2} \ar[ddl]|\hole & (S'_2, \mathcal{O}_{S'_2}) \ar[ddl] \\ (X_1, \mathcal{O}_{X_1}) \ar[r]_{i_1} \ar[d]_{f_1} & (X'_1, \mathcal{O}_{X'_1}) \ar[d]^{f'_1} \\ (S_1, \mathcal{O}_{S_1}) \ar[r]^{t_1} & (S'_1, \mathcal{O}_{S'_1}) } } \end{equation}\] whose horizontal arrows are first order thickenings. Set \(\mathcal{G}_j = \Ker(i_j^\sharp)\) and assume given a \(g\)-map \(\nu : \mathcal{G}_1 \to \mathcal{G}_2\) of modules giving rise to the commutative diagram [08UA]\[\begin{equation} \vcenter{ \xymatrix{ & 0 \ar[r] & \mathcal{G}_2 \ar[r] & \mathcal{O}_{X'_2} \ar[r] & \mathcal{O}_{X_2} \ar[r] & 0 \\ & 0 \ar[r]|\hole & \mathcal{J}_2 \ar[u]_{c_2} \ar[r] & \mathcal{O}_{S'_2} \ar[u] \ar[r]|\hole & \mathcal{O}_{S_2} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G}_1 \ar[ruu] \ar[r] & \mathcal{O}_{X'_1} \ar[r] & \mathcal{O}_{X_1} \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & \mathcal{O}_{S'_1} \ar[ruu]|\hole \ar[u] \ar[r] & \mathcal{O}_{S_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0 } } \end{equation}\] with front and back solutions to (08U7).

1. There exist a canonical element in \(\Ext^1_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2)\) whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed spaces \(X'_2 \to X'_1\) fitting into (08U9) compatibly with \(\nu\).

2. If there exists a morphism \(X'_2 \to X'_1\) fitting into (08U9) compatibly with \(\nu\) the set of all such morphisms is a principal homogeneous space under \[\Hom_{\mathcal{O}_{X_1}}(\Omega_{X_1/S_1}, g_*\mathcal{G}_2) = \Hom_{\mathcal{O}_{X_2}}(g^*\Omega_{X_1/S_1}, \mathcal{G}_2) = \Ext^0_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2).\]

Proof

The naive cotangent complex \(\NL_{X_1/S_1}\) is defined in Modules, Definition 08TN. The equalities in the last statement of the lemma follow from the fact that \(g^*\) is adjoint to \(g_*\), the fact that \(H^0(\NL_{X_1/S_1}) = \Omega_{X_1/S_1}\) (by construction of the naive cotangent complex) and the fact that \(Lg^*\) is the left derived functor of \(g^*\). Thus we will work with the groups \(\Ext^k_{\mathcal{O}_{X_2}}(Lg^*\NL_{X_1/S_1}, \mathcal{G}_2)\), \(k = 0, 1\) in the rest of the proof. We first argue that we can reduce to the case where the underlying topological spaces of all ringed spaces in the lemma is the same.

To do this, observe that \(g^{-1}\NL_{X_1/S_1}\) is equal to the naive cotangent complex of the homomorphism of sheaves of rings \(g^{-1}f_1^{-1}\mathcal{O}_{S_1} \to g^{-1}\mathcal{O}_{X_1}\), see Modules, Lemma 08TM. Moreover, the degree \(0\) term of \(\NL_{X_1/S_1}\) is a flat \(\mathcal{O}_{X_1}\)-module, hence the canonical map \[Lg^*\NL_{X_1/S_1} \longrightarrow g^{-1}\NL_{X_1/S_1} \otimes_{g^{-1}\mathcal{O}_{X_1}} \mathcal{O}_{X_2}\] induces an isomorphism on cohomology sheaves in degrees \(0\) and \(-1\). Thus we may replace the Ext groups of the lemma with \[\Ext^k_{g^{-1}\mathcal{O}_{X_1}}(g^{-1}\NL_{X_1/S_1}, \mathcal{G}_2) = \Ext^k_{g^{-1}\mathcal{O}_{X_1}}( \NL_{g^{-1}\mathcal{O}_{X_1}/g^{-1}f_1^{-1}\mathcal{O}_{S_1}}, \mathcal{G}_2)\] The set of morphism of ringed spaces \(X'_2 \to X'_1\) fitting into (08U9) compatibly with \(\nu\) is in one-to-one bijection with the set of homomorphisms of \(g^{-1}f_1^{-1}\mathcal{O}_{S'_1}\)-algebras \(g^{-1}\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}\) which are compatible with \(f^\sharp\) and \(\nu\). In this way we see that we may assume we have a diagram (08UA) of sheaves on \(X\) and we are looking to find a homomorphism of sheaves of rings \(\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}\) fitting into it.

In the rest of the proof of the lemma we assume all underlying topological spaces are the same, i.e., we have a diagram (08UA) of sheaves on a space \(X\) and we are looking for homomorphisms of sheaves of rings \(\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}\) fitting into it. As ext groups we will use \(\Ext^k_{\mathcal{O}_{X_1}}( \NL_{\mathcal{O}_{X_1}/\mathcal{O}_{S_1}}, \mathcal{G}_2)\), \(k = 0, 1\).

Step 1. Construction of the obstruction class. Consider the sheaf of sets \[\mathcal{E} = \mathcal{O}_{X'_1} \times_{\mathcal{O}_{X_2}} \mathcal{O}_{X'_2}\] This comes with a surjective map \(\alpha : \mathcal{E} \to \mathcal{O}_{X_1}\) and hence we can use \(\NL(\alpha)\) instead of \(\NL_{\mathcal{O}_{X_1}/\mathcal{O}_{S_1}}\), see Modules, Lemma 08TL. Set \[\mathcal{I}' = \Ker(\mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X_1}) \quad\text{and}\quad \mathcal{I} = \Ker(\mathcal{O}_{S_1}[\mathcal{E}] \to \mathcal{O}_{X_1})\] There is a surjection \(\mathcal{I}' \to \mathcal{I}\) whose kernel is \(\mathcal{J}_1\mathcal{O}_{S'_1}[\mathcal{E}]\). We obtain two homomorphisms of \(\mathcal{O}_{S'_1}\)-algebras \[a : \mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X'_1} \quad\text{and}\quad b : \mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X'_2}\] which induce maps \(a|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_1\) and \(b|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_2\). Both \(a\) and \(b\) annihilate \((\mathcal{I}')^2\). Moreover \(a\) and \(b\) agree on \(\mathcal{J}_1\mathcal{O}_{S'_1}[\mathcal{E}]\) as maps into \(\mathcal{G}_2\) because the left hand square of (08UA) is commutative. Thus the difference \(b|_{\mathcal{I}'} - \nu \circ a|_{\mathcal{I}'}\) induces a well defined \(\mathcal{O}_{X_1}\)-linear map \[\xi : \mathcal{I}/\mathcal{I}^2 \longrightarrow \mathcal{G}_2\] which sends the class of a local section \(f\) of \(\mathcal{I}\) to \(\nu(a(f')) - b(f')\) where \(f'\) is a lift of \(f\) to a local section of \(\mathcal{I}'\). We let \([\xi] \in \Ext^1_{\mathcal{O}_{X_1}}(\NL(\alpha), \mathcal{G}_2)\) be the image (see below).

Step 2. Vanishing of \([\xi]\) is necessary. Let us write \(\Omega = \Omega_{\mathcal{O}_{S_1}[\mathcal{E}]/\mathcal{O}_{S_1}} \otimes_{\mathcal{O}_{S_1}[\mathcal{E}]} \mathcal{O}_{X_1}\). Observe that \(\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega)\) fits into a distinguished triangle \[\Omega[0] \to \NL(\alpha) \to \mathcal{I}/\mathcal{I}^2[1] \to \Omega[1]\] Thus we see that \([\xi]\) is zero if and only if \(\xi\) is a composition \(\mathcal{I}/\mathcal{I}^2 \to \Omega \to \mathcal{G}_2\) for some map \(\Omega \to \mathcal{G}_2\). Suppose there exists a homomorphisms of sheaves of rings \(\varphi : \mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}\) fitting into (08UA). In this case consider the map \(\mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{G}_2\), \(f' \mapsto b(f') - \varphi(a(f'))\). A calculation shows this annihilates \(\mathcal{J}_1\mathcal{O}_{S'_1}[\mathcal{E}]\) and induces a derivation \(\mathcal{O}_{S_1}[\mathcal{E}] \to \mathcal{G}_2\). The resulting linear map \(\Omega \to \mathcal{G}_2\) witnesses the fact that \([\xi] = 0\) in this case.

Step 3. Vanishing of \([\xi]\) is sufficient. Let \(\theta : \Omega \to \mathcal{G}_2\) be a \(\mathcal{O}_{X_1}\)-linear map such that \(\xi\) is equal to \(\theta \circ (\mathcal{I}/\mathcal{I}^2 \to \Omega)\). Then a calculation shows that \[b + \theta \circ d : \mathcal{O}_{S'_1}[\mathcal{E}] \to \mathcal{O}_{X'_2}\] restricted to \(\mathcal{I}'\) agrees with \(\nu \circ a : \mathcal{I}' \to \mathcal{G}_2\). Since \(\mathcal{O}_{X'_1}\) is the pushout of \(\mathcal{I}' \to \mathcal{O}_{S'_1}[\mathcal{E}]\) and \(\mathcal{I}' \to \mathcal{G}_1\) the maps \(b + \theta \circ d\) and \(a\) define a map \(\mathcal{O}_{X'_1} \to \mathcal{O}_{X'_2}\) fitting into (08UA).

Proof of (2) in the special case above. Omitted. Hint: This is exactly the same as the proof of (2) of Lemma 08S5.

Lemma

Let \(X\) be a topological space. Let \(\mathcal{A} \to \mathcal{B}\) be a homomorphism of sheaves of rings. Let \(\mathcal{G}\) be a \(\mathcal{B}\)-module. Let \(\xi \in \Ext^1_\mathcal{B}(\NL_{\mathcal{B}/\mathcal{A}}, \mathcal{G})\). There exists a map of sheaves of sets \(\alpha : \mathcal{E} \to \mathcal{B}\) such that \(\xi \in \Ext^1_\mathcal{B}(\NL(\alpha), \mathcal{G})\) is the class of a map \(\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}\) (see proof for notation).

Proof

Recall that given \(\alpha : \mathcal{E} \to \mathcal{B}\) such that \(\mathcal{A}[\mathcal{E}] \to \mathcal{B}\) is surjective with kernel \(\mathcal{I}\) the complex \(\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B})\) is canonically isomorphic to \(\NL_{\mathcal{B}/\mathcal{A}}\), see Modules, Lemma 08TL. Observe moreover, that \(\Omega = \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B}\) is the sheaf associated to the presheaf \(U \mapsto \bigoplus_{e \in \mathcal{E}(U)} \mathcal{B}(U)\). In other words, \(\Omega\) is the free \(\mathcal{B}\)-module on the sheaf of sets \(\mathcal{E}\) and in particular there is a canonical map \(\mathcal{E} \to \Omega\).

Having said this, pick some \(\mathcal{E}\) (for example \(\mathcal{E} = \mathcal{B}\) as in the definition of the naive cotangent complex). The obstruction to writing \(\xi\) as the class of a map \(\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}\) is an element in \(\Ext^1_\mathcal{B}(\Omega, \mathcal{G})\). Say this is represented by the extension \(0 \to \mathcal{G} \to \mathcal{H} \to \Omega \to 0\) of \(\mathcal{B}\)-modules. Consider the sheaf of sets \(\mathcal{E}' = \mathcal{E} \times_\Omega \mathcal{H}\) which comes with an induced map \(\alpha' : \mathcal{E}' \to \mathcal{B}\). Let \(\mathcal{I}' = \Ker(\mathcal{A}[\mathcal{E}'] \to \mathcal{B})\) and \(\Omega' = \Omega_{\mathcal{A}[\mathcal{E}']/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}']} \mathcal{B}\). The pullback of \(\xi\) under the quasi-isomorphism \(\NL(\alpha') \to \NL(\alpha)\) maps to zero in \(\Ext^1_\mathcal{B}(\Omega', \mathcal{G})\) because the pullback of the extension \(\mathcal{H}\) by the map \(\Omega' \to \Omega\) is split as \(\Omega'\) is the free \(\mathcal{B}\)-module on the sheaf of sets \(\mathcal{E}'\) and since by construction there is a commutative diagram \[\xymatrix{ \mathcal{E}' \ar[r] \ar[d] & \mathcal{E} \ar[d] \\ \mathcal{H} \ar[r] & \Omega }\] This finishes the proof.

Lemma

If there exists a solution to (08U7), then the set of isomorphism classes of solutions is principal homogeneous under \(\Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})\).

Proof

We observe right away that given two solutions \(X'_1\) and \(X'_2\) to (08U7) we obtain by Lemma 08U8 an obstruction element \(o(X'_1, X'_2) \in \Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})\) to the existence of a map \(X'_1 \to X'_2\). Clearly, this element is the obstruction to the existence of an isomorphism, hence separates the isomorphism classes. To finish the proof it therefore suffices to show that given a solution \(X'\) and an element \(\xi \in \Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})\) we can find a second solution \(X'_\xi\) such that \(o(X', X'_\xi) = \xi\).

Pick \(\alpha : \mathcal{E} \to \mathcal{O}_X\) as in Lemma 08UB for the class \(\xi\). Consider the surjection \(f^{-1}\mathcal{O}_S[\mathcal{E}] \to \mathcal{O}_X\) with kernel \(\mathcal{I}\) and corresponding naive cotangent complex \(\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega_{f^{-1}\mathcal{O}_S[\mathcal{E}]/f^{-1}\mathcal{O}_S} \otimes_{f^{-1}\mathcal{O}_S[\mathcal{E}]} \mathcal{O}_X)\). By the lemma \(\xi\) is the class of a morphism \(\delta : \mathcal{I}/\mathcal{I}^2 \to \mathcal{G}\). After replacing \(\mathcal{E}\) by \(\mathcal{E} \times_{\mathcal{O}_X} \mathcal{O}_{X'}\) we may also assume that \(\alpha\) factors through a map \(\alpha' : \mathcal{E} \to \mathcal{O}_{X'}\).

These choices determine an \(f^{-1}\mathcal{O}_{S'}\)-algebra map \(\varphi : f^{-1}\mathcal{O}_{S'}[\mathcal{E}] \to \mathcal{O}_{X'}\). Let \(\mathcal{I}' = \Ker(f^{-1}\mathcal{O}_{S'}[\mathcal{E}] \to \mathcal{O}_X)\). Observe that \(\varphi\) induces a map \(\varphi|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}\) and that \(\mathcal{O}_{X'}\) is the pushout, as in the following diagram \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & \mathcal{I}' \ar[u]^{\varphi|_{\mathcal{I}'}} \ar[r] & f^{-1}\mathcal{O}_{S'}[\mathcal{E}] \ar[u] \ar[r] & \mathcal{O}_X \ar[u]_{=} \ar[r] & 0 }\] Let \(\psi : \mathcal{I}' \to \mathcal{G}\) be the sum of the map \(\varphi|_{\mathcal{I}'}\) and the composition \[\mathcal{I}' \to \mathcal{I}'/(\mathcal{I}')^2 \to \mathcal{I}/\mathcal{I}^2 \xrightarrow{\delta} \mathcal{G}.\] Then the pushout along \(\psi\) is an other ring extension \(\mathcal{O}_{X'_\xi}\) fitting into a diagram as above. A calculation (omitted) shows that \(o(X', X'_\xi) = \xi\) as desired.

Lemma

Let \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\) be a morphism of ringed spaces. Let \(\mathcal{G}\) be a \(\mathcal{O}_X\)-module. The set of isomorphism classes of extensions of \(f^{-1}\mathcal{O}_S\)-algebras \[0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\] where \(\mathcal{G}\) is an ideal of square zero¹ is canonically bijective to \(\Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})\).

Proof

To prove this we apply the previous results to the case where (08U7) is given by the diagram \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & \mathcal{O}_S \ar[u] \ar[r]^{\text{id}} & \mathcal{O}_S \ar[u] \ar[r] & 0 }\] Thus our lemma follows from Lemma 08UC and the fact that there exists a solution, namely \(\mathcal{G} \oplus \mathcal{O}_X\). (See remark below for a direct construction of the bijection.)

Remark

Let \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\) and \(\mathcal{G}\) be as in Lemma 0GPZ. Consider an extension \(0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\) as in the lemma. We can choose a sheaf of sets \(\mathcal{E}\) and a commutative diagram \[\xymatrix{ \mathcal{E} \ar[d]_{\alpha'} \ar[rd]^\alpha \\ \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X }\] such that \(f^{-1}\mathcal{O}_S[\mathcal{E}] \to \mathcal{O}_X\) is surjective with kernel \(\mathcal{J}\). (For example you can take any sheaf of sets surjecting onto \(\mathcal{O}_{X'}\).) Then \[\NL_{X/S} \cong \NL(\alpha) = \left( \mathcal{J}/\mathcal{J}^2 \longrightarrow \Omega_{f^{-1}\mathcal{O}_S[\mathcal{E}]/f^{-1}\mathcal{O}_S} \otimes_{f^{-1}\mathcal{O}_S[\mathcal{E}]} \mathcal{O}_X\right)\] See Modules, Section 08TG and in particular Lemma 08TL. Of course \(\alpha'\) determines a map \(f^{-1}\mathcal{O}_S[\mathcal{E}] \to \mathcal{O}_{X'}\) which in turn determines a map \[\mathcal{J}/\mathcal{J}^2 \longrightarrow \mathcal{G}\] which in turn determines the element of \(\Ext^1_{\mathcal{O}_X}(\NL(\alpha), \mathcal{G}) = \Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})\) corresponding to \(\mathcal{O}_{X'}\) by the bijection of the lemma.

Lemma

Let \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\) and \(g : (Y, \mathcal{O}_Y) \to (X, \mathcal{O}_X)\) be morphisms of ringed spaces. Let \(\mathcal{F}\) be a \(\mathcal{O}_X\)-module. Let \(\mathcal{G}\) be a \(\mathcal{O}_Y\)-module. Let \(c : \mathcal{F} \to \mathcal{G}\) be a \(g\)-map. Finally, consider

1. \(0 \to \mathcal{F} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\) an extension of \(f^{-1}\mathcal{O}_S\)-algebras corresponding to \(\xi \in \Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{F})\), and

2. \(0 \to \mathcal{G} \to \mathcal{O}_{Y'} \to \mathcal{O}_Y \to 0\) an extension of \(g^{-1}f^{-1}\mathcal{O}_S\)-algebras corresponding to \(\zeta \in \Ext^1_{\mathcal{O}_Y}(\NL_{Y/S}, \mathcal{G})\).

See Lemma 0GPZ. Then there is an \(S\)-morphism \(g' : Y' \to X'\) compatible with \(g\) and \(c\) if and only if \(\xi\) and \(\zeta\) map to the same element of \(\Ext^1_{\mathcal{O}_Y}(Lg^*\NL_{X/S}, \mathcal{G})\).

Proof

The stament makes sense as we have the maps \[\Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{F}) \to \Ext^1_{\mathcal{O}_Y}(Lg^*\NL_{X/S}, Lg^*\mathcal{F}) \to \Ext^1_{\mathcal{O}_Y}(Lg^*\NL_{X/S}, \mathcal{G})\] using the map \(Lg^*\mathcal{F} \to g^*\mathcal{F} \xrightarrow{c} \mathcal{G}\) and \[\Ext^1_{\mathcal{O}_Y}(\NL_{Y/S}, \mathcal{G}) \to \Ext^1_{\mathcal{O}_Y}(Lg^*\NL_{X/S}, \mathcal{G})\] using the map \(Lg^*\NL_{X/S} \to \NL_{Y/S}\). The statement of the lemma can be deduced from Lemma 08U8 applied to the diagram \[\xymatrix{ & 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{Y'} \ar[r] & \mathcal{O}_Y \ar[r] & 0 \\ & 0 \ar[r]|\hole & 0 \ar[u] \ar[r] & \mathcal{O}_S \ar[u] \ar[r]|\hole & \mathcal{O}_S \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[ruu] \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[ruu]|\hole \ar[u] \ar[r] & \mathcal{O}_S \ar[ruu]|\hole \ar[u] \ar[r] & \mathcal{O}_S \ar[ruu]|\hole \ar[u] \ar[r] & 0 }\] and a compatibility between the constructions in the proofs of Lemmas 0GPZ and 08U8 whose statement and proof we omit. (See remark below for a direct argument.)

Remark

Let \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\), \(g : (Y, \mathcal{O}_Y) \to (X, \mathcal{O}_X)\), \(\mathcal{F}\), \(\mathcal{G}\), \(c : \mathcal{F} \to \mathcal{G}\), \(0 \to \mathcal{F} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\), \(\xi \in \Ext^1_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{F})\), \(0 \to \mathcal{G} \to \mathcal{O}_{Y'} \to \mathcal{O}_Y \to 0\), and \(\zeta \in \Ext^1_{\mathcal{O}_Y}(\NL_{Y/S}, \mathcal{G})\) be as in Lemma 0GQ1. Using pushout along \(c : g^{-1}\mathcal{F} \to \mathcal{G}\) we can construct an extension \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}'_1 \ar[r] & g^{-1}\mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{F} \ar[u]^c \ar[r] & g^{-1}\mathcal{O}_{X'} \ar[u] \ar[r] & g^{-1}\mathcal{O}_X \ar@{=}[u] \ar[r] & 0 }\] Using pullback along \(g^\sharp : g^{-1}\mathcal{O}_X \to \mathcal{O}_Y\) we can construct an extension \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{Y'} \ar[r] & \mathcal{O}_Y \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G} \ar@{=}[u] \ar[r] & \mathcal{O}'_2 \ar[u] \ar[r] & g^{-1}\mathcal{O}_X \ar[u] \ar[r] & 0 }\] A diagram chase tells us that there exists an \(S\)-morphism \(Y' \to X'\) compatible with \(g\) and \(c\) if and only if \(\mathcal{O}'_1\) is isomorphic to \(\mathcal{O}'_2\) as \(g^{-1}f^{-1}\mathcal{O}_S\)-algebra extensions of \(g^{-1}\mathcal{O}_X\) by \(\mathcal{G}\). By Lemma 0GPZ these extensions are classified by the LHS of \[\Ext^1_{g^{-1}\mathcal{O}_X}( \NL_{g^{-1}\mathcal{O}_X/g^{-1}f^{-1}\mathcal{O}_S}, \mathcal{G}) = \Ext^1_{\mathcal{O}_Y}(Lg^*\NL_{X/S}, \mathcal{G})\] Here the equality comes from tensor-hom adjunction and the equalities \[\NL_{g^{-1}\mathcal{O}_X/g^{-1}f^{-1}\mathcal{O}_S} = g^{-1}\NL_{X/S} \quad\text{and}\quad Lg^*\NL_{X/S} = g^{-1}\NL_{X/S} \otimes_{g^{-1}\mathcal{O}_X}^\mathbf{L} \mathcal{O}_Y\] For the first of these see Modules, Lemma 08TM; the second follows from the definition of derived pullback. Thus, in order to see that Lemma 0GQ1 is true, it suffices to show that \(\mathcal{O}'_1\) corresponds to the image of \(\xi\) and that \(\mathcal{O}'_2\) correspond to the image of \(\zeta\). The correspondence between \(\xi\) and \(\mathcal{O}'_1\) is immediate from the construction of the class \(\xi\) in Remark 0GQ0. For the correspondence between \(\zeta\) and \(\mathcal{O}'_2\), we first choose a commutative diagram \[\xymatrix{ \mathcal{E} \ar[d]_{\beta'} \ar[rd]^\beta \\ \mathcal{O}_{Y'} \ar[r] & \mathcal{O}_Y }\] such that \(g^{-1}f^{-1}\mathcal{O}_S[\mathcal{E}] \to \mathcal{O}_Y\) is surjective with kernel \(\mathcal{K}\). Next choose a commutative diagram \[\xymatrix{ \mathcal{E} \ar[d]_{\beta'} & \mathcal{E}' \ar[l]^\varphi \ar[d]_{\alpha'} \ar[rd]^\alpha \\ \mathcal{O}_{Y'} & \mathcal{O}'_2 \ar[l] \ar[r] & g^{-1}\mathcal{O}_X }\] such that \(g^{-1}f^{-1}\mathcal{O}_S[\mathcal{E}'] \to g^{-1}\mathcal{O}_X\) is surjective with kernel \(\mathcal{J}\). (For example just take \(\mathcal{E}' = \mathcal{E} \amalg \mathcal{O}'_2\) as a sheaf of sets.) The map \(\varphi\) induces a map of complexes \(\NL(\alpha) \to \NL(\beta)\) (notation as in Modules, Section 08TG) and in particular \(\bar\varphi : \mathcal{J}/\mathcal{J}^2 \to \mathcal{K}/\mathcal{K}^2\). Then \(\NL(\alpha) \cong \NL_{Y/S}\) and \(\NL(\beta) \cong \NL_{g^{-1}\mathcal{O}_X/g^{-1}f^{-1}\mathcal{O}_S}\) and the map of complexes \(\NL(\alpha) \to \NL(\beta)\) represents the map \(Lg^*\NL_{X/S} \to \NL_{Y/S}\) used in the statement of Lemma 0GQ1 (see first part of its proof). Now \(\zeta\) corresponds to the class of the map \(\mathcal{K}/\mathcal{K}^2 \to \mathcal{G}\) induced by \(\beta'\), see Remark 0GQ0. Similarly, the extension \(\mathcal{O}'_2\) corresponds to the map \(\mathcal{J}/\mathcal{J}^2 \to \mathcal{G}\) induced by \(\alpha'\). The commutative diagram above shows that this map is the composition of the map \(\mathcal{K}/\mathcal{K}^2 \to \mathcal{G}\) induced by \(\beta'\) with the map \(\bar\varphi : \mathcal{J}/\mathcal{J}^2 \to \mathcal{K}/\mathcal{K}^2\). This proves the compatibility we were looking for.

Lemma

Let \(t : (S, \mathcal{O}_S) \to (S', \mathcal{O}_{S'})\), \(\mathcal{J} = \Ker(t^\sharp)\), \(f : (X, \mathcal{O}_X) \to (S, \mathcal{O}_S)\), \(\mathcal{G}\), and \(c : \mathcal{J} \to \mathcal{G}\) be as in (08U7). Denote \(\xi \in \Ext^1_{\mathcal{O}_S}(\NL_{S/S'}, \mathcal{J})\) the element corresponding to the extension \(\mathcal{O}_{S'}\) of \(\mathcal{O}_S\) by \(\mathcal{J}\) via Lemma 0GPZ. The set of isomorphism classes of solutions is canonically bijective to the fibre of \[\Ext^1_{\mathcal{O}_X}(\NL_{X/S'}, \mathcal{G}) \to \Ext^1_{\mathcal{O}_X}(Lf^*\NL_{S/S'}, \mathcal{G})\] over the image of \(\xi\).

Proof

By Lemma 0GPZ applied to \(X \to S'\) and the \(\mathcal{O}_X\)-module \(\mathcal{G}\) we see that elements \(\zeta\) of \(\Ext^1_{\mathcal{O}_X}(\NL_{X/S'}, \mathcal{G})\) parametrize extensions \(0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_X \to 0\) of \(f^{-1}\mathcal{O}_{S'}\)-algebras. By Lemma 0GQ1 applied to \(X \to S \to S'\) and \(c : \mathcal{J} \to \mathcal{G}\) we see that there is an \(S'\)-morphism \(X' \to S'\) compatible with \(c\) and \(f : X \to S\) if and only if \(\zeta\) maps to \(\xi\). Of course this is the same thing as saying \(\mathcal{O}_{X'}\) is a solution of (08U7).

Remark

In the situation of Lemma 0GQ3 we have maps of complexes \[Lf^*\NL_{S'/S} \to \NL_{X/S'} \to \NL_{X/S}\] These maps are closed to forming a distinguished triangle, see Modules, Lemma 0E1Z. If it were a distinguished triangle we would conclude that the image of \(\xi\) in \(\Ext^2_{\mathcal{O}_X}(\NL_{X/S}, \mathcal{G})\) would be the obstruction to the existence of a solution to (08U7).

Deformations of schemes

In this section we spell out what the results in Section 08U6 mean for deformations of schemes.

Lemma

Let \(S \subset S'\) be a first order thickening of schemes. Let \(f : X \to S\) be a flat morphism of schemes. If there exists a flat morphism \(f' : X' \to S'\) of schemes and an isomorphism \(a : X \to X' \times_{S'} S\) over \(S\), then

1. the set of isomorphism classes of pairs \((f' : X' \to S', a)\) is principal homogeneous under \(\Ext^1_{\mathcal{O}_X}(\NL_{X/S}, f^*\mathcal{C}_{S/S'})\), and

2. the set of automorphisms of \(\varphi : X' \to X'\) over \(S'\) which reduce to the identity on \(X' \times_{S'} S\) is \(\Ext^0_{\mathcal{O}_X}(\NL_{X/S}, f^*\mathcal{C}_{S/S'})\).

Proof

First we observe that thickenings of schemes as defined in More on Morphisms, Section 04EW are the same things as morphisms of schemes which are thickenings in the sense of Section 08KY. We may think of \(X\) as a closed subscheme of \(X'\) so that \((f, f') : (X \subset X') \to (S \subset S')\) is a morphism of first order thickenings. Then we see from More on Morphisms, Lemma 063Y (or from the more general Lemma 08LI) that the ideal sheaf of \(X\) in \(X'\) is equal to \(f^*\mathcal{C}_{S/S'}\). Hence we have a commutative diagram \[\xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{S/S'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & \mathcal{C}_{S/S'} \ar[u] \ar[r] & \mathcal{O}_{S'} \ar[u] \ar[r] & \mathcal{O}_S \ar[u] \ar[r] & 0 }\] where the vertical arrows are \(f\)-maps; please compare with (08U7). Thus part (1) follows from Lemma 08UC and part (2) from part (2) of Lemma 08U8. (Note that \(\NL_{X/S}\) as defined for a morphism of schemes in More on Morphisms, Section 0D0G agrees with \(\NL_{X/S}\) as used in Section 08U6.)

Lemma

Let \((A, \mathfrak m, k)\) be a complete Noetherian local ring. Let \(X \to \Spec(A)\) be a proper flat morphism. If for some \(n \geq 0\) \[X \times_{\Spec(A)} \Spec(k) \cong \mathbf{P}^n_k,\] then \(X \cong \mathbf{P}^n_A\) as schemes over \(A\).

Proof

If \(n = 0\), then \(X \to \Spec(A)\) is finite by More on Morphisms, Lemma 02UP. Indeed, the open neighbourhood in that lemma contains the closed point of the spectrum of a local ring and hence is the whole spectrum. Write \(X = \Spec(B)\). The \(A\)-module \(B\) is finite flat and \(B/\mathfrak mB \cong k\). Thus \(B\) is free of rank \(1\) over \(A\) by Algebra, Lemma 00NZ, and the unit map \(A \to B\), which is an isomorphism modulo \(\mathfrak m\), is an isomorphism. This proves the result in this case.

Assume \(n \geq 1\). For \(q \geq 1\), set \[A_q = A/\mathfrak m^q, \qquad S_q = \Spec(A_q), \qquad X_q = X \times_{\Spec(A)} S_q, \qquad P_q = \mathbf{P}^n_{A_q}.\] Choose an isomorphism \(a_1 : X_1 \to P_1\). We claim that it can be extended inductively to a compatible system of isomorphisms \(a_q : X_q \to P_q\). Suppose that \(a_q\) has been constructed. The schemes \(X_{q + 1}\) and \(P_{q + 1}\), together with the identifications induced by \(a_q\), are two flat deformations of \(P_q\) over the first order thickening \[\Spec(A_q) \subset \Spec(A_{q + 1}).\] By Lemma 0D14, the set of isomorphism classes of such deformations is principal homogeneous under \[\Ext^1_{\mathcal{O}_{P_q}} \left( \NL_{P_q/A_q}, \mathcal{O}_{P_q} \otimes_{A_q} \mathfrak m^q/\mathfrak m^{q + 1} \right).\] The morphism \(P_q \to \Spec(A_q)\) is smooth. Hence More on Morphisms, Lemma 0D0N identifies the first argument with \(\Omega_{P_q/A_q}\). Since this sheaf is finite locally free, the displayed group is \[H^1\left(P_q, \SheafHom_{\mathcal{O}_{P_q}} \left(\Omega_{P_q/A_q}, \mathcal{O}_{P_q} \otimes_{A_q} \mathfrak m^q/\mathfrak m^{q + 1}\right)\right).\] Put \(V_q = \mathfrak m^q/\mathfrak m^{q + 1}\) and let \(i : P_1 \to P_q\) be the closed immersion. The vector space \(V_q\) is finite dimensional over \(k\). Base change for differentials and the fact that \(\Omega_{P_q/A_q}\) is finite locally free show that the sheaf in the last display is \[i_*\left( \SheafHom_{\mathcal{O}_{P_1}} (\Omega_{P_1/k}, \mathcal{O}_{P_1}) \otimes_k V_q \right).\] Thus the displayed group is \[H^1\left(P_1, \SheafHom_{\mathcal{O}_{P_1}} (\Omega_{P_1/k}, \mathcal{O}_{P_1})\right) \otimes_k V_q,\] which vanishes by de Rham Cohomology, Lemma derham-lemma-H1-tangent-projective-space. Hence this principal homogeneous space has at most one element. The two deformations are isomorphic compatibly with \(a_q\), proving the claim.

The compatible morphisms \(a_q\) algebraize uniquely to a morphism \(a : X \to \mathbf{P}^n_A\) by Cohomology of Schemes, Lemma 0A42. Applying the same lemma to the compatible inverses \(a_q^{-1}\) gives a morphism \(b : \mathbf{P}^n_A \to X\). The uniqueness assertion of that lemma applied to the two compositions shows that \(a \circ b\) and \(b \circ a\) are the respective identity morphisms. Hence \(a\) is an isomorphism.

Lemma

Let \((A, \mathfrak m, k)\) be a complete Noetherian local ring. Let \(Y\) be a smooth projective scheme over \(k\). Set \[\mathcal{T}_{Y/k} = \SheafHom_{\mathcal{O}_Y}(\Omega_{Y/k}, \mathcal{O}_Y).\] If \[H^2(Y, \mathcal{O}_Y) = H^2(Y, \mathcal{T}_{Y/k}) = 0,\] then there exists a smooth projective morphism \(X \to \Spec(A)\) and an isomorphism \(X \times_{\Spec(A)} \Spec(k) \cong Y\).

Proof

For \(q \geq 1\), set \[A_q = A/\mathfrak m^q, \qquad S_q = \Spec(A_q), \qquad V_q = \mathfrak m^q/\mathfrak m^{q + 1}.\] We will construct a compatible system of smooth proper morphisms \(X_q \to S_q\), starting with \(X_1 = Y\).

Suppose \(X_q\) has been constructed. Apply Cotangent Complex, Lemma 08UZ to the first order thickening \(S_q \subset S_{q + 1}\) and the coefficient module \(\mathcal{O}_{X_q} \otimes_{A_q} V_q\), using the canonical map from the ideal of \(S_q\) in \(S_{q + 1}\). The obstruction to lifting \(X_q\) belongs to \[\Ext^2_{\mathcal{O}_{X_q}} \left(L_{X_q/S_q}, \mathcal{O}_{X_q} \otimes_{A_q} V_q\right).\] The morphism \(X_q \to S_q\) is smooth, so Cotangent Complex, Lemma 08R5 identifies \(L_{X_q/S_q}\) with \(\Omega_{X_q/S_q}[0]\). The finite dimensional \(k\)-vector space \(V_q\) is annihilated by \(\mathfrak m\), and the coefficient module is the pushforward from \(Y = X_1\) of \(\mathcal{O}_Y \otimes_k V_q\). Hence the obstruction group is \[H^2(Y, \mathcal{T}_{Y/k}) \otimes_k V_q = 0.\] We obtain a first order thickening \(X_q \subset X_{q + 1}\) over \(S_q \subset S_{q + 1}\). It is a scheme by More on Morphisms, Lemma 05YV, and it is flat over \(S_{q + 1}\) by More on Morphisms, Lemma 063Y. It is locally of finite presentation and proper by More on Morphisms, Lemma 09ZW. Since its only fibre is \(Y\), it is smooth by Morphisms, Lemma 01V8. This completes the induction.

By Cohomology of Schemes, Lemma coherent-lemma-algebraize-flat-formal-scheme-H2, the system \((X_q)\) algebraizes to a projective flat morphism \(X \to \Spec(A)\). The smooth locus is an open neighbourhood of the closed fibre by Morphisms, Lemma 02V4. Applying the same proper-image argument to its complement shows that \(X \to \Spec(A)\) is smooth.

Lemma

Let \((A, \mathfrak m, k)\) be a complete Noetherian local ring. If \(Y\) is a smooth proper scheme over \(k\) of dimension at most \(1\), then there exists a smooth projective morphism \(X \to \Spec(A)\) whose special fibre is isomorphic to \(Y\).

Proof

The scheme \(Y\) is projective by Varieties, Lemma 0A26. Moreover, \[H^2(Y, \mathcal{O}_Y) = H^2(Y, \mathcal{T}_{Y/k}) = 0\] by Cohomology, Proposition 02UZ. Thus the result follows from Lemma defos-lemma-projective-lift-complete-local.

Thickenings of ringed topoi

This section is the analogue of Section 08KY for ringed topoi. In the following few sections we will use the following notions:

1. A sheaf of ideals \(\mathcal{I} \subset \mathcal{O}'\) on a ringed topos \((\Sh(\mathcal{D}), \mathcal{O}')\) is locally nilpotent if any local section of \(\mathcal{I}\) is locally nilpotent.

2. A thickening of ringed topoi is a morphism \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) of ringed topoi such that

   1. \(i_*\) is an equivalence \(\Sh(\mathcal{C}) \to \Sh(\mathcal{D})\),

   2. the map \(i^\sharp : \mathcal{O}' \to i_*\mathcal{O}\) is surjective, and

   3. the kernel of \(i^\sharp\) is a locally nilpotent sheaf of ideals.

3. A first order thickening of ringed topoi is a thickening \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) of ringed topoi such that \(\Ker(i^\sharp)\) has square zero.

4. It is clear how to define morphisms of thickenings of ringed topoi, morphisms of thickenings of ringed topoi over a base ringed topos, etc.

If \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) is a thickening of ringed topoi then we identify the underlying topoi and think of \(\mathcal{O}\), \(\mathcal{O}'\), and \(\mathcal{I} = \Ker(i^\sharp)\) as sheaves on \(\mathcal{C}\). We obtain a short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0\] of \(\mathcal{O}'\)-modules. By Modules on Sites, Lemma 08M3 the category of \(\mathcal{O}\)-modules is equivalent to the category of \(\mathcal{O}'\)-modules annihilated by \(\mathcal{I}\). In particular, if \(i\) is a first order thickening, then \(\mathcal{I}\) is a \(\mathcal{O}\)-module.

Situation

A morphism of thickenings of ringed topoi \((f, f')\) is given by a commutative diagram [08M8]\[\begin{equation} \vcenter{ \xymatrix{ (\Sh(\mathcal{C}), \mathcal{O}) \ar[r]_i \ar[d]_f & (\Sh(\mathcal{D}), \mathcal{O}') \ar[d]^{f'} \\ (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \ar[r]^t & (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) } } \end{equation}\] of ringed topoi whose horizontal arrows are thickenings. In this situation we set \(\mathcal{I} = \Ker(i^\sharp) \subset \mathcal{O}'\) and \(\mathcal{J} = \Ker(t^\sharp) \subset \mathcal{O}_{\mathcal{B}'}\). As \(f = f'\) on underlying topoi we will identify the pullback functors \(f^{-1}\) and \((f')^{-1}\). Observe that \((f')^\sharp : f^{-1}\mathcal{O}_{\mathcal{B}'} \to \mathcal{O}'\) induces in particular a map \(f^{-1}\mathcal{J} \to \mathcal{I}\) and therefore a map of \(\mathcal{O}'\)-modules \[(f')^*\mathcal{J} \longrightarrow \mathcal{I}\] If \(i\) and \(t\) are first order thickenings, then \((f')^*\mathcal{J} = f^*\mathcal{J}\) and the map above becomes a map \(f^*\mathcal{J} \to \mathcal{I}\).

Definition

In Situation 08M7 we say that \((f, f')\) is a strict morphism of thickenings if the map \((f')^*\mathcal{J} \longrightarrow \mathcal{I}\) is surjective.

Modules on first order thickenings of ringed topoi

In this section we discuss some preliminaries to the deformation theory of modules. Let \(i : (\Sh(\mathcal{C}, \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. We will freely use the notation introduced in Section 08M6, in particular we will identify the underlying topological topoi. In this section we consider short exact sequences [08MB]\[\begin{equation} 0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \end{equation}\] of \(\mathcal{O}'\)-modules, where \(\mathcal{F}\), \(\mathcal{K}\) are \(\mathcal{O}\)-modules and \(\mathcal{F}'\) is an \(\mathcal{O}'\)-module. In this situation we have a canonical \(\mathcal{O}\)-module map \[c_{\mathcal{F}'} : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \longrightarrow \mathcal{K}\] where \(\mathcal{I} = \Ker(i^\sharp)\). Namely, given local sections \(f\) of \(\mathcal{I}\) and \(s\) of \(\mathcal{F}\) we set \(c_{\mathcal{F}'}(f \otimes s) = fs'\) where \(s'\) is a local section of \(\mathcal{F}'\) lifting \(s\).

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. Assume given extensions \[0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0\] as in (08MB) and maps \(\varphi : \mathcal{F} \to \mathcal{G}\) and \(\psi : \mathcal{K} \to \mathcal{L}\).

1. If there exists an \(\mathcal{O}'\)-module map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\), then the diagram \[\xymatrix{ \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_\mathcal{O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} }\] is commutative.

2. The set of \(\mathcal{O}'\)-module maps \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\) is, if nonempty, a principal homogeneous space under \(\Hom_\mathcal{O}(\mathcal{F}, \mathcal{L})\).

Proof

Part (1) is immediate from the description of the maps. For (2), if \(\varphi'\) and \(\varphi''\) are two maps \(\mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\), then \(\varphi' - \varphi''\) factors as \[\mathcal{F}' \to \mathcal{F} \to \mathcal{L} \to \mathcal{G}'\] The map in the middle comes from a unique element of \(\Hom_\mathcal{O}(\mathcal{F}, \mathcal{L})\) by Modules on Sites, Lemma 08M3. Conversely, given an element \(\alpha\) of this group we can add the composition (as displayed above with \(\alpha\) in the middle) to \(\varphi'\). Some details omitted.

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. Assume given extensions \[0 \to \mathcal{K} \to \mathcal{F}' \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{L} \to \mathcal{G}' \to \mathcal{G} \to 0\] as in (08MB) and maps \(\varphi : \mathcal{F} \to \mathcal{G}\) and \(\psi : \mathcal{K} \to \mathcal{L}\). Assume the diagram \[\xymatrix{ \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_{\mathcal{F}'}} \ar[d]_{1 \otimes \varphi} & \mathcal{K} \ar[d]^\psi \\ \mathcal{I} \otimes_\mathcal{O} \mathcal{G} \ar[r]^-{c_{\mathcal{G}'}} & \mathcal{L} }\] is commutative. Then there exists an element \[o(\varphi, \psi) \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{L})\] whose vanishing is a necessary and sufficient condition for the existence of a map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) compatible with \(\varphi\) and \(\psi\).

Proof

We can construct explicitly an extension \[0 \to \mathcal{L} \to \mathcal{H} \to \mathcal{F} \to 0\] by taking \(\mathcal{H}\) to be the cohomology of the complex \[\mathcal{K} \xrightarrow{1, - \psi} \mathcal{F}' \oplus \mathcal{G}' \xrightarrow{\varphi, 1} \mathcal{G}\] in the middle (with obvious notation). A calculation with local sections using the assumption that the diagram of the lemma commutes shows that \(\mathcal{H}\) is annihilated by \(\mathcal{I}\). Hence \(\mathcal{H}\) defines a class in \[\Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{L}) \subset \Ext^1_{\mathcal{O}'}(\mathcal{F}, \mathcal{L})\] Finally, the class of \(\mathcal{H}\) is the difference of the pushout of the extension \(\mathcal{F}'\) via \(\psi\) and the pullback of the extension \(\mathcal{G}'\) via \(\varphi\) (calculations omitted). Thus the vanishing of the class of \(\mathcal{H}\) is equivalent to the existence of a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d]_{\psi} & \mathcal{F}' \ar[r] \ar[d]_{\varphi'} & \mathcal{F} \ar[r] \ar[d]_\varphi & 0\\ 0 \ar[r] & \mathcal{L} \ar[r] & \mathcal{G}' \ar[r] & \mathcal{G} \ar[r] & 0 }\] as desired.

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. Assume given \(\mathcal{O}\)-modules \(\mathcal{F}\), \(\mathcal{K}\) and an \(\mathcal{O}\)-linear map \(c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}\). If there exists a sequence (08MB) with \(c_{\mathcal{F}'} = c\) then the set of isomorphism classes of these extensions is principal homogeneous under \(\Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K})\).

Proof

Assume given extensions \[0 \to \mathcal{K} \to \mathcal{F}'_1 \to \mathcal{F} \to 0 \quad\text{and}\quad 0 \to \mathcal{K} \to \mathcal{F}'_2 \to \mathcal{F} \to 0\] with \(c_{\mathcal{F}'_1} = c_{\mathcal{F}'_2} = c\). Then the difference (in the extension group, see Homology, Section 010I) is an extension \[0 \to \mathcal{K} \to \mathcal{E} \to \mathcal{F} \to 0\] where \(\mathcal{E}\) is annihilated by \(\mathcal{I}\) (local computation omitted). Hence the sequence is an extension of \(\mathcal{O}\)-modules, see Modules on Sites, Lemma 08M3. Conversely, given such an extension \(\mathcal{E}\) we can add the extension \(\mathcal{E}\) to the \(\mathcal{O}'\)-extension \(\mathcal{F}'\) without affecting the map \(c_{\mathcal{F}'}\). Some details omitted.

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. Assume given \(\mathcal{O}\)-modules \(\mathcal{F}\), \(\mathcal{K}\) and an \(\mathcal{O}\)-linear map \(c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}\). Then there exists an element \[o(\mathcal{F}, \mathcal{K}, c) \in \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K})\] whose vanishing is a necessary and sufficient condition for the existence of a sequence (08MB) with \(c_{\mathcal{F}'} = c\).

Proof

We first show that if \(\mathcal{K}\) is an injective \(\mathcal{O}\)-module, then there does exist a sequence (08MB) with \(c_{\mathcal{F}'} = c\). To do this, choose a flat \(\mathcal{O}'\)-module \(\mathcal{H}'\) and a surjection \(\mathcal{H}' \to \mathcal{F}\) (Modules on Sites, Lemma 03EW). Let \(\mathcal{J} \subset \mathcal{H}'\) be the kernel. Since \(\mathcal{H}'\) is flat we have \[\mathcal{I} \otimes_{\mathcal{O}'} \mathcal{H}' = \mathcal{I}\mathcal{H}' \subset \mathcal{J} \subset \mathcal{H}'\] Observe that the map \[\mathcal{I}\mathcal{H}' = \mathcal{I} \otimes_{\mathcal{O}'} \mathcal{H}' \longrightarrow \mathcal{I} \otimes_{\mathcal{O}'} \mathcal{F} = \mathcal{I} \otimes_\mathcal{O} \mathcal{F}\] annihilates \(\mathcal{I}\mathcal{J}\). Namely, if \(f\) is a local section of \(\mathcal{I}\) and \(s\) is a local section of \(\mathcal{H}\), then \(fs\) is mapped to \(f \otimes \overline{s}\) where \(\overline{s}\) is the image of \(s\) in \(\mathcal{F}\). Thus we obtain \[\xymatrix{ \mathcal{I}\mathcal{H}'/\mathcal{I}\mathcal{J} \ar@{^{(}->}[r] \ar[d] & \mathcal{J}/\mathcal{I}\mathcal{J} \ar@{..>}[d]_\gamma \\ \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \ar[r]^-c & \mathcal{K} }\] a diagram of \(\mathcal{O}\)-modules. If \(\mathcal{K}\) is injective as an \(\mathcal{O}\)-module, then we obtain the dotted arrow. Denote \(\gamma' : \mathcal{J} \to \mathcal{K}\) the composition of \(\gamma\) with \(\mathcal{J} \to \mathcal{J}/\mathcal{I}\mathcal{J}\). A local calculation shows the pushout \[\xymatrix{ 0 \ar[r] & \mathcal{J} \ar[r] \ar[d]_{\gamma'} & \mathcal{H}' \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{F}' \ar[r] & \mathcal{F} \ar[r] & 0 }\] is a solution to the problem posed by the lemma.

General case. Choose an embedding \(\mathcal{K} \subset \mathcal{K}'\) with \(\mathcal{K}'\) an injective \(\mathcal{O}\)-module. Let \(\mathcal{Q}\) be the quotient, so that we have an exact sequence \[0 \to \mathcal{K} \to \mathcal{K}' \to \mathcal{Q} \to 0\] Denote \(c' : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}'\) be the composition. By the paragraph above there exists a sequence \[0 \to \mathcal{K}' \to \mathcal{E}' \to \mathcal{F} \to 0\] as in (08MB) with \(c_{\mathcal{E}'} = c'\). Note that \(c'\) composed with the map \(\mathcal{K}' \to \mathcal{Q}\) is zero, hence the pushout of \(\mathcal{E}'\) by \(\mathcal{K}' \to \mathcal{Q}\) is an extension \[0 \to \mathcal{Q} \to \mathcal{D}' \to \mathcal{F} \to 0\] as in (08MB) with \(c_{\mathcal{D}'} = 0\). This means exactly that \(\mathcal{D}'\) is annihilated by \(\mathcal{I}\), in other words, the \(\mathcal{D}'\) is an extension of \(\mathcal{O}\)-modules, i.e., defines an element \[o(\mathcal{F}, \mathcal{K}, c) \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{Q}) = \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K})\] (the equality holds by the long exact cohomology sequence associated to the exact sequence above and the vanishing of higher ext groups into the injective module \(\mathcal{K}'\)). If \(o(\mathcal{F}, \mathcal{K}, c) = 0\), then we can choose a splitting \(s : \mathcal{F} \to \mathcal{D}'\) and we can set \[\mathcal{F}' = \Ker(\mathcal{E}' \to \mathcal{D}'/s(\mathcal{F}))\] so that we obtain the following diagram \[\xymatrix{ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d] & \mathcal{F}' \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] & \mathcal{E}' \ar[r] & \mathcal{F} \ar[r] & 0 }\] with exact rows which shows that \(c_{\mathcal{F}'} = c\). Conversely, if \(\mathcal{F}'\) exists, then the pushout of \(\mathcal{F}'\) by the map \(\mathcal{K} \to \mathcal{K}'\) is isomorphic to \(\mathcal{E}'\) by Lemma 08ME and the vanishing of higher ext groups into the injective module \(\mathcal{K}'\). This gives a diagram as above, which implies that \(\mathcal{D}'\) is split as an extension, i.e., the class \(o(\mathcal{F}, \mathcal{K}, c)\) is zero.

Remark

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. A first order thickening \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) is said to be trivial if there exists a morphism of ringed topoi \(\pi : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) which is a left inverse to \(i\). The choice of such a morphism \(\pi\) is called a trivialization of the first order thickening. Given \(\pi\) we obtain a splitting [08MH]\[\begin{equation} \mathcal{O}' = \mathcal{O} \oplus \mathcal{I} \end{equation}\] as sheaves of algebras on \(\mathcal{C}\) by using \(\pi^\sharp\) to split the surjection \(\mathcal{O}' \to \mathcal{O}\). Conversely, such a splitting determines a morphism \(\pi\). The category of trivialized first order thickenings of \((\Sh(\mathcal{C}), \mathcal{O})\) is equivalent to the category of \(\mathcal{O}\)-modules.

Remark

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a trivial first order thickening of ringed topoi and let \(\pi : (\Sh(\mathcal{D}), \mathcal{O}') \to (\Sh(\mathcal{C}), \mathcal{O})\) be a trivialization. Then given any triple \((\mathcal{F}, \mathcal{K}, c)\) consisting of a pair of \(\mathcal{O}\)-modules and a map \(c : \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}\) we may set \[\mathcal{F}'_{c, triv} = \mathcal{F} \oplus \mathcal{K}\] and use the splitting (08MH) associated to \(\pi\) and the map \(c\) to define the \(\mathcal{O}'\)-module structure and obtain an extension (08MB). We will call \(\mathcal{F}'_{c, triv}\) the trivial extension of \(\mathcal{F}\) by \(\mathcal{K}\) corresponding to \(c\) and the trivialization \(\pi\). Given any extension \(\mathcal{F}'\) as in (08MB) we can use \(\pi^\sharp : \mathcal{O} \to \mathcal{O}'\) to think of \(\mathcal{F}'\) as an \(\mathcal{O}\)-module extension, hence a class \(\xi_{\mathcal{F}'}\) in \(\Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K})\). Lemma 08ME assures that \(\mathcal{F}' \mapsto \xi_{\mathcal{F}'}\) induces a bijection \[\left\{ \begin{matrix} \text{isomorphism classes of extensions}\\ \mathcal{F}'\text{ as in (\href{defos.html#defos-equation-extension-ringed-topoi}{08MB}) with } c = c_{\mathcal{F}'} \end{matrix} \right\} \longrightarrow \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K})\] Moreover, the trivial extension \(\mathcal{F}'_{c, triv}\) maps to the zero class.

Remark

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \((\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)\), \(i = 1, 2\) be first order thickenings with ideal sheaves \(\mathcal{I}_i\). Let \(h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2)\) be a morphism of first order thickenings of \((\Sh(\mathcal{C}), \mathcal{O})\). Picture \[\xymatrix{ & (\Sh(\mathcal{C}), \mathcal{O}) \ar[ld] \ar[rd] & \\ (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \ar[rr]^h & & (\Sh(\mathcal{D}_2), \mathcal{O}'_2) }\] Observe that \(h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1\) in particular induces an \(\mathcal{O}\)-module map \(\mathcal{I}_2 \to \mathcal{I}_1\). Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2\) be a pair consisting of an \(\mathcal{O}\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i\). Assume furthermore given a map of \(\mathcal{O}\)-modules \(\mathcal{K}_2 \to \mathcal{K}_1\) such that \[\xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 }\] is commutative. Then there is a canonical functoriality \[\left\{ \begin{matrix} \mathcal{F}'_2\text{ as in (\href{defos.html#defos-equation-extension-ringed-topoi}{08MB}) with }\\ c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2 \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \mathcal{F}'_1\text{ as in (\href{defos.html#defos-equation-extension-ringed-topoi}{08MB}) with }\\ c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1 \end{matrix} \right\}\] Namely, thinking of all sheaves \(\mathcal{O}\), \(\mathcal{O}'_i\), \(\mathcal{F}\), \(\mathcal{K}_i\), etc as sheaves on \(\mathcal{C}\), we set given \(\mathcal{F}'_2\) the sheaf \(\mathcal{F}'_1\) equal to the pushout, i.e., fitting into the following diagram of extensions \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K}_1 \ar[r] & \mathcal{F}'_1 \ar[r] & \mathcal{F} \ar[r] & 0 }\] We omit the construction of the \(\mathcal{O}'_1\)-module structure on the pushout (this uses the commutativity of the diagram involving \(c_1\) and \(c_2\)).

Remark

Let \((\Sh(\mathcal{C}), \mathcal{O})\), \((\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)\), \(\mathcal{I}_i\), and \(h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2)\) be as in Remark 08MJ. Assume that we are given trivializations \(\pi_i : (\Sh(\mathcal{D}_i), \mathcal{O}'_i) \to (\Sh(\mathcal{C}), \mathcal{O})\) such that \(\pi_1 = h \circ \pi_2\). In other words, assume \(h\) is a morphism of trivialized first order thickenings of \((\Sh(\mathcal{C}), \mathcal{O})\). Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2\) be a pair consisting of an \(\mathcal{O}\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i\). Assume furthermore given a map of \(\mathcal{O}\)-modules \(\mathcal{K}_2 \to \mathcal{K}_1\) such that \[\xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 }\] is commutative. In this situation the construction of Remark 08MI induces a commutative diagram \[\xymatrix{ \{\mathcal{F}'_2\text{ as in (\href{defos.html#defos-equation-extension-ringed-topoi}{08MB}) with } c_2 = c_{\mathcal{F}'_2}\text{ and }\mathcal{K} = \mathcal{K}_2\} \ar[d] \ar[rr] & & \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_2) \ar[d] \\ \{\mathcal{F}'_1\text{ as in (\href{defos.html#defos-equation-extension-ringed-topoi}{08MB}) with } c_1 = c_{\mathcal{F}'_1}\text{ and }\mathcal{K} = \mathcal{K}_1\} \ar[rr] & & \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1) }\] where the vertical map on the right is given by functoriality of \(\Ext\) and the map \(\mathcal{K}_2 \to \mathcal{K}_1\) and the vertical map on the left is the one from Remark 08MJ.

Remark

Let \((\Sh(\mathcal{C}), \mathcal{O})\), \((\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_i), \mathcal{O}'_i)\), \(\mathcal{I}_i\), and \(h : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2)\) be as in Remark 08MJ. Observe that \(h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1\) in particular induces an \(\mathcal{O}\)-module map \(\mathcal{I}_2 \to \mathcal{I}_1\). Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2\) be a pair consisting of an \(\mathcal{O}\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i\). Assume furthermore given a map of \(\mathcal{O}\)-modules \(\mathcal{K}_2 \to \mathcal{K}_1\) such that \[\xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 }\] is commutative. Then we claim the map \[\Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_2) \longrightarrow \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_1)\] sends \(o(\mathcal{F}, \mathcal{K}_2, c_2)\) to \(o(\mathcal{F}, \mathcal{K}_1, c_1)\).

To prove this claim choose an embedding \(j_2 : \mathcal{K}_2 \to \mathcal{K}_2'\) where \(\mathcal{K}_2'\) is an injective \(\mathcal{O}\)-module. As in the proof of Lemma 08MF we can choose an extension of \(\mathcal{O}_2\)-modules \[0 \to \mathcal{K}_2' \to \mathcal{E}_2 \to \mathcal{F} \to 0\] such that \(c_{\mathcal{E}_2} = j_2 \circ c_2\). The proof of Lemma 08MF constructs \(o(\mathcal{F}, \mathcal{K}_2, c_2)\) as the Yoneda extension class (in the sense of Derived Categories, Section 06XP) of the exact sequence of \(\mathcal{O}\)-modules \[0 \to \mathcal{K}_2 \to \mathcal{K}_2' \to \mathcal{E}_2/\mathcal{K}_2 \to \mathcal{F} \to 0\] Let \(\mathcal{K}_1'\) be the cokernel of \(\mathcal{K}_2 \to \mathcal{K}_1 \oplus \mathcal{K}_2'\). There is an injection \(j_1 : \mathcal{K}_1 \to \mathcal{K}_1'\) and a map \(\mathcal{K}_2' \to \mathcal{K}_1'\) forming a commutative square. We form the pushout: \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2' \ar[r] \ar[d] & \mathcal{E}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}_1' \ar[r] & \mathcal{E}_1 \ar[r] & \mathcal{F} \ar[r] & 0 }\] There is a canonical \(\mathcal{O}_1\)-module structure on \(\mathcal{E}_1\) and for this structure we have \(c_{\mathcal{E}_1} = j_1 \circ c_1\) (this uses the commutativity of the diagram involving \(c_1\) and \(c_2\) above). The procedure of Lemma 08MF tells us that \(o(\mathcal{F}, \mathcal{K}_1, c_1)\) is the Yoneda extension class of the exact sequence of \(\mathcal{O}\)-modules \[0 \to \mathcal{K}_1 \to \mathcal{K}_1' \to \mathcal{E}_1/\mathcal{K}_1 \to \mathcal{F} \to 0\] Since we have maps of exact sequences \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[d] \ar[r] & \mathcal{K}_2' \ar[d] \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}_2 \ar[r] & \mathcal{K}_2' \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] & \mathcal{F} \ar[r] & 0 }\] we conclude that the claim is true.

Remark

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. We define a sequence of morphisms of first order thickenings \[(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)\] of \((\Sh(\mathcal{C}), \mathcal{O})\) to be a complex if the corresponding maps between the ideal sheaves \(\mathcal{I}_i\) give a complex of \(\mathcal{O}\)-modules \(\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1\) (i.e., the composition is zero). In this case the composition \((\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)\) factors through \((\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)\), i.e., the first order thickening \((\Sh(\mathcal{D}_1), \mathcal{O}'_1)\) of \((\Sh(\mathcal{C}), \mathcal{O})\) is trivial and comes with a canonical trivialization \(\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})\).

We say a sequence of morphisms of first order thickenings \[(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)\] of \((\Sh(\mathcal{C}), \mathcal{O})\) is a short exact sequence if the corresponding maps between ideal sheaves is a short exact sequence \[0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0\] of \(\mathcal{O}\)-modules.

Remark

Let \((\Sh(\mathcal{C}), \mathcal{O})\) be a ringed topos. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Let \[(\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{D}_2), \mathcal{O}'_2) \to (\Sh(\mathcal{D}_3), \mathcal{O}'_3)\] be a complex first order thickenings of \((\Sh(\mathcal{C}), \mathcal{O})\), see Remark 08ML. Let \((\mathcal{K}_i, c_i)\), \(i = 1, 2, 3\) be pairs consisting of an \(\mathcal{O}\)-module \(\mathcal{K}_i\) and a map \(c_i : \mathcal{I}_i \otimes_\mathcal{O} \mathcal{F} \to \mathcal{K}_i\). Assume given a short exact sequence of \(\mathcal{O}\)-modules \[0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0\] such that \[\vcenter{ \xymatrix{ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } } \quad\text{and}\quad \vcenter{ \xymatrix{ \mathcal{I}_3 \otimes_\mathcal{O} \mathcal{F} \ar[r]_-{c_3} \ar[d] & \mathcal{K}_3 \ar[d] \\ \mathcal{I}_2 \otimes_\mathcal{O} \mathcal{F} \ar[r]^-{c_2} & \mathcal{K}_2 } }\] are commutative. Finally, assume given an extension \[0 \to \mathcal{K}_2 \to \mathcal{F}'_2 \to \mathcal{F} \to 0\] as in (08MB) with \(\mathcal{K} = \mathcal{K}_2\) of \(\mathcal{O}'_2\)-modules with \(c_{\mathcal{F}'_2} = c_2\). In this situation we can apply the functoriality of Remark 08MJ to obtain an extension \(\mathcal{F}'_1\) of \(\mathcal{O}'_1\)-modules (we’ll describe \(\mathcal{F}'_1\) in this special case below). By Remark 08MI using the canonical splitting \(\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})\) of Remark 08ML we obtain \(\xi_{\mathcal{F}'_1} \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1)\). Finally, we have the obstruction \[o(\mathcal{F}, \mathcal{K}_3, c_3) \in \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_3)\] see Lemma 08MF. In this situation we claim that the canonical map \[\partial : \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{K}_1) \longrightarrow \Ext^2_\mathcal{O}(\mathcal{F}, \mathcal{K}_3)\] coming from the short exact sequence \(0 \to \mathcal{K}_3 \to \mathcal{K}_2 \to \mathcal{K}_1 \to 0\) sends \(\xi_{\mathcal{F}'_1}\) to the obstruction class \(o(\mathcal{F}, \mathcal{K}_3, c_3)\).

To prove this claim choose an embedding \(j : \mathcal{K}_3 \to \mathcal{K}\) where \(\mathcal{K}\) is an injective \(\mathcal{O}\)-module. We can lift \(j\) to a map \(j' : \mathcal{K}_2 \to \mathcal{K}\). Set \(\mathcal{E}'_2 = j'_*\mathcal{F}'_2\) equal to the pushout of \(\mathcal{F}'_2\) by \(j'\) so that \(c_{\mathcal{E}'_2} = j' \circ c_2\). Picture: \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[r] \ar[d]_{j'} & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] & \mathcal{E}'_2 \ar[r] & \mathcal{F} \ar[r] & 0 }\] Set \(\mathcal{E}'_3 = \mathcal{E}'_2\) but viewed as an \(\mathcal{O}'_3\)-module via \(\mathcal{O}'_3 \to \mathcal{O}'_2\). Then \(c_{\mathcal{E}'_3} = j \circ c_3\). The proof of Lemma 08MF constructs \(o(\mathcal{F}, \mathcal{K}_3, c_3)\) as the boundary of the class of the extension of \(\mathcal{O}\)-modules \[0 \to \mathcal{K}/\mathcal{K}_3 \to \mathcal{E}'_3/\mathcal{K}_3 \to \mathcal{F} \to 0\] On the other hand, note that \(\mathcal{F}'_1 = \mathcal{F}'_2/\mathcal{K}_3\) hence the class \(\xi_{\mathcal{F}'_1}\) is the class of the extension \[0 \to \mathcal{K}_2/\mathcal{K}_3 \to \mathcal{F}'_2/\mathcal{K}_3 \to \mathcal{F} \to 0\] seen as a sequence of \(\mathcal{O}\)-modules using \(\pi^\sharp\) where \(\pi : (\Sh(\mathcal{D}_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})\) is the canonical splitting. Thus finally, the claim follows from the fact that we have a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{K}_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F}'_2/\mathcal{K}_3 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}/\mathcal{K}_3 \ar[r] & \mathcal{E}'_3/\mathcal{K}_3 \ar[r] & \mathcal{F} \ar[r] & 0 }\] which is \(\mathcal{O}\)-linear (with the \(\mathcal{O}\)-module structures given above).

Infinitesimal deformations of modules on ringed topoi

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. We freely use the notation introduced in Section 08M6. Let \(\mathcal{F}'\) be an \(\mathcal{O}'\)-module and set \(\mathcal{F} = i^*\mathcal{F}'\). In this situation we have a short exact sequence \[0 \to \mathcal{I}\mathcal{F}' \to \mathcal{F}' \to \mathcal{F} \to 0\] of \(\mathcal{O}'\)-modules. Since \(\mathcal{I}^2 = 0\) the \(\mathcal{O}'\)-module structure on \(\mathcal{I}\mathcal{F}'\) comes from a unique \(\mathcal{O}\)-module structure. Thus the sequence above is an extension as in (08MB). As a special case, if \(\mathcal{F}' = \mathcal{O}'\) we have \(i^*\mathcal{O}' = \mathcal{O}\) and \(\mathcal{I}\mathcal{O}' = \mathcal{I}\) and we recover the sequence of structure sheaves \[0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0\]

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}'\)-modules. Set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}\)-linear map. The set of lifts of \(\varphi\) to an \(\mathcal{O}'\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) is, if nonempty, a principal homogeneous space under \(\Hom_\mathcal{O}(\mathcal{F}, \mathcal{I}\mathcal{G}')\).

Proof

This is a special case of Lemma 08MC but we also give a direct proof. We have short exact sequences of modules \[0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0 \quad\text{and}\quad 0 \to \mathcal{I}\mathcal{G}' \to \mathcal{G}' \to \mathcal{G} \to 0\] and similarly for \(\mathcal{F}'\). Since \(\mathcal{I}\) has square zero the \(\mathcal{O}'\)-module structure on \(\mathcal{I}\) and \(\mathcal{I}\mathcal{G}'\) comes from a unique \(\mathcal{O}\)-module structure. It follows that \[\Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \Hom_\mathcal{O}(\mathcal{F}, \mathcal{I}\mathcal{G}') \quad\text{and}\quad \Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) = \Hom_\mathcal{O}(\mathcal{F}, \mathcal{G})\] The lemma now follows from the exact sequence \[0 \to \Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') \to \Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G}') \to \Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G})\] see Homology, Lemma 05AA.

Lemma

Let \((f, f')\) be a morphism of first order thickenings of ringed topoi as in Situation 08M7. Let \(\mathcal{F}'\) be an \(\mathcal{O}'\)-module and set \(\mathcal{F} = i^*\mathcal{F}'\). Assume that \(\mathcal{F}\) is flat over \(\mathcal{O}_\mathcal{B}\) and that \((f, f')\) is a strict morphism of thickenings (Definition 08M9). Then the following are equivalent

1. \(\mathcal{F}'\) is flat over \(\mathcal{O}_{\mathcal{B}'}\), and

2. the canonical map \(f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I}\mathcal{F}'\) is an isomorphism.

Moreover, in this case the maps \[f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I}\mathcal{F}'\] are isomorphisms.

Proof

The map \(f^*\mathcal{J} \to \mathcal{I}\) is surjective as \((f, f')\) is a strict morphism of thickenings. Hence the final statement is a consequence of (2).

Proof of the equivalence of (1) and (2). By definition flatness over \(\mathcal{O}_\mathcal{B}\) means flatness over \(f^{-1}\mathcal{O}_\mathcal{B}\). Similarly for flatness over \(f^{-1}\mathcal{O}_{\mathcal{B}'}\). Note that the strictness of \((f, f')\) and the assumption that \(\mathcal{F} = i^*\mathcal{F}'\) imply that \[\mathcal{F} = \mathcal{F}'/(f^{-1}\mathcal{J})\mathcal{F}'\] as sheaves on \(\mathcal{C}\). Moreover, observe that \(f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} = f^{-1}\mathcal{J} \otimes_{f^{-1}\mathcal{O}_\mathcal{B}} \mathcal{F}\). Hence the equivalence of (1) and (2) follows from Modules on Sites, Lemma 08M4.

Lemma

Let \((f, f')\) be a morphism of first order thickenings of ringed topoi as in Situation 08M7. Let \(\mathcal{F}'\) be an \(\mathcal{O}'\)-module and set \(\mathcal{F} = i^*\mathcal{F}'\). Assume that \(\mathcal{F}'\) is flat over \(\mathcal{O}_{\mathcal{B}'}\) and that \((f, f')\) is a strict morphism of thickenings. Then the following are equivalent

1. \(\mathcal{F}'\) is an \(\mathcal{O}'\)-module of finite presentation, and

2. \(\mathcal{F}\) is an \(\mathcal{O}\)-module of finite presentation.

Proof

The implication (1) \(\Rightarrow\) (2) follows from Modules on Sites, Lemma 03DO. For the converse, assume \(\mathcal{F}\) of finite presentation. We may and do assume that \(\mathcal{C} = \mathcal{C}'\). By Lemma 08MQ we have a short exact sequence \[0 \to \mathcal{I} \otimes_{\mathcal{O}_X} \mathcal{F} \to \mathcal{F}' \to \mathcal{F} \to 0\] Let \(U\) be an object of \(\mathcal{C}\) such that \(\mathcal{F}|_U\) has a presentation \[\mathcal{O}_U^{\oplus m} \to \mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U \to 0\] After replacing \(U\) by the members of a covering we may assume the map \(\mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U\) lifts to a map \((\mathcal{O}'_U)^{\oplus n} \to \mathcal{F}'|_U\). The induced map \(\mathcal{I}^{\oplus n} \to \mathcal{I} \otimes \mathcal{F}\) is surjective by right exactness of \(\otimes\). Thus after replacing \(U\) by the members of a covering we can find a lift \((\mathcal{O}'|_U)^{\oplus m} \to (\mathcal{O}'|_U)^{\oplus n}\) of the given map \(\mathcal{O}_U^{\oplus m} \to \mathcal{O}_U^{\oplus n}\) such that \[(\mathcal{O}'_U)^{\oplus m} \to (\mathcal{O}'_U)^{\oplus n} \to \mathcal{F}'|_U \to 0\] is a complex. Using right exactness of \(\otimes\) once more it is seen that this complex is exact.

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08M7. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}'\)-modules and set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}\)-linear map. Assume that \(\mathcal{G}'\) is flat over \(\mathcal{O}_{\mathcal{B}'}\) and that \((f, f')\) is a strict morphism of thickenings. The set of lifts of \(\varphi\) to an \(\mathcal{O}'\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\) is, if nonempty, a principal homogeneous space under \[\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J})\]

Proof

Combine Lemmas 08MP and 08MQ.

Lemma

Let \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{D}), \mathcal{O}')\) be a first order thickening of ringed topoi. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}'\)-modules and set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}\)-linear map. There exists an element \[o(\varphi) \in \Ext^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')\] whose vanishing is a necessary and sufficient condition for the existence of a lift of \(\varphi\) to an \(\mathcal{O}'\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\).

Proof

It is clear from the proof of Lemma 08MP that the vanishing of the boundary of \(\varphi\) via the map \[\Hom_\mathcal{O}(\mathcal{F}, \mathcal{G}) = \Hom_{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) \longrightarrow \Ext^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}')\] is a necessary and sufficient condition for the existence of a lift. We conclude as \[\Ext^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \Ext^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')\] the adjointness of \(i_* = Ri_*\) and \(Li^*\) on the derived category (Cohomology on Sites, Lemma 07A6).

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08M7. Let \(\mathcal{F}'\), \(\mathcal{G}'\) be \(\mathcal{O}'\)-modules and set \(\mathcal{F} = i^*\mathcal{F}'\) and \(\mathcal{G} = i^*\mathcal{G}'\). Let \(\varphi : \mathcal{F} \to \mathcal{G}\) be an \(\mathcal{O}\)-linear map. Assume that \(\mathcal{F}'\) and \(\mathcal{G}'\) are flat over \(\mathcal{O}_{\mathcal{B}'}\) and that \((f, f')\) is a strict morphism of thickenings. There exists an element \[o(\varphi) \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J})\] whose vanishing is a necessary and sufficient condition for the existence of a lift of \(\varphi\) to an \(\mathcal{O}'\)-linear map \(\varphi' : \mathcal{F}' \to \mathcal{G}'\).

Proof

This follows from Lemma 08MS as we claim that under the assumptions of the lemma we have \[\Ext^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') = \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J})\] Namely, we have \(\mathcal{I}\mathcal{G}' = \mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J}\) by Lemma 08MQ. On the other hand, observe that \[H^{-1}(Li^*\mathcal{F}') = \text{Tor}_1^{\mathcal{O}'}(\mathcal{F}', \mathcal{O})\] (local computation omitted). Using the short exact sequence \[0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0\] we see that this \(\text{Tor}_1\) is computed by the kernel of the map \(\mathcal{I} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I}\mathcal{F}'\) which is zero by the final assertion of Lemma 08MQ. Thus \(\tau_{\geq -1}Li^*\mathcal{F}' = \mathcal{F}\). On the other hand, we have \[\Ext^1_\mathcal{O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') = \Ext^1_\mathcal{O}(\tau_{\geq -1}Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')\] by the dual of Derived Categories, Lemma 05TC.

Proof

We can apply Lemma 08MD as follows. Note that \(\mathcal{K} = \mathcal{I} \otimes_\mathcal{O} \mathcal{F}\) and \(\mathcal{L} = \mathcal{I} \otimes_\mathcal{O} \mathcal{G}\) by Lemma 08MQ, that \(c_{\mathcal{F}'} = 1 \otimes 1\) and \(c_{\mathcal{G}'} = 1 \otimes 1\) and taking \(\psi = 1 \otimes \varphi\) the diagram of the lemma commutes. Thus \(o(\varphi) = o(\varphi, 1 \otimes \varphi)\) works.

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08M7. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Assume \((f, f')\) is a strict morphism of thickenings and \(\mathcal{F}\) flat over \(\mathcal{O}_\mathcal{B}\). If there exists a pair \((\mathcal{F}', \alpha)\) consisting of an \(\mathcal{O}'\)-module \(\mathcal{F}'\) flat over \(\mathcal{O}_{\mathcal{B}'}\) and an isomorphism \(\alpha : i^*\mathcal{F}' \to \mathcal{F}\), then the set of isomorphism classes of such pairs is principal homogeneous under \(\Ext^1_\mathcal{O}( \mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F})\).

Proof

If we assume there exists one such module, then the canonical map \[f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I} \otimes_\mathcal{O} \mathcal{F}\] is an isomorphism by Lemma 08MQ. Apply Lemma 08ME with \(\mathcal{K} = \mathcal{I} \otimes_\mathcal{O} \mathcal{F}\) and \(c = 1\). By Lemma 08MQ the corresponding extensions \(\mathcal{F}'\) are all flat over \(\mathcal{O}_{\mathcal{B}'}\).

Lemma

Let \((f, f')\) be a morphism of first order thickenings as in Situation 08M7. Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Assume \((f, f')\) is a strict morphism of thickenings and \(\mathcal{F}\) flat over \(\mathcal{O}_\mathcal{B}\). There exists an \(\mathcal{O}'\)-module \(\mathcal{F}'\) flat over \(\mathcal{O}_{\mathcal{B}'}\) with \(i^*\mathcal{F}' \cong \mathcal{F}\), if and only if

1. the canonical map \(f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F} \to \mathcal{I} \otimes_\mathcal{O} \mathcal{F}\) is an isomorphism, and

2. the class \(o(\mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F}, 1) \in \Ext^2_\mathcal{O}( \mathcal{F}, \mathcal{I} \otimes_\mathcal{O} \mathcal{F})\) of Lemma 08MF is zero.

Proof

This follows immediately from the characterization of \(\mathcal{O}'\)-modules flat over \(\mathcal{O}_{\mathcal{B}'}\) of Lemma 08MQ and Lemma 08MF.

Application to flat modules on flat thickenings of ringed topoi

Consider a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}), \mathcal{O}) \ar[r]_i \ar[d]_f & (\Sh(\mathcal{D}), \mathcal{O}') \ar[d]^{f'} \\ (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \ar[r]^t & (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'}) }\] of ringed topoi whose horizontal arrows are first order thickenings as in Situation 08M7. Set \(\mathcal{I} = \Ker(i^\sharp) \subset \mathcal{O}'\) and \(\mathcal{J} = \Ker(t^\sharp) \subset \mathcal{O}_{\mathcal{B}'}\). Let \(\mathcal{F}\) be an \(\mathcal{O}\)-module. Assume that

1. \((f, f')\) is a strict morphism of thickenings,

2. \(f'\) is flat, and

3. \(\mathcal{F}\) is flat over \(\mathcal{O}_\mathcal{B}\).

Note that (1) \(+\) (2) imply that \(\mathcal{I} = f^*\mathcal{J}\) (apply Lemma 08MQ to \(\mathcal{O}'\)). The theory of the preceding section is especially nice under these assumptions. We summarize the results already obtained in the following lemma.

Lemma

In the situation above.

1. There exists an \(\mathcal{O}'\)-module \(\mathcal{F}'\) flat over \(\mathcal{O}_{\mathcal{B}'}\) with \(i^*\mathcal{F}' \cong \mathcal{F}\), if and only if the class \(o(\mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F}, 1) \in \Ext^2_\mathcal{O}( \mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})\) of Lemma 08MF is zero.

2. If such a module exists, then the set of isomorphism classes of lifts is principal homogeneous under \(\Ext^1_\mathcal{O}( \mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})\).

3. Given a lift \(\mathcal{F}'\), the set of automorphisms of \(\mathcal{F}'\) which pull back to \(\text{id}_\mathcal{F}\) is canonically isomorphic to \(\Ext^0_\mathcal{O}( \mathcal{F}, f^*\mathcal{J} \otimes_\mathcal{O} \mathcal{F})\).

Proof

Part (1) follows from Lemma 08MV as we have seen above that \(\mathcal{I} = f^*\mathcal{J}\). Part (2) follows from Lemma 08MU. Part (3) follows from Lemma 08MR.

Situation

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\) be a morphism of ringed topoi. Consider a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \ar[r]_h \ar[d]_{f'_1} & (\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \ar[d]_{f'_2} \\ (\Sh(\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) \ar[r] & (\Sh(\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) }\] where \(h\) is a morphism of first order thickenings of \((\Sh(\mathcal{C}), \mathcal{O})\), the lower horizontal arrow is a morphism of first order thickenings of \((\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\), each \(f'_i\) restricts to \(f\), both pairs \((f, f_i')\) are strict morphisms of thickenings, and both \(f'_i\) are flat. Finally, let \(\mathcal{F}\) be an \(\mathcal{O}\)-module flat over \(\mathcal{O}_\mathcal{B}\).

Lemma

In Situation 0CYD the obstruction class \(o(\mathcal{F}, f^*\mathcal{J}_2 \otimes_\mathcal{O} \mathcal{F}, 1)\) maps to the obstruction class \(o(\mathcal{F}, f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F}, 1)\) under the canonical map \[\Ext^2_\mathcal{O}( \mathcal{F}, f^*\mathcal{J}_2 \otimes_\mathcal{O} \mathcal{F}) \to \Ext^2_\mathcal{O}( \mathcal{F}, f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F})\]

Proof

Follows from Remark 0CYC.

Situation

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\) be a morphism of ringed topoi. Consider a commutative diagram \[\xymatrix{ (\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \ar[r]_h \ar[d]_{f'_1} & (\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \ar[r] \ar[d]_{f'_2} & (\Sh(\mathcal{C}'_3), \mathcal{O}'_3) \ar[d]_{f'_3} \\ (\Sh(\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) \ar[r] & (\Sh(\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[r] & (\Sh(\mathcal{B}'_3), \mathcal{O}_{\mathcal{B}'_3}) }\] where (a) the top row is a short exact sequence of first order thickenings of \((\Sh(\mathcal{C}), \mathcal{O})\), (b) the lower row is a short exact sequence of first order thickenings of \((\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\), (c) each \(f'_i\) restricts to \(f\), (d) each pair \((f, f_i')\) is a strict morphism of thickenings, and (e) each \(f'_i\) is flat. Finally, let \(\mathcal{F}'_2\) be an \(\mathcal{O}'_2\)-module flat over \(\mathcal{O}_{\mathcal{B}'_2}\) and set \(\mathcal{F} = \mathcal{F}'_2 \otimes \mathcal{O}\). Let \(\pi : (\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \to (\Sh(\mathcal{C}), \mathcal{O})\) be the canonical splitting (Remark 08ML).

Lemma

In Situation 08VX the modules \(\pi^*\mathcal{F}\) and \(h^*\mathcal{F}'_2\) are \(\mathcal{O}'_1\)-modules flat over \(\mathcal{O}_{\mathcal{B}'_1}\) restricting to \(\mathcal{F}\) on \((\Sh(\mathcal{C}), \mathcal{O})\). Their difference (Lemma 08VW) is an element \(\theta\) of \(\Ext^1_\mathcal{O}(\mathcal{F}, f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F})\) whose boundary in \(\Ext^2_\mathcal{O}(\mathcal{F}, f^*\mathcal{J}_3 \otimes_\mathcal{O} \mathcal{F})\) equals the obstruction (Lemma 08VW) to lifting \(\mathcal{F}\) to an \(\mathcal{O}'_3\)-module flat over \(\mathcal{O}_{\mathcal{B}'_3}\).

Proof

Note that both \(\pi^*\mathcal{F}\) and \(h^*\mathcal{F}'_2\) restrict to \(\mathcal{F}\) on \((\Sh(\mathcal{C}), \mathcal{O})\) and that the kernels of \(\pi^*\mathcal{F} \to \mathcal{F}\) and \(h^*\mathcal{F}'_2 \to \mathcal{F}\) are given by \(f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F}\). Hence flatness by Lemma 08MQ. Taking the boundary makes sense as the sequence of modules \[0 \to f^*\mathcal{J}_3 \otimes_\mathcal{O} \mathcal{F} \to f^*\mathcal{J}_2 \otimes_\mathcal{O} \mathcal{F} \to f^*\mathcal{J}_1 \otimes_\mathcal{O} \mathcal{F} \to 0\] is short exact due to the assumptions in Situation 08VX and the fact that \(\mathcal{F}\) is flat over \(\mathcal{O}_\mathcal{B}\). The statement on the obstruction class is a direct translation of the result of Remark 08MM to this particular situation.

Deformations of ringed topoi and the naive cotangent complex

In this section we use the naive cotangent complex to do a little bit of deformation theory. We start with a first order thickening \(t : (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \to (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})\) of ringed topoi. We denote \(\mathcal{J} = \Ker(t^\sharp)\) and we identify the underlying topoi of \(\mathcal{B}\) and \(\mathcal{B}'\). Moreover we assume given a morphism of ringed topoi \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\), an \(\mathcal{O}\)-module \(\mathcal{G}\), and a map \(f^{-1}\mathcal{J} \to \mathcal{G}\) of sheaves of \(f^{-1}\mathcal{O}_\mathcal{B}\)-modules. In this section we ask ourselves whether we can find the question mark fitting into the following diagram [08UF]\[\begin{equation} \vcenter{ \xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O} \ar[r] & 0 \\ 0 \ar[r] & f^{-1}\mathcal{J} \ar[u]^c \ar[r] & f^{-1}\mathcal{O}_{\mathcal{B}'} \ar[u] \ar[r] & f^{-1}\mathcal{O}_\mathcal{B} \ar[u] \ar[r] & 0 } } \end{equation}\] and moreover how unique the solution is (if it exists). More precisely, we look for a first order thickening \(i : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')\) and a morphism of thickenings \((f, f')\) as in (08M8) where \(\Ker(i^\sharp)\) is identified with \(\mathcal{G}\) such that \((f')^\sharp\) induces the given map \(c\). We will say \((\Sh(\mathcal{C}'), \mathcal{O}')\) is a solution to (08UF).

Lemma

Assume given a commutative diagram of morphisms ringed topoi [08UH]\[\begin{equation} \vcenter{ \xymatrix{ & (\Sh(\mathcal{C}_2), \mathcal{O}_2) \ar[r]_{i_2} \ar[d]_{f_2} \ar[ddl]_g & (\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \ar[d]^{f'_2} \\ & (\Sh(\mathcal{B}_2), \mathcal{O}_{\mathcal{B}_2}) \ar[r]^{t_2} \ar[ddl]|\hole & (\Sh(\mathcal{B}'_2), \mathcal{O}_{\mathcal{B}'_2}) \ar[ddl] \\ (\Sh(\mathcal{C}_1), \mathcal{O}_1) \ar[r]_{i_1} \ar[d]_{f_1} & (\Sh(\mathcal{C}'_1), \mathcal{O}'_1) \ar[d]^{f'_1} \\ (\Sh(\mathcal{B}_1), \mathcal{O}_{\mathcal{B}_1}) \ar[r]^{t_1} & (\Sh(\mathcal{B}'_1), \mathcal{O}_{\mathcal{B}'_1}) } } \end{equation}\] whose horizontal arrows are first order thickenings. Set \(\mathcal{G}_j = \Ker(i_j^\sharp)\) and assume given a map of \(g^{-1}\mathcal{O}_1\)-modules \(\nu : g^{-1}\mathcal{G}_1 \to \mathcal{G}_2\) giving rise to the commutative diagram [08UI]\[\begin{equation} \vcenter{ \xymatrix{ & 0 \ar[r] & \mathcal{G}_2 \ar[r] & \mathcal{O}'_2 \ar[r] & \mathcal{O}_2 \ar[r] & 0 \\ & 0 \ar[r]|\hole & f_2^{-1}\mathcal{J}_2 \ar[u]_{c_2} \ar[r] & f_2^{-1}\mathcal{O}_{\mathcal{B}'_2} \ar[u] \ar[r]|\hole & f_2^{-1}\mathcal{O}_{\mathcal{B}_2} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G}_1 \ar[ruu] \ar[r] & \mathcal{O}'_1 \ar[r] & \mathcal{O}_1 \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & f_1^{-1}\mathcal{J}_1 \ar[ruu]|\hole \ar[u]^{c_1} \ar[r] & f_1^{-1}\mathcal{O}_{\mathcal{B}'_1} \ar[ruu]|\hole \ar[u] \ar[r] & f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \ar[ruu]|\hole \ar[u] \ar[r] & 0 } } \end{equation}\] with front and back solutions to (08UF). (The north-north-west arrows are maps on \(\mathcal{C}_2\) after applying \(g^{-1}\) to the source.)

1. There exist a canonical element in \(\Ext^1_{\mathcal{O}_2}( Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)\) whose vanishing is a necessary and sufficient condition for the existence of a morphism of ringed topoi \((\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to (\Sh(\mathcal{C}'_1), \mathcal{O}'_1)\) fitting into (08UH) compatibly with \(\nu\).

2. If there exists a morphism \((\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to (\Sh(\mathcal{C}'_1), \mathcal{O}'_1)\) fitting into (08UH) compatibly with \(\nu\) the set of all such morphisms is a principal homogeneous space under \[\Hom_{\mathcal{O}_1}( \Omega_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, g_*\mathcal{G}_2) = \Hom_{\mathcal{O}_2}( g^*\Omega_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) = \Ext^0_{\mathcal{O}_2}( Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2).\]

Proof

The proof of this lemma is identical to the proof of Lemma 08U8. We urge the reader to read that proof instead of this one. We will identify the underlying topoi for every thickening in sight (we have already used this convention in the statement). The equalities in the last statement of the lemma are immediate from the definitions. Thus we will work with the groups \(\Ext^k_{\mathcal{O}_2}( Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)\), \(k = 0, 1\) in the rest of the proof. We first argue that we can reduce to the case where the underlying topos of all ringed topoi in the lemma is the same.

To do this, observe that \(g^{-1}\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}\) is equal to the naive cotangent complex of the homomorphism of sheaves of rings \(g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}_1} \to g^{-1}\mathcal{O}_1\), see Modules on Sites, Lemma 08TQ. Moreover, the degree \(0\) term of \(\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}\) is a flat \(\mathcal{O}_1\)-module, hence the canonical map \[Lg^*\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}} \longrightarrow g^{-1}\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}} \otimes_{g^{-1}\mathcal{O}_1} \mathcal{O}_2\] induces an isomorphism on cohomology sheaves in degrees \(0\) and \(-1\). Thus we may replace the Ext groups of the lemma with \[\Ext^k_{g^{-1}\mathcal{O}_1}( g^{-1}\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2) = \Ext^k_{g^{-1}\mathcal{O}_1}( \NL_{g^{-1}\mathcal{O}_1/g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)\] The set of morphism of ringed topoi \((\Sh(\mathcal{C}'_2), \mathcal{O}'_2) \to (\Sh(\mathcal{C}'_1), \mathcal{O}'_1)\) fitting into (08UH) compatibly with \(\nu\) is in one-to-one bijection with the set of homomorphisms of \(g^{-1}f_1^{-1}\mathcal{O}_{\mathcal{B}'_1}\)-algebras \(g^{-1}\mathcal{O}'_1 \to \mathcal{O}'_2\) which are compatible with \(f^\sharp\) and \(\nu\). In this way we see that we may assume we have a diagram (08UI) of sheaves on a site \(\mathcal{C}\) (with \(f_1 = f_2 = \text{id}\) on underlying topoi) and we are looking to find a homomorphism of sheaves of rings \(\mathcal{O}'_1 \to \mathcal{O}'_2\) fitting into it.

In the rest of the proof of the lemma we assume all underlying topological spaces are the same, i.e., we have a diagram (08UI) of sheaves on a site \(\mathcal{C}\) (with \(f_1 = f_2 = \text{id}\) on underlying topoi) and we are looking for homomorphisms of sheaves of rings \(\mathcal{O}'_1 \to \mathcal{O}'_2\) fitting into it. As ext groups we will use \(\Ext^k_{\mathcal{O}_1}( \NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}, \mathcal{G}_2)\), \(k = 0, 1\).

Step 1. Construction of the obstruction class. Consider the sheaf of sets \[\mathcal{E} = \mathcal{O}'_1 \times_{\mathcal{O}_2} \mathcal{O}'_2\] This comes with a surjective map \(\alpha : \mathcal{E} \to \mathcal{O}_1\) and hence we can use \(\NL(\alpha)\) instead of \(\NL_{\mathcal{O}_1/\mathcal{O}_{\mathcal{B}_1}}\), see Modules on Sites, Lemma 08TY. Set \[\mathcal{I}' = \Ker(\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}_1) \quad\text{and}\quad \mathcal{I} = \Ker(\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}] \to \mathcal{O}_1)\] There is a surjection \(\mathcal{I}' \to \mathcal{I}\) whose kernel is \(\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]\). We obtain two homomorphisms of \(\mathcal{O}_{\mathcal{B}'_2}\)-algebras \[a : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}'_1 \quad\text{and}\quad b : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \to \mathcal{O}'_2\] which induce maps \(a|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_1\) and \(b|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}_2\). Both \(a\) and \(b\) annihilate \((\mathcal{I}')^2\). Moreover \(a\) and \(b\) agree on \(\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]\) as maps into \(\mathcal{G}_2\) because the left hand square of (08UI) is commutative. Thus the difference \(b|_{\mathcal{I}'} - \nu \circ a|_{\mathcal{I}'}\) induces a well defined \(\mathcal{O}_1\)-linear map \[\xi : \mathcal{I}/\mathcal{I}^2 \longrightarrow \mathcal{G}_2\] which sends the class of a local section \(f\) of \(\mathcal{I}\) to \(a(f') - \nu(b(f'))\) where \(f'\) is a lift of \(f\) to a local section of \(\mathcal{I}'\). We let \([\xi] \in \Ext^1_{\mathcal{O}_1}(\NL(\alpha), \mathcal{G}_2)\) be the image (see below).

Step 2. Vanishing of \([\xi]\) is necessary. Let us write \(\Omega = \Omega_{\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}]/\mathcal{O}_{\mathcal{B}_1}} \otimes_{\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}]} \mathcal{O}_1\). Observe that \(\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega)\) fits into a distinguished triangle \[\Omega[0] \to \NL(\alpha) \to \mathcal{I}/\mathcal{I}^2[1] \to \Omega[1]\] Thus we see that \([\xi]\) is zero if and only if \(\xi\) is a composition \(\mathcal{I}/\mathcal{I}^2 \to \Omega \to \mathcal{G}_2\) for some map \(\Omega \to \mathcal{G}_2\). Suppose there exists a homomorphisms of sheaves of rings \(\varphi : \mathcal{O}'_1 \to \mathcal{O}'_2\) fitting into (08UI). In this case consider the map \(\mathcal{O}'_1[\mathcal{E}] \to \mathcal{G}_2\), \(f' \mapsto b(f') - \varphi(a(f'))\). A calculation shows this annihilates \(\mathcal{J}_1\mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}]\) and induces a derivation \(\mathcal{O}_{\mathcal{B}_1}[\mathcal{E}] \to \mathcal{G}_2\). The resulting linear map \(\Omega \to \mathcal{G}_2\) witnesses the fact that \([\xi] = 0\) in this case.

Step 3. Vanishing of \([\xi]\) is sufficient. Let \(\theta : \Omega \to \mathcal{G}_2\) be a \(\mathcal{O}_1\)-linear map such that \(\xi\) is equal to \(\theta \circ (\mathcal{I}/\mathcal{I}^2 \to \Omega)\). Then a calculation shows that \[b + \theta \circ d : \mathcal{O}_{\mathcal{B}'_1}[\mathcal{E}] \longrightarrow \mathcal{O}'_2\] annihilates \(\mathcal{I}'\) and hence defines a map \(\mathcal{O}'_1 \to \mathcal{O}'_2\) fitting into (08UI).

Proof of (2) in the special case above. Omitted. Hint: This is exactly the same as the proof of (2) of Lemma 08S5.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{A} \to \mathcal{B}\) be a homomorphism of sheaves of rings on \(\mathcal{C}\). Let \(\mathcal{G}\) be a \(\mathcal{B}\)-module. Let \(\xi \in \Ext^1_\mathcal{B}(\NL_{\mathcal{B}/\mathcal{A}}, \mathcal{G})\). There exists a map of sheaves of sets \(\alpha : \mathcal{E} \to \mathcal{B}\) such that \(\xi \in \Ext^1_\mathcal{B}(\NL(\alpha), \mathcal{G})\) is the class of a map \(\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}\) (see proof for notation).

Proof

Recall that given \(\alpha : \mathcal{E} \to \mathcal{B}\) such that \(\mathcal{A}[\mathcal{E}] \to \mathcal{B}\) is surjective with kernel \(\mathcal{I}\) the complex \(\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B})\) is canonically isomorphic to \(\NL_{\mathcal{B}/\mathcal{A}}\), see Modules on Sites, Lemma 08TY. Observe moreover, that \(\Omega = \Omega_{\mathcal{A}[\mathcal{E}]/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}]} \mathcal{B}\) is the sheaf associated to the presheaf \(U \mapsto \bigoplus_{e \in \mathcal{E}(U)} \mathcal{B}(U)\). In other words, \(\Omega\) is the free \(\mathcal{B}\)-module on the sheaf of sets \(\mathcal{E}\) and in particular there is a canonical map \(\mathcal{E} \to \Omega\).

Having said this, pick some \(\mathcal{E}\) (for example \(\mathcal{E} = \mathcal{B}\) as in the definition of the naive cotangent complex). The obstruction to writing \(\xi\) as the class of a map \(\mathcal{I}/\mathcal{I}^2 \to \mathcal{G}\) is an element in \(\Ext^1_\mathcal{B}(\Omega, \mathcal{G})\). Say this is represented by the extension \(0 \to \mathcal{G} \to \mathcal{H} \to \Omega \to 0\) of \(\mathcal{B}\)-modules. Consider the sheaf of sets \(\mathcal{E}' = \mathcal{E} \times_\Omega \mathcal{H}\) which comes with an induced map \(\alpha' : \mathcal{E}' \to \mathcal{B}\). Let \(\mathcal{I}' = \Ker(\mathcal{A}[\mathcal{E}'] \to \mathcal{B})\) and \(\Omega' = \Omega_{\mathcal{A}[\mathcal{E}']/\mathcal{A}} \otimes_{\mathcal{A}[\mathcal{E}']} \mathcal{B}\). The pullback of \(\xi\) under the quasi-isomorphism \(\NL(\alpha') \to \NL(\alpha)\) maps to zero in \(\Ext^1_\mathcal{B}(\Omega', \mathcal{G})\) because the pullback of the extension \(\mathcal{H}\) by the map \(\Omega' \to \Omega\) is split as \(\Omega'\) is the free \(\mathcal{B}\)-module on the sheaf of sets \(\mathcal{E}'\) and since by construction there is a commutative diagram \[\xymatrix{ \mathcal{E}' \ar[r] \ar[d] & \mathcal{E} \ar[d] \\ \mathcal{H} \ar[r] & \Omega }\] This finishes the proof.

Lemma

If there exists a solution to (08UF), then the set of isomorphism classes of solutions is principal homogeneous under \(\Ext^1_\mathcal{O}( \NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\).

Proof

We observe right away that given two solutions \(\mathcal{O}'_1\) and \(\mathcal{O}'_2\) to (08UF) we obtain by Lemma 08UG an obstruction element \(o(\mathcal{O}'_1, \mathcal{O}'_2) \in \Ext^1_\mathcal{O}( \NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\) to the existence of a map \(\mathcal{O}'_1 \to \mathcal{O}'_2\). Clearly, this element is the obstruction to the existence of an isomorphism, hence separates the isomorphism classes. To finish the proof it therefore suffices to show that given a solution \(\mathcal{O}'\) and an element \(\xi \in \Ext^1_\mathcal{O}( \NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\) we can find a second solution \(\mathcal{O}'_\xi\) such that \(o(\mathcal{O}', \mathcal{O}'_\xi) = \xi\).

Pick \(\alpha : \mathcal{E} \to \mathcal{O}\) as in Lemma 08UJ for the class \(\xi\). Consider the surjection \(f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}] \to \mathcal{O}\) with kernel \(\mathcal{I}\) and corresponding naive cotangent complex \(\NL(\alpha) = (\mathcal{I}/\mathcal{I}^2 \to \Omega_{f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}]/ f^{-1}\mathcal{O}_\mathcal{B}} \otimes_{f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}]} \mathcal{O})\). By the lemma \(\xi\) is the class of a morphism \(\delta : \mathcal{I}/\mathcal{I}^2 \to \mathcal{G}\). After replacing \(\mathcal{E}\) by \(\mathcal{E} \times_\mathcal{O} \mathcal{O}'\) we may also assume that \(\alpha\) factors through a map \(\alpha' : \mathcal{E} \to \mathcal{O}'\).

These choices determine an \(f^{-1}\mathcal{O}_{\mathcal{B}'}\)-algebra map \(\varphi : \mathcal{O}_{\mathcal{B}'}[\mathcal{E}] \to \mathcal{O}'\). Let \(\mathcal{I}' = \Ker(\varphi)\). Observe that \(\varphi\) induces a map \(\varphi|_{\mathcal{I}'} : \mathcal{I}' \to \mathcal{G}\) and that \(\mathcal{O}'\) is the pushout, as in the following diagram \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}' \ar[r] & \mathcal{O} \ar[r] & 0 \\ 0 \ar[r] & \mathcal{I}' \ar[u]^{\varphi|_{\mathcal{I}'}} \ar[r] & f^{-1}\mathcal{O}_{\mathcal{B}'}[\mathcal{E}] \ar[u] \ar[r] & \mathcal{O} \ar[u]_{=} \ar[r] & 0 }\] Let \(\psi : \mathcal{I}' \to \mathcal{G}\) be the sum of the map \(\varphi|_{\mathcal{I}'}\) and the composition \[\mathcal{I}' \to \mathcal{I}'/(\mathcal{I}')^2 \to \mathcal{I}/\mathcal{I}^2 \xrightarrow{\delta} \mathcal{G}.\] Then the pushout along \(\psi\) is an other ring extension \(\mathcal{O}'_\xi\) fitting into a diagram as above. A calculation (omitted) shows that \(o(\mathcal{O}', \mathcal{O}'_\xi) = \xi\) as desired.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\) be a morphism of ringed topoi. Let \(\mathcal{G}\) be an \(\mathcal{O}\)-module. The set of isomorphism classes of extensions of \(f^{-1}\mathcal{O}_\mathcal{B}\)-algebras \[0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O} \to 0\] where \(\mathcal{G}\) is an ideal of square zero² is canonically bijective to \(\Ext^1_\mathcal{O}(\NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\).

Proof

To prove this we apply the previous results to the case where (08UF) is given by the diagram \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O} \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & f^{-1}\mathcal{O}_\mathcal{B} \ar[u] \ar[r]^{\text{id}} & f^{-1}\mathcal{O}_\mathcal{B} \ar[u] \ar[r] & 0 }\] Thus our lemma follows from Lemma 08UK and the fact that there exists a solution, namely \(\mathcal{G} \oplus \mathcal{O}\). (See remark below for a direct construction of the bijection.)

Remark

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\mathcal{B}, \mathcal{O}_\mathcal{B})\) and \(\mathcal{G}\) be as in Lemma 0GQ5. Consider an extension \(0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O} \to 0\) as in the lemma. We can choose a sheaf of sets \(\mathcal{E}\) and a commutative diagram \[\xymatrix{ \mathcal{E} \ar[d]_{\alpha'} \ar[rd]^\alpha \\ \mathcal{O}' \ar[r] & \mathcal{O} }\] such that \(f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}] \to \mathcal{O}\) is surjective with kernel \(\mathcal{J}\). (For example you can take any sheaf of sets surjecting onto \(\mathcal{O}'\).) Then \[\NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}} \cong \NL(\alpha) = \left( \mathcal{J}/\mathcal{J}^2 \longrightarrow \Omega_{f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}]/ f^{-1}\mathcal{O}_\mathcal{B}} \otimes_{f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}]} \mathcal{O}\right)\] See Modules on Sites, Section 08TT and in particular Lemma 08TY. Of course \(\alpha'\) determines a map \(f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}] \to \mathcal{O}'\) which in turn determines a map \[\mathcal{J}/\mathcal{J}^2 \longrightarrow \mathcal{G}\] which in turn determines the element of \(\Ext^1_\mathcal{O}(\NL(\alpha), \mathcal{G}) = \Ext^1_\mathcal{O}(\NL_{\mathcal{O}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\) corresponding to \(\mathcal{O}'\) by the bijection of the lemma.

Lemma

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\) and \(g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\) be morphisms of ringed topoi. Let \(\mathcal{F}\) be a \(\mathcal{O}_\mathcal{C}\)-module. Let \(\mathcal{G}\) be a \(\mathcal{O}_\mathcal{D}\)-module. Let \(c : g^*\mathcal{F} \to \mathcal{G}\) be a \(\mathcal{O}_\mathcal{D}\)-linear map. Finally, consider

1. \(0 \to \mathcal{F} \to \mathcal{O}_{\mathcal{C}'} \to \mathcal{O}_\mathcal{C} \to 0\) an extension of \(f^{-1}\mathcal{O}_\mathcal{B}\)-algebras corresponding to \(\xi \in \Ext^1_{\mathcal{O}_\mathcal{C}}( \NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{F})\), and

2. \(0 \to \mathcal{G} \to \mathcal{O}_{\mathcal{D}'} \to \mathcal{O}_\mathcal{D} \to 0\) an extension of \(g^{-1}f^{-1}\mathcal{O}_\mathcal{B}\)-algebras corresponding to \(\zeta \in \Ext^1_{\mathcal{O}_\mathcal{D}}( \NL_{\mathcal{O}_\mathcal{D}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\).

See Lemma 0GQ5. Then there is a morphism \[g' : (\Sh(\mathcal{D}), \mathcal{O}_{\mathcal{D}'}) \longrightarrow (\Sh(\mathcal{C}), \mathcal{O}_{\mathcal{C}'})\] of ringed topoi over \((\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\) compatible with \(g\) and \(c\) if and only if \(\xi\) and \(\zeta\) map to the same element of \(\Ext^1_{\mathcal{O}_\mathcal{D}}( Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\).

Proof

The stament makes sense as we have the maps \[\Ext^1_{\mathcal{O}_\mathcal{C}}( \NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{F}) \to \Ext^1_{\mathcal{O}_\mathcal{D}}( Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, Lg^*\mathcal{F}) \to \Ext^1_{\mathcal{O}_\mathcal{D}} (Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\] using the map \(Lg^*\mathcal{F} \to g^*\mathcal{F} \xrightarrow{c} \mathcal{G}\) and \[\Ext^1_{\mathcal{O}_Y}( \NL_{\mathcal{O}_\mathcal{D}/\mathcal{O}_\mathcal{B}}, \mathcal{G}) \to \Ext^1_{\mathcal{O}_Y}( Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\] using the map \(Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}} \to \NL_{\mathcal{O}_\mathcal{D}/\mathcal{O}_\mathcal{B}}\). The statement of the lemma can be deduced from Lemma 08UG applied to the diagram \[\xymatrix{ & 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{\mathcal{D}'} \ar[r] & \mathcal{O}_\mathcal{D} \ar[r] & 0 \\ & 0 \ar[r]|\hole & 0 \ar[u] \ar[r] & g^{-1}f^{-1}\mathcal{O}_\mathcal{B} \ar[u] \ar[r]|\hole & g^{-1}f^{-1}\mathcal{O}_\mathcal{B} \ar[u] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{F} \ar[ruu] \ar[r] & \mathcal{O}_{\mathcal{C}'} \ar[r] & \mathcal{O}_\mathcal{C} \ar[ruu] \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[ruu]|\hole \ar[u] \ar[r] & f^{-1}\mathcal{O}_\mathcal{B} \ar[ruu]|\hole \ar[u] \ar[r] & f^{-1}\mathcal{O}_\mathcal{B} \ar[ruu]|\hole \ar[u] \ar[r] & 0 }\] and a compatibility between the constructions in the proofs of Lemmas 0GQ5 and 08UG whose statement and proof we omit. (See remark below for a direct argument.)

Remark

Let \(f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\), \(g : (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) \to (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C})\), \(\mathcal{F}\), \(\mathcal{G}\), \(c : g^*\mathcal{F} \to \mathcal{G}\), \(0 \to \mathcal{F} \to \mathcal{O}_{\mathcal{C}'} \to \mathcal{O}_\mathcal{C} \to 0\), \(\xi \in \Ext^1_{\mathcal{O}_\mathcal{C}}( \NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{F})\), \(0 \to \mathcal{G} \to \mathcal{O}_{\mathcal{D}'} \to \mathcal{O}_\mathcal{D} \to 0\), and \(\zeta \in \Ext^1_{\mathcal{O}_\mathcal{D}}( \NL_{\mathcal{O}_\mathcal{D}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\) be as in Lemma 0GQ7. Using pushout along \(c : g^{-1}\mathcal{F} \to \mathcal{G}\) we can construct an extension \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}'_1 \ar[r] & g^{-1}\mathcal{O}_\mathcal{C} \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{F} \ar[u]^c \ar[r] & g^{-1}\mathcal{O}_{\mathcal{C}'} \ar[u] \ar[r] & g^{-1}\mathcal{O}_\mathcal{C} \ar@{=}[u] \ar[r] & 0 }\] Using pullback along \(g^\sharp : g^{-1}\mathcal{O}_\mathcal{C} \to \mathcal{O}_\mathcal{D}\) we can construct an extension \[\xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & \mathcal{O}_{\mathcal{D}'} \ar[r] & \mathcal{O}_\mathcal{D} \ar[r] & 0 \\ 0 \ar[r] & \mathcal{G} \ar@{=}[u] \ar[r] & \mathcal{O}'_2 \ar[u] \ar[r] & g^{-1}\mathcal{O}_\mathcal{C} \ar[u] \ar[r] & 0 }\] A diagram chase tells us that there exists a morphism \(g' : (\Sh(\mathcal{D}), \mathcal{O}_{\mathcal{D}'}) \to (\Sh(\mathcal{C}), \mathcal{O}_{\mathcal{C}'})\) over \((\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\) compatible with \(g\) and \(c\) if and only if \(\mathcal{O}'_1\) is isomorphic to \(\mathcal{O}'_2\) as \(g^{-1}f^{-1}\mathcal{O}_\mathcal{B}\)-algebra extensions of \(g^{-1}\mathcal{O}_\mathcal{C}\) by \(\mathcal{G}\). By Lemma 0GQ5 these extensions are classified by the LHS of \[\Ext^1_{g^{-1}\mathcal{O}_\mathcal{C}}( \NL_{g^{-1}\mathcal{O}_\mathcal{C}/g^{-1}f^{-1}\mathcal{O}_\mathcal{B}}, \mathcal{G}) = \Ext^1_{\mathcal{O}_\mathcal{D}}( Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}}, \mathcal{G})\] Here the equality comes from tensor-hom adjunction and the equalities \[\NL_{g^{-1}\mathcal{O}_\mathcal{C}/g^{-1}f^{-1}\mathcal{O}_\mathcal{B}} = g^{-1}\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}} \quad\text{and}\quad Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}} = g^{-1}\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}} \otimes_{g^{-1}\mathcal{O}_X}^\mathbf{L} \mathcal{O}_Y\] For the first of these see Modules on Sites, Lemma 08TZ; the second follows from the definition of derived pullback. Thus, in order to see that Lemma 0GQ7 is true, it suffices to show that \(\mathcal{O}'_1\) corresponds to the image of \(\xi\) and that \(\mathcal{O}'_2\) correspond to the image of \(\zeta\). The correspondence between \(\xi\) and \(\mathcal{O}'_1\) is immediate from the construction of the class \(\xi\) in Remark 0GQ6. For the correspondence between \(\zeta\) and \(\mathcal{O}'_2\), we first choose a commutative diagram \[\xymatrix{ \mathcal{E} \ar[d]_{\beta'} \ar[rd]^\beta \\ \mathcal{O}_{\mathcal{D}'} \ar[r] & \mathcal{O}_\mathcal{D} }\] such that \(g^{-1}f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}] \to \mathcal{O}_\mathcal{D}\) is surjective with kernel \(\mathcal{K}\). Next choose a commutative diagram \[\xymatrix{ \mathcal{E} \ar[d]_{\beta'} & \mathcal{E}' \ar[l]^\varphi \ar[d]_{\alpha'} \ar[rd]^\alpha \\ \mathcal{O}_{\mathcal{D}'} & \mathcal{O}'_2 \ar[l] \ar[r] & g^{-1}\mathcal{O}_\mathcal{C} }\] such that \(g^{-1}f^{-1}\mathcal{O}_\mathcal{B}[\mathcal{E}'] \to g^{-1}\mathcal{O}_\mathcal{C}\) is surjective with kernel \(\mathcal{J}\). (For example just take \(\mathcal{E}' = \mathcal{E} \amalg \mathcal{O}'_2\) as a sheaf of sets.) The map \(\varphi\) induces a map of complexes \(\NL(\alpha) \to \NL(\beta)\) (notation as in Modules, Section 08TG) and in particular \(\bar\varphi : \mathcal{J}/\mathcal{J}^2 \to \mathcal{K}/\mathcal{K}^2\). Then \(\NL(\alpha) \cong \NL_{\mathcal{O}_\mathcal{D}/\mathcal{O}_\mathcal{B}}\) and \(\NL(\beta) \cong \NL_{g^{-1}\mathcal{O}_\mathcal{C}/g^{-1}f^{-1}\mathcal{O}_\mathcal{B}}\) and the map of complexes \(\NL(\alpha) \to \NL(\beta)\) represents the map \(Lg^*\NL_{\mathcal{O}_\mathcal{C}/\mathcal{O}_\mathcal{B}} \to \NL_{\mathcal{O}_\mathcal{D}/\mathcal{O}_\mathcal{B}}\) used in the statement of Lemma 0GQ7 (see first part of its proof). Now \(\zeta\) corresponds to the class of the map \(\mathcal{K}/\mathcal{K}^2 \to \mathcal{G}\) induced by \(\beta'\), see Remark 0GQ6. Similarly, the extension \(\mathcal{O}'_2\) corresponds to the map \(\mathcal{J}/\mathcal{J}^2 \to \mathcal{G}\) induced by \(\alpha'\). The commutative diagram above shows that this map is the composition of the map \(\mathcal{K}/\mathcal{K}^2 \to \mathcal{G}\) induced by \(\beta'\) with the map \(\bar\varphi : \mathcal{J}/\mathcal{J}^2 \to \mathcal{K}/\mathcal{K}^2\). This proves the compatibility we were looking for.

Lemma

Let \(t : (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \to (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})\), \(\mathcal{J} = \Ker(t^\sharp)\), \(f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B})\), \(\mathcal{G}\), and \(c : \mathcal{J} \to \mathcal{G}\) be as in (08UF). Denote \(\xi \in \Ext^1_{\mathcal{O}_\mathcal{B}}( \NL_{\mathcal{O}_\mathcal{B}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{J})\) the element corresponding to the extension \(\mathcal{O}_{\mathcal{B}'}\) of \(\mathcal{O}_\mathcal{B}\) by \(\mathcal{J}\) via Lemma 0GQ5. The set of isomorphism classes of solutions is canonically bijective to the fibre of \[\Ext^1_\mathcal{O}(\NL_{\mathcal{O}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G})\to \Ext^1_\mathcal{O}( Lf^*\NL_{\mathcal{O}_\mathcal{B}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G})\] over the image of \(\xi\).

Proof

By Lemma 0GQ5 applied to \(t \circ f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})\) and the \(\mathcal{O}\)-module \(\mathcal{G}\) we see that elements \(\zeta\) of \(\Ext^1_\mathcal{O}(\NL_{\mathcal{O}/\mathcal{O}_{\mathcal{B}'}}, \mathcal{G})\) parametrize extensions \(0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O} \to 0\) of \(f^{-1}\mathcal{O}_{\mathcal{B}'}\)-algebras. By Lemma 0GQ7 applied to \[(\Sh(\mathcal{C}), \mathcal{O}) \xrightarrow{f} (\Sh(\mathcal{B}), \mathcal{O}_\mathcal{B}) \xrightarrow{t} (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})\] and \(c : \mathcal{J} \to \mathcal{G}\) we see that there is an morphism \[f' : (\Sh(\mathcal{C}), \mathcal{O}') \longrightarrow (\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})\] over \((\Sh(\mathcal{B}'), \mathcal{O}_{\mathcal{B}'})\) compatible with \(c\) and \(f\) if and only if \(\zeta\) maps to \(\xi\). Of course this is the same thing as saying \(\mathcal{O}'\) is a solution of (08UF).

Deformations of algebraic spaces

In this section we spell out what the results in Section 08UE mean for deformations of algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(i : Z \to Z'\) be a morphism of algebraic spaces over \(S\). The following are equivalent

1. \(i\) is a thickening of algebraic spaces as defined in More on Morphisms of Spaces, Section 05ZJ, and

2. the associated morphism \(i_{small} : (\Sh(Z_\etale), \mathcal{O}_Z) \to (\Sh(Z'_\etale), \mathcal{O}_{Z'})\) of ringed topoi (Properties of Spaces, Lemma 03G8) is a thickening in the sense of Section 08M6.

Proof

We stress that this is not a triviality.

Assume (1). By More on Morphisms of Spaces, Lemma 05ZS the morphism \(i\) induces an equivalence of small étale sites and in particular of topoi. Of course \(i^\sharp\) is surjective with locally nilpotent kernel by definition of thickenings.

Assume (2). (This direction is less important and more of a curiosity.) For any étale morphism \(Y' \to Z'\) we see that \(Y = Z \times_{Z'} Y'\) has the same étale topos as \(Y'\). In particular, \(Y'\) is quasi-compact if and only if \(Y\) is quasi-compact because being quasi-compact is a topos theoretic notion (Sites, Lemma 0D06). Having said this we see that \(Y'\) is quasi-compact and quasi-separated if and only if \(Y\) is quasi-compact and quasi-separated (because you can characterize \(Y'\) being quasi-separated by saying that for all \(Y'_1, Y'_2\) quasi-compact algebraic spaces étale over \(Y'\) we have that \(Y'_1 \times_{Y'} Y'_2\) is quasi-compact). Take \(Y'\) affine. Then the algebraic space \(Y\) is quasi-compact and quasi-separated. For any quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{F}\) we have \(H^q(Y, \mathcal{F}) = H^q(Y', (Y \to Y')_*\mathcal{F})\) because the étale topoi are the same. Then \(H^q(Y', (Y \to Y')_*\mathcal{F}) = 0\) because the pushforward is quasi-coherent (Morphisms of Spaces, Lemma 03M9) and \(Y\) is affine. It follows that \(Y'\) is affine by Cohomology of Spaces, Proposition 07V6 (there surely is a proof of this direction of the lemma avoiding this reference). Hence \(i\) is an affine morphism. In the affine case it follows easily from the conditions in Section 08M6 that \(i\) is a thickening of algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(Y \subset Y'\) be a first order thickening of algebraic spaces over \(S\). Let \(f : X \to Y\) be a flat morphism of algebraic spaces over \(S\). If there exists a flat morphism \(f' : X' \to Y'\) of algebraic spaces over \(S\) and an isomorphsm \(a : X \to X' \times_{Y'} Y\) over \(Y\), then

1. the set of isomorphism classes of pairs \((f' : X' \to Y', a)\) is principal homogeneous under \(\Ext^1_{\mathcal{O}_X}(\NL_{X/Y}, f^*\mathcal{C}_{Y/Y'})\), and

2. the set of automorphisms of \(\varphi : X' \to X'\) over \(Y'\) which reduce to the identity on \(X' \times_{Y'} Y\) is \(\Ext^0_{\mathcal{O}_X}(\NL_{X/Y}, f^*\mathcal{C}_{Y/Y'})\).

Proof

We will apply the material on deformations of ringed topoi to the small étale topoi of the algebraic spaces in the lemma. We may think of \(X\) as a closed subspace of \(X'\) so that \((f, f') : (X \subset X') \to (Y \subset Y')\) is a morphism of first order thickenings. By Lemma 0D16 this translates into a morphism of thickenings of ringed topoi. Then we see from More on Morphisms of Spaces, Lemma 06BH (or from the more general Lemma 08MQ) that the ideal sheaf of \(X\) in \(X'\) is equal to \(f^*\mathcal{C}_{Y'/Y}\) and this is in fact equivalent to flatness of \(X'\) over \(Y'\). Hence we have a commutative diagram \[\xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{Y/Y'} \ar[r] & \mathcal{O}_{X'} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & f_{small}^{-1}\mathcal{C}_{Y/Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_{Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_Y \ar[u] \ar[r] & 0 }\] Please compare with (08UF). Observe that automorphisms \(\varphi\) as in (2) give automorphisms \(\varphi^\sharp : \mathcal{O}_{X'} \to \mathcal{O}_{X'}\) fitting in the diagram above. Conversely, an automorphism \(\alpha : \mathcal{O}_{X'} \to \mathcal{O}_{X'}\) fitting into the diagram of sheaves above is equal to \(\varphi^\sharp\) for some automorphism \(\varphi\) as in (2) by More on Morphisms of Spaces, Lemma 05ZN. Finally, by More on Morphisms of Spaces, Lemma 05ZT if we find another sheaf of rings \(\mathcal{A}\) on \(X_\etale\) fitting into the diagram \[\xymatrix{ 0 \ar[r] & f^*\mathcal{C}_{Y/Y'} \ar[r] & \mathcal{A} \ar[r] & \mathcal{O}_X \ar[r] & 0 \\ 0 \ar[r] & f_{small}^{-1}\mathcal{C}_{Y/Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_{Y'} \ar[u] \ar[r] & f_{small}^{-1}\mathcal{O}_Y \ar[u] \ar[r] & 0 }\] then there exists a first order thickening \(X \subset X''\) with \(\mathcal{O}_{X''} = \mathcal{A}\) and applying More on Morphisms of Spaces, Lemma 05ZN once more, we obtain a morphism \((f, f'') : (X \subset X'') \to (Y \subset Y')\) with all the desired properties. Thus part (1) follows from Lemma 08UK and part (2) from part (2) of Lemma 08UG. (Note that \(\NL_{X/Y}\) as defined for a morphism of algebraic spaces in More on Morphisms of Spaces, Section 0D0U agrees with \(\NL_{X/Y}\) as used in Section 08UE.)

Let \(S\) be a scheme. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F} \to \mathcal{G}\) be a homomorphism of \(\mathcal{O}_X\)-modules (not necessarily quasi-coherent). Consider the functor \[F : \left\{ \begin{matrix} \text{extensions of }f^{-1}\mathcal{O}_B\text{ algebras}\\ 0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_X \to 0\\ \text{where }\mathcal{F}\text{ is an ideal of square zero} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{extensions of }f^{-1}\mathcal{O}_B\text{ algebras}\\ 0 \to \mathcal{G} \to \mathcal{O}' \to \mathcal{O}_X \to 0\\ \text{where }\mathcal{G}\text{ is an ideal of square zero} \end{matrix} \right\}\] given by pushout.

Lemma

In the situation above assume that \(X\) is quasi-compact and quasi-separated and that \(DQ_X(\mathcal{F}) \to DQ_X(\mathcal{G})\) (Derived Categories of Spaces, Section 0CR3) is an isomorphism. Then the functor \(F\) is an equivalence of categories.

Proof

Recall that \(\NL_{X/B}\) is an object of \(D_\QCoh(\mathcal{O}_X)\), see More on Morphisms of Spaces, Lemma 0D0Y. Hence our assumption implies the maps \[\Ext^i_X(\NL_{X/B}, \mathcal{F}) \longrightarrow \Ext^i_X(\NL_{X/B}, \mathcal{G})\] are isomorphisms for all \(i\). This implies our functor is fully faithful by Lemma 08UG. On the other hand, the functor is essentially surjective by Lemma 08UK because we have the solutions \(\mathcal{O}_X \oplus \mathcal{F}\) and \(\mathcal{O}_X \oplus \mathcal{G}\) in both categories.

Let \(S\) be a scheme. Let \(B \subset B'\) be a first order thickening of algebraic spaces over \(S\) with ideal sheaf \(\mathcal{J}\) which we view either as a quasi-coherent \(\mathcal{O}_B\)-module or as a quasi-coherent sheaf of ideals on \(B'\), see More on Morphisms of Spaces, Section 05ZJ. Let \(f : X \to B\) be a morphism of algebraic spaces over \(S\). Let \(\mathcal{F} \to \mathcal{G}\) be a homomorphism of \(\mathcal{O}_X\)-modules (not necessarily quasi-coherent). Let \(c : f^{-1}\mathcal{J} \to \mathcal{F}\) be a map of \(f^{-1}\mathcal{O}_B\)-modules and denote \(c' : f^{-1}\mathcal{J} \to \mathcal{G}\) the composition. Consider the functor \[FT : \{\text{solutions to }(\href{defos.html#defos-equation-to-solve-ringed-topoi}{08UF}) \text{ for }\mathcal{F}\text{ and }c\} \longrightarrow \{\text{solutions to }(\href{defos.html#defos-equation-to-solve-ringed-topoi}{08UF}) \text{ for }\mathcal{G}\text{ and }c'\}\] given by pushout.

Lemma

In the situation above assume that \(X\) is quasi-compact and quasi-separated and that \(DQ_X(\mathcal{F}) \to DQ_X(\mathcal{G})\) (Derived Categories of Spaces, Section 0CR3) is an isomorphism. Then the functor \(FT\) is an equivalence of categories.

Proof

A solution of (08UF) for \(\mathcal{F}\) in particular gives an extension of \(f^{-1}\mathcal{O}_{B'}\)-algebras \[0 \to \mathcal{F} \to \mathcal{O}' \to \mathcal{O}_X \to 0\] where \(\mathcal{F}\) is an ideal of square zero. Similarly for \(\mathcal{G}\). Moreover, given such an extension, we obtain a map \(c_{\mathcal{O}'} : f^{-1}\mathcal{J} \to \mathcal{F}\). Thus we are looking at the full subcategory of such extensions of \(f^{-1}\mathcal{O}_{B'}\)-algebras with \(c = c_{\mathcal{O}'}\). Clearly, if \(\mathcal{O}'' = F(\mathcal{O}')\) where \(F\) is the equivalence of Lemma 0D3P (applied to \(X \to B'\) this time), then \(c_{\mathcal{O}''}\) is the composition of \(c_{\mathcal{O}'}\) and the map \(\mathcal{F} \to \mathcal{G}\). This proves the lemma.

Deformations of complexes

This section is a warmup for the next one. We will use as much as possible the material in the chapters on commutative algebra.

Lemma

Let \(R' \to R\) be a surjection of rings whose kernel is an ideal \(I\) of square zero. For every \(K \in D^-(R)\) there is a canonical map \[\omega(K) : K \longrightarrow K \otimes_R^\mathbf{L} I[2]\] in \(D(R)\) with the following properties

1. \(\omega(K) = 0\) if and only if there exists \(K' \in D(R')\) with \(K' \otimes_{R'}^\mathbf{L} R = K\),

2. given \(K \to L\) in \(D^-(R)\) the diagram \[\xymatrix{ K \ar[d] \ar[rr]_-{\omega(K)} & & K \otimes^\mathbf{L}_R I[2] \ar[d] \\ L \ar[rr]^-{\omega(L)} & & L \otimes^\mathbf{L}_R I[2] }\] commutes, and

3. formation of \(\omega(K)\) is compatible with ring maps \(R' \to S'\) (see proof for a precise statement).

Proof

Choose a bounded above complex \(K^\bullet\) of free \(R\)-modules representing \(K\). Then we can choose free \(R'\)-modules \((K')^n\) lifting \(K^n\). We can choose \(R'\)-module maps \((d')^n_K : (K')^n \to (K')^{n + 1}\) lifting the differentials \(d^n_K : K^n \to K^{n + 1}\) of \(K^\bullet\). Although the compositions \[(d')^{n + 1}_K \circ (d')^n_K : (K')^n \to (K')^{n + 2}\] may not be zero, they do factor as \[(K')^n \to K^n \xrightarrow{\omega^n_K} K^{n + 2} \otimes_R I = I(K')^{n + 2} \to (K')^{n + 2}\] because \(d^{n + 1} \circ d^n = 0\). A calculation shows that \(\omega^n_K\) defines a map of complexes. This map of complexes defines \(\omega(K)\).

Let us prove this construction is compatible with a map of complexes \(\alpha^\bullet : K^\bullet \to L^\bullet\) of bounded above free \(R\)-modules and given choices of lifts \((K')^n, (L')^n, (d')^n_K, (d')^n_L\). Namely, choose \((\alpha')^n : (K')^n \to (L')^n\) lifting the components \(\alpha^n : K^n \to L^n\). As before we get a factorization \[(K')^n \to K^n \xrightarrow{h^n} L^{n + 1} \otimes_R I = I(L')^{n + 1} \to (L')^{n + 2}\] of \((d')^n_L \circ (\alpha')^n - (\alpha')^{n + 1} \circ (d')_K^n\). Then it is an pleasant calculation to show that \[\omega^n_L \circ \alpha^n = (d_L^{n + 1} \otimes \text{id}_I) \circ h^n + h^{n + 1} \circ d_K^n + (\alpha^{n + 2} \otimes \text{id}_I) \circ \omega^n_K\] This proves the commutativity of the diagram in (2) of the lemma in this particular case. Using this for two different choices of bounded above free complexes representing \(K\), we find that \(\omega(K)\) is well defined! And of course (2) holds in general as well.

If \(K\) lifts to \(K'\) in \(D^-(R')\), then we can represent \(K'\) by a bounded above complex of free \(R'\)-modules and we see immediately that \(\omega(K) = 0\). Conversely, going back to our choices \(K^\bullet\), \((K')^n\), \((d')^n_K\), if \(\omega(K) = 0\), then we can find \(g^n : K^n \to K^{n + 1} \otimes_R I\) with \[\omega^n = (d_K^{n + 1} \otimes \text{id}_I) \circ g^n + g^{n + 1} \circ d_K^n\] This means that with differentials \((d')^n_K - g^n : (K')^n \to (K')^{n + 1}\) we obtain a complex of free \(R'\)-modules lifting \(K^\bullet\). This proves (1).

Finally, part (3) means the following: Let \(R' \to S'\) be a map of rings. Set \(S = S' \otimes_{R'} R\) and denote \(J = IS' \subset S'\) the square zero kernel of \(S' \to S\). Then given \(K \in D^-(R)\) the statement is that we get a commutative diagram \[\xymatrix{ K \otimes_R^\mathbf{L} S \ar[d] \ar[rr]_-{\omega(K) \otimes \text{id}} & & (K \otimes^\mathbf{L}_R I[2]) \otimes_R^\mathbf{L} S \ar[d] \\ K \otimes_R^\mathbf{L} S \ar[rr]^-{\omega(K \otimes_R^\mathbf{L} S)} & & (K \otimes_R^\mathbf{L} S) \otimes^\mathbf{L}_S J[2] }\] Here the right vertical arrow comes from \[(K \otimes^\mathbf{L}_R I[2]) \otimes_R^\mathbf{L} S = (K \otimes_R^\mathbf{L} S) \otimes_S^\mathbf{L} (I \otimes_R^\mathbf{L} S)[2] \longrightarrow (K \otimes_R^\mathbf{L} S) \otimes_S^\mathbf{L} J[2]\] Choose \(K^\bullet\), \((K')^n\), and \((d')^n_K\) as above. Then we can use \(K^\bullet \otimes_R S\), \((K')^n \otimes_{R'} S'\), and \((d')^n_K \otimes \text{id}_{S'}\) for the construction of \(\omega(K \otimes_R^\mathbf{L} S)\). With these choices commutativity is immediately verified on the level of maps of complexes.

Deformations of complexes on ringed topoi

This material is taken from [lieblich-complexes].

The material in this section works in the setting of a first order thickening of ringed topoi as defined in Section 08M6. However, in order to simplify the notation we will assume the underlying sites \(\mathcal{C}\) and \(\mathcal{D}\) are the same. Moreover, the surjective homomorphism \(\mathcal{O}' \to \mathcal{O}\) of sheaves of rings will be denoted \(\mathcal{O} \to \mathcal{O}_0\) as is perhaps more customary in the literature.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}_0\) be a surjection of sheaves of rings. Assume given the following data

1. flat \(\mathcal{O}\)-modules \(\mathcal{G}^n\),

2. maps of \(\mathcal{O}\)-modules \(\mathcal{G}^n \to \mathcal{G}^{n + 1}\),

3. a complex \(\mathcal{K}_0^\bullet\) of \(\mathcal{O}_0\)-modules,

4. maps of \(\mathcal{O}\)-modules \(\mathcal{G}^n \to \mathcal{K}_0^n\)

such that

1. \(H^n(\mathcal{K}_0^\bullet) = 0\) for \(n \gg 0\),

2. \(\mathcal{G}^n = 0\) for \(n \gg 0\),

3. with \(\mathcal{G}^n_0 = \mathcal{G}^n \otimes_\mathcal{O} \mathcal{O}_0\) the induced maps determine a complex \(\mathcal{G}_0^\bullet\) and a map of complexes \(\mathcal{G}_0^\bullet \to \mathcal{K}_0^\bullet\).

Then there exist

1. flat \(\mathcal{O}\)-modules \(\mathcal{F}^n\),

2. maps of \(\mathcal{O}\)-modules \(\mathcal{F}^n \to \mathcal{F}^{n + 1}\),

3. maps of \(\mathcal{O}\)-modules \(\mathcal{F}^n \to \mathcal{K}_0^n\),

4. maps of \(\mathcal{O}\)-modules \(\mathcal{G}^n \to \mathcal{F}^n\),

such that \(\mathcal{F}^n = 0\) for \(n \gg 0\), such that the diagrams \[\xymatrix{ \mathcal{G}^n \ar[r] \ar[d] & \mathcal{G}^{n + 1} \ar[d] \\ \mathcal{F}^n \ar[r] & \mathcal{F}^{n + 1} }\] commute for all \(n\), such that the composition \(\mathcal{G}^n \to \mathcal{F}^n \to \mathcal{K}_0^n\) is the given map \(\mathcal{G}^n \to \mathcal{K}_0^n\), and such that with \(\mathcal{F}^n_0 = \mathcal{F}^n \otimes_\mathcal{O} \mathcal{O}_0\) we obtain a complex \(\mathcal{F}_0^\bullet\) and map of complexes \(\mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet\) which is a quasi-isomorphism.

Proof

We will prove by descending induction on \(e\) that we can find \(\mathcal{F}^n\), \(\mathcal{G}^n \to \mathcal{F}^n\), and \(\mathcal{F}^n \to \mathcal{F}^{n + 1}\) for \(n \geq e\) fitting into a commutative diagram \[\xymatrix{ \ldots \ar[r] & \mathcal{G}^{e - 1} \ar[r] \ar@/_2pc/[dd] & \mathcal{G}^e \ar[d] \ar[r] \ar@/_2pc/[dd] & \mathcal{G}^{e + 1} \ar[d] \ar[r] \ar@/_2pc/[dd]|\hole & \ldots \\ & & \mathcal{F}^e \ar[d] \ar[r] & \mathcal{F}^{e + 1} \ar[d] \ar[r] & \ldots \\ \ldots \ar[r] & \mathcal{K}_0^{e - 1} \ar[r] & \mathcal{K}_0^e \ar[r] & \mathcal{K}_0^{e + 1} \ar[r] & \ldots }\] such that \(\mathcal{F}_0^\bullet\) is a complex, the induced map \(\mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet\) induces an isomorphism on \(H^n\) for \(n > e\) and a surjection for \(n = e\). For \(e \gg 0\) this is true because we can take \(\mathcal{F}^n = 0\) for \(n \geq e\) in that case by assumptions (a) and (b).

Induction step. We have to construct \(\mathcal{F}^{e - 1}\) and the maps \(\mathcal{G}^{e - 1} \to \mathcal{F}^{e - 1}\), \(\mathcal{F}^{e - 1} \to \mathcal{F}^e\), and \(\mathcal{F}^{e - 1} \to \mathcal{K}_0^{e - 1}\). We will choose \(\mathcal{F}^{e - 1} = A \oplus B \oplus C\) as a direct sum of three pieces.

For the first we take \(A = \mathcal{G}^{e - 1}\) and we choose our map \(\mathcal{G}^{e - 1} \to \mathcal{F}^{e - 1}\) to be the inclusion of the first summand. The maps \(A \to \mathcal{K}^{e - 1}_0\) and \(A \to \mathcal{F}^e\) will be the obvious ones.

To choose \(B\) we consider the surjection (by induction hypothesis) \[\gamma : \Ker(\mathcal{F}^e_0 \to \mathcal{F}^{e + 1}_0) \longrightarrow \Ker(\mathcal{K}^e_0 \to \mathcal{K}^{e + 1}_0)/ \Im(\mathcal{K}^{e - 1}_0 \to \mathcal{K}^e_0)\] We can choose a set \(I\), for each \(i \in I\) an object \(U_i\) of \(\mathcal{C}\), and sections \(s_i \in \mathcal{F}^e(U_i)\), \(t_i \in \mathcal{K}^{e - 1}_0(U_i)\) such that

1. \(s_i\) maps to a section of \(\Ker(\gamma) \subset \Ker(\mathcal{F}^e_0 \to \mathcal{F}^{e + 1}_0)\),

2. \(s_i\) and \(t_i\) map to the same section of \(\mathcal{K}^e_0\),

3. the sections \(s_i\) generate \(\Ker(\gamma)\) as an \(\mathcal{O}_0\)-module.

We omit giving the full justification for this; one uses that \(\mathcal{F}^e \to \mathcal{F}^e_0\) is a surjective maps of sheaves of sets. Then we set to put \[B = \bigoplus\nolimits_{i \in I} j_{U_i!}\mathcal{O}_{U_i}\] and define the maps \(B \to \mathcal{F}^e\) and \(B \to \mathcal{K}_0^{e - 1}\) by using \(s_i\) and \(t_i\) to determine where to send the summand \(j_{U_i!}\mathcal{O}_{U_i}\).

With \(\mathcal{F}^{e - 1} = A \oplus B\) and maps as above, this produces a diagram as above for \(e - 1\) such that \(\mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet\) induces an isomorphism on \(H^n\) for \(n \geq e\). To get the map to be surjective on \(H^{e - 1}\) we choose the summand \(C\) as follows. Choose a set \(J\), for each \(j \in J\) an object \(U_j\) of \(\mathcal{C}\) and a section \(t_j\) of \(\Ker(\mathcal{K}^{e - 1}_0 \to \mathcal{K}^e_0)\) over \(U_j\) such that these sections generate this kernel over \(\mathcal{O}_0\). Then we put \[C = \bigoplus\nolimits_{j \in J} j_{U_j!}\mathcal{O}_{U_j}\] and the zero map \(C \to \mathcal{F}^e\) and the map \(C \to \mathcal{K}_0^{e - 1}\) by using \(s_j\) to determine where to the summand \(j_{U_j!}\mathcal{O}_{U_j}\). This finishes the induction step by taking \(\mathcal{F}^{e - 1} = A \oplus B \oplus C\) and maps as indicated.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}_0\) be a surjection of sheaves of rings whose kernel is an ideal sheaf \(\mathcal{I}\) of square zero. For every object \(K_0\) in \(D^-(\mathcal{O}_0)\) there is a canonical map \[\omega(K_0) : K_0 \longrightarrow K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}[2]\] in \(D(\mathcal{O}_0)\) such that for any map \(K_0 \to L_0\) in \(D^-(\mathcal{O}_0)\) the diagram \[\xymatrix{ K_0 \ar[d] \ar[rr]_-{\omega(K_0)} & & (K_0 \otimes^\mathbf{L}_{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ L_0 \ar[rr]^-{\omega(L_0)} & & (L_0 \otimes^\mathbf{L}_{\mathcal{O}_0} \mathcal{I})[2] }\] commutes.

Proof

Represent \(K_0\) by any complex \(\mathcal{K}_0^\bullet\) of \(\mathcal{O}_0\)-modules. Apply Lemma 0DIT with \(\mathcal{G}^n = 0\) for all \(n\). Denote \(d : \mathcal{F}^n \to \mathcal{F}^{n + 1}\) the maps produced by the lemma. Then we see that \(d \circ d : \mathcal{F}^n \to \mathcal{F}^{n + 2}\) is zero modulo \(\mathcal{I}\). Since \(\mathcal{F}^n\) is flat, we see that \(\mathcal{I}\mathcal{F}^n = \mathcal{F}^n \otimes_{\mathcal{O}} \mathcal{I} = \mathcal{F}^n_0 \otimes_{\mathcal{O}_0} \mathcal{I}\). Hence we obtain a canonical map of complexes \[d \circ d : \mathcal{F}_0^\bullet \longrightarrow (\mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2]\] Since \(\mathcal{F}_0^\bullet\) is a bounded above complex of flat \(\mathcal{O}_0\)-modules, it is K-flat and may be used to compute derived tensor product. Moreover, the map of complexes \(\mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet\) is a quasi-isomorphism by construction. Therefore the source and target of the map just constructed represent \(K_0\) and \(K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}[2]\) and we obtain our map \(\omega(K_0)\).

Let us show that this procedure is compatible with maps of complexes. Namely, let \(\mathcal{L}_0^\bullet\) represent another object of \(D^-(\mathcal{O}_0)\) and suppose that \[\mathcal{K}_0^\bullet \longrightarrow \mathcal{L}_0^\bullet\] is a map of complexes. Apply Lemma 0DIT for the complex \(\mathcal{L}_0^\bullet\), the flat modules \(\mathcal{F}^n\), the maps \(\mathcal{F}^n \to \mathcal{F}^{n + 1}\), and the compositions \(\mathcal{F}^n \to \mathcal{K}_0^n \to \mathcal{L}_0^n\) (we apologize for the reversal of letters used). We obtain flat modules \(\mathcal{G}^n\), maps \(\mathcal{F}^n \to \mathcal{G}^n\), maps \(\mathcal{G}^n \to \mathcal{G}^{n + 1}\), and maps \(\mathcal{G}^n \to \mathcal{L}_0^n\) with all properties as in the lemma. Then it is clear that \[\xymatrix{ \mathcal{F}_0^\bullet \ar[d] \ar[r] & (\mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ \mathcal{G}_0^\bullet \ar[r] & (\mathcal{G}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2] }\] is a commutative diagram of complexes.

To see that \(\omega(K_0)\) is well defined, suppose that we have two complexes \(\mathcal{K}_0^\bullet\) and \((\mathcal{K}'_0)^\bullet\) of \(\mathcal{O}_0\)-modules representing \(K_0\) and two systems \((\mathcal{F}^n, d : \mathcal{F}^n \to \mathcal{F}^{n + 1}, \mathcal{F}^n \to \mathcal{K}_0^n)\) and \(((\mathcal{F}')^n, d : (\mathcal{F}')^n \to (\mathcal{F}')^{n + 1}, (\mathcal{F}')^n \to \mathcal{K}_0^n)\) as above. Then we can choose a complex \((\mathcal{K}''_0)^\bullet\) and quasi-isomorphisms \(\mathcal{K}_0^\bullet \to (\mathcal{K}''_0)^\bullet\) and \((\mathcal{K}'_0)^\bullet \to (\mathcal{K}''_0)^\bullet\) realizing the fact that both complexes represent \(K_0\) in the derived category. Next, we apply the result of the previous paragraph to \[(\mathcal{K}_0)^\bullet \oplus (\mathcal{K}'_0)^\bullet \longrightarrow (\mathcal{K}''_0)^\bullet\] This produces a commutative diagram \[\xymatrix{ \mathcal{F}_0^\bullet \oplus (\mathcal{F}'_0)^\bullet \ar[d] \ar[r] & (\mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2] \oplus ((\mathcal{F}'_0)^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ \mathcal{G}_0^\bullet \ar[r] & (\mathcal{G}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2] }\] Since the vertical arrows give quasi-isomorphisms on the summands we conclude the desired commutativity in \(D(\mathcal{O}_0)\).

Having established well-definedness, the statement on compatibility with maps is a consequence of the result in the second paragraph.

Lemma

Let \((\mathcal{C}, \mathcal{O})\) be a ringed site. Let \(\alpha : K \to L\) be a map of \(D^-(\mathcal{O})\). Let \(\mathcal{F}\) be a sheaf of \(\mathcal{O}\)-modules. Let \(n \in \mathbf{Z}\).

1. If \(H^i(\alpha)\) is an isomorphism for \(i \geq n\), then \(H^i(\alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}_\mathcal{F})\) is an isomorphism for \(i \geq n\).

2. If \(H^i(\alpha)\) is an isomorphism for \(i > n\) and surjective for \(i = n\), then \(H^i(\alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}_\mathcal{F})\) is an isomorphism for \(i > n\) and surjective for \(i = n\).

Proof

Choose a distinguished triangle \[K \to L \to C \to K[1]\] In case (2) we see that \(H^i(C) = 0\) for \(i \geq n\). Hence \(H^i(C \otimes_\mathcal{O}^\mathbf{L} \mathcal{F}) = 0\) for \(i \geq n\) by (the dual of) Derived Categories, Lemma 05TC. This in turn shows that \(H^i(\alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}_\mathcal{F})\) is an isomorphism for \(i > n\) and surjective for \(i = n\). In case (1) we moreover see that \(H^{n - 1}(L) \to H^{n - 1}(C)\) is surjective. Considering the diagram \[\xymatrix{ H^{n - 1}(L) \otimes_\mathcal{O} \mathcal{F} \ar[r] \ar[d] & H^{n - 1}(C) \otimes_\mathcal{O} \mathcal{F} \ar@{=}[d] \\ H^{n - 1}(L \otimes_\mathcal{O}^\mathbf{L} \mathcal{F}) \ar[r] & H^{n - 1}(C \otimes_\mathcal{O}^\mathbf{L} \mathcal{F}) }\] we conclude the lower horizontal arrow is surjective. Combined with what was said before this implies that \(H^n(\alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}_\mathcal{F})\) is an isomorphism.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}_0\) be a surjection of sheaves of rings whose kernel is an ideal sheaf \(\mathcal{I}\) of square zero. For every object \(K_0\) in \(D^-(\mathcal{O}_0)\) the following are equivalent

1. the class \(\omega(K_0) \in \Ext^2_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0} \mathcal{I})\) constructed in Lemma 0DIU is zero,

2. there exists \(K \in D^-(\mathcal{O})\) with \(K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0 = K_0\) in \(D(\mathcal{O}_0)\).

Proof

Let \(K\) be as in (2). Then we can represent \(K\) by a bounded above complex \(\mathcal{F}^\bullet\) of flat \(\mathcal{O}\)-modules. Then \(\mathcal{F}_0^\bullet = \mathcal{F}^\bullet \otimes_{\mathcal{O}} \mathcal{O}_0\) represents \(K_0\) in \(D(\mathcal{O}_0)\). Since \(d_{\mathcal{F}^\bullet} \circ d_{\mathcal{F}^\bullet} = 0\) as \(\mathcal{F}^\bullet\) is a complex, we see from the very construction of \(\omega(K_0)\) that it is zero.

Assume (1). Let \(\mathcal{F}^n\), \(d : \mathcal{F}^n \to \mathcal{F}^{n + 1}\) be as in the construction of \(\omega(K_0)\). The nullity of \(\omega(K_0)\) implies that the map \[\omega = d \circ d : \mathcal{F}_0^\bullet \longrightarrow (\mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I})[2]\] is zero in \(D(\mathcal{O}_0)\). By definition of the derived category as the localization of the homotopy category of complexes of \(\mathcal{O}_0\)-modules, there exists a quasi-isomorphism \(\alpha : \mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet\) such that there exist \(\mathcal{O}_0\)-modules maps \(h^n : \mathcal{G}_0^n \to \mathcal{F}_0^{n + 1} \otimes_\mathcal{O} \mathcal{I}\) with \[\omega \circ \alpha = d_{\mathcal{F}_0^\bullet \otimes \mathcal{I}} \circ h + h \circ d_{\mathcal{G}_0^\bullet}\] We set \[\mathcal{H}^n = \mathcal{F}^n \times_{\mathcal{F}^n_0} \mathcal{G}_0^n\] and we define \[d' : \mathcal{H}^n \longrightarrow \mathcal{H}^{n + 1},\quad (f^n, g_0^n) \longmapsto (d(f^n) - h^n(g_0^n), d(g_0^n))\] with obvious notation using that \(\mathcal{F}_0^{n + 1} \otimes_{\mathcal{O}_0} \mathcal{I} = \mathcal{F}^{n + 1} \otimes_\mathcal{O} \mathcal{I} = \mathcal{I}\mathcal{F}^{n + 1} \subset \mathcal{F}^{n + 1}\). Then one checks \(d' \circ d' = 0\) by our choice of \(h^n\) and definition of \(\omega\). Hence \(\mathcal{H}^\bullet\) defines an object in \(D(\mathcal{O})\). On the other hand, there is a short exact sequence of complexes of \(\mathcal{O}\)-modules \[0 \to \mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I} \to \mathcal{H}^\bullet \to \mathcal{G}_0^\bullet \to 0\] We still have to show that \(\mathcal{H}^\bullet \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0\) is isomorphic to \(K_0\). Choose a quasi-isomorphism \(\mathcal{E}^\bullet \to \mathcal{H}^\bullet\) where \(\mathcal{E}^\bullet\) is a bounded above complex of flat \(\mathcal{O}\)-modules. We obtain a commutative diagram \[\xymatrix{ 0 \ar[r] & \mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{I} \ar[d]^\beta \ar[r] & \mathcal{E}^\bullet \ar[d]^\gamma \ar[r] & \mathcal{E}_0^\bullet \ar[d]^\delta \ar[r] & 0 \\ 0 \ar[r] & \mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I} \ar[r] & \mathcal{H}^\bullet \ar[r] & \mathcal{G}_0^\bullet \ar[r] & 0 }\] We claim that \(\delta\) is a quasi-isomorphism. Since \(H^i(\delta)\) is an isomorphism for \(i \gg 0\), we can use descending induction on \(n\) such that \(H^i(\delta)\) is an isomorphism for \(i \geq n\). Observe that \(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{I}\) represents \(\mathcal{E}_0^\bullet \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}\), that \(\mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I}\) represents \(\mathcal{G}_0^\bullet \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I}\), and that \(\beta = \delta \otimes_{\mathcal{O}_0}^\mathbf{L} \text{id}_\mathcal{I}\) as maps in \(D(\mathcal{O}_0)\). This is true because \(\beta = (\alpha \otimes \text{id}_\mathcal{I}) \circ (\delta \otimes \text{id}_\mathcal{I})\). Suppose that \(H^i(\delta)\) is an isomorphism in degrees \(\geq n\). Then the same is true for \(\beta\) by what we just said and Lemma 0DIV. Then we can look at the diagram \[\xymatrix{ H^{n - 1}(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{I}) \ar[r] \ar[d]^{H^{n - 1}(\beta)} & H^{n - 1}(\mathcal{E}^\bullet) \ar[r] \ar[d] & H^{n - 1}(\mathcal{E}_0^\bullet) \ar[r] \ar[d]^{H^{n - 1}(\delta)} & H^n(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{I}) \ar[r] \ar[d]^{H^n(\beta)} & H^n(\mathcal{E}^\bullet) \ar[d] \\ H^{n - 1}(\mathcal{F}_0^\bullet \otimes_\mathcal{O} \mathcal{I}) \ar[r] & H^{n - 1}(\mathcal{H}^\bullet) \ar[r] & H^{n - 1}(\mathcal{G}_0^\bullet) \ar[r] & H^n(\mathcal{F}_0^\bullet \otimes_\mathcal{O} \mathcal{I}) \ar[r] & H^n(\mathcal{H}^\bullet) }\] Using Homology, Lemma 05QA we see that \(H^{n - 1}(\delta)\) is surjective. This in turn implies that \(H^{n - 1}(\beta)\) is surjective by Lemma 0DIV. Using Homology, Lemma 05QA again we see that \(H^{n - 1}(\delta)\) is an isomorphism. The claim holds by induction, so \(\delta\) is a quasi-isomorphism which is what we wanted to show.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}_0\) be a surjection of sheaves of rings. Assume given the following data

1. a complex of \(\mathcal{O}\)-modules \(\mathcal{F}^\bullet\),

2. a complex \(\mathcal{K}_0^\bullet\) of \(\mathcal{O}_0\)-modules,

3. a quasi-isomorphism \(\mathcal{K}_0^\bullet \to \mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{O}_0\),

Then there exist a quasi-isomorphism \(\mathcal{G}^\bullet \to \mathcal{F}^\bullet\) such that the map of complexes \(\mathcal{G}^\bullet \otimes_\mathcal{O} \mathcal{O}_0 \to \mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{O}_0\) factors through \(\mathcal{K}_0^\bullet\) in the homotopy category of complexes of \(\mathcal{O}_0\)-modules.

Proof

Set \(\mathcal{F}_0^\bullet = \mathcal{F}^\bullet \otimes_\mathcal{O} \mathcal{O}_0\). By Derived Categories, Lemma 0642 there exists a factorization \[\mathcal{K}_0^\bullet \to \mathcal{L}_0^\bullet \to \mathcal{F}_0^\bullet\] of the given map such that the first arrow has an inverse up to homotopy and the second arrow is termwise split surjective. Hence we may assume that \(\mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet\) is termwise surjective. In that case we take \[\mathcal{G}^n = \mathcal{F}^n \times_{\mathcal{F}^n_0} \mathcal{K}_0^n\] and everything is clear.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}_0\) be a surjection of sheaves of rings whose kernel is an ideal sheaf \(\mathcal{I}\) of square zero. Let \(K, L \in D^-(\mathcal{O})\). Set \(K_0 = K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0\) and \(L_0 = L \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0\) in \(D^-(\mathcal{O}_0)\). Given \(\alpha_0 : K_0 \to L_0\) in \(D(\mathcal{O}_0)\) there is a canonical element \[o(\alpha_0) \in \Ext^1_{\mathcal{O}_0}(K_0, L_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})\] whose vanishing is necessary and sufficient for the existence of a map \(\alpha : K \to L\) in \(D(\mathcal{O})\) with \(\alpha_0 = \alpha \otimes_\mathcal{O}^\mathbf{L} \text{id}\).

Proof

Finding \(\alpha : K \to L\) lifing \(\alpha_0\) is the same as finding \(\alpha : K \to L\) such that the composition \(K \xrightarrow{\alpha} L \to L_0\) is equal to the composition \(K \to K_0 \xrightarrow{\alpha_0} L_0\). The short exact sequence \(0 \to \mathcal{I} \to \mathcal{O} \to \mathcal{O}_0 \to 0\) gives rise to a canonical distinguished triangle \[L \otimes_\mathcal{O}^\mathbf{L} \mathcal{I} \to L \to L_0 \to (L \otimes_\mathcal{O}^\mathbf{L} \mathcal{I})[1]\] in \(D(\mathcal{O})\). By Derived Categories, Lemma 0149 the composition \[K \to K_0 \xrightarrow{\alpha_0} L_0 \to (L \otimes_\mathcal{O}^\mathbf{L} \mathcal{I})[1]\] is zero if and only if we can find \(\alpha : K \to L\) lifting \(\alpha_0\). The composition is an element in \[\Hom_{D(\mathcal{O})}(K, (L \otimes_\mathcal{O}^\mathbf{L} \mathcal{I})[1]) = \Hom_{D(\mathcal{O}_0)}(K_0, (L \otimes_\mathcal{O}^\mathbf{L} \mathcal{I})[1]) = \Ext^1_{\mathcal{O}_0}(K_0, L_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})\] by adjunction.

Lemma

Let \(\mathcal{C}\) be a site. Let \(\mathcal{O} \to \mathcal{O}_0\) be a surjection of sheaves of rings whose kernel is an ideal sheaf \(\mathcal{I}\) of square zero. Let \(K_0 \in D^-(\mathcal{O})\). A lift of \(K_0\) is a pair \((K, \alpha_0)\) consisting of an object \(K\) in \(D^-(\mathcal{O})\) and an isomorphism \(\alpha_0 : K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0 \to K_0\) in \(D(\mathcal{O}_0)\).

1. Given a lift \((K, \alpha)\) the group of automorphism of the pair is canonically the cokernel of a map \[\Ext^{-1}_{\mathcal{O}_0}(K_0, K_0) \longrightarrow \Hom_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})\]

2. If there is a lift, then the set of isomorphism classes of lifts is principal homogenenous under \(\Ext^1_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})\).

Proof

An automorphism of \((K, \alpha)\) is a map \(\varphi : K \to K\) in \(D(\mathcal{O})\) with \(\varphi \otimes_\mathcal{O} \text{id}_{\mathcal{O}_0} = \text{id}\). This is the same thing as saying that \[K \xrightarrow{\varphi - \text{id}} K \to K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0\] is zero. We conclude the group of automorphisms is the cokernel of a map \[\Hom_\mathcal{O}(K, K_0[-1]) \longrightarrow \Hom_\mathcal{O}(K, K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})\] by the distinguished triangle \[K \otimes_\mathcal{O}^\mathbf{L} \mathcal{I} \to K \to K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0 \to (K \otimes_\mathcal{O}^\mathbf{L} \mathcal{I})[1]\] in \(D(\mathcal{O})\) and Derived Categories, Lemma 0149. To translate into the groups in the lemma use adjunction of the restriction functor \(D(\mathcal{O}_0) \to D(\mathcal{O})\) and \(- \otimes_\mathcal{O} \mathcal{O}_0 : D(\mathcal{O}) \to D(\mathcal{O}_0)\). This proves (1).

Proof of (2). Assume that \(K_0 = K \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0\) in \(D(\mathcal{O})\). By Lemma 0DIY the map sending a lift \((K', \alpha_0)\) to the obstruction \(o(\alpha_0)\) to lifting \(\alpha_0\) defines a canonical injective map from the set of isomomorphism classes of pairs to \(\Ext^1_{\mathcal{O}_0}(K_0, K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})\). To finish the proof we show that it is surjective. Pick \(\xi : K_0 \to (K_0 \otimes_{\mathcal{O}_0}^\mathbf{L} \mathcal{I})[1]\) in the \(\Ext^1\) of the lemma. Choose a bounded above complex \(\mathcal{F}^\bullet\) of flat \(\mathcal{O}\)-modules representing \(K\). The map \(\xi\) can be represented as \(t \circ s^{-1}\) where \(s : \mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet\) is a quasi-isomorphism and \(t : \mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I}[1]\) is a map of complexes. By Lemma 0DIX we can assume there exists a quasi-isomorphism \(\mathcal{G}^\bullet \to \mathcal{F}^\bullet\) of complexes of \(\mathcal{O}\)-modules such that \(\mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet\) factors through \(s\) up to homotopy. We may and do replace \(\mathcal{G}^\bullet\) by a bounded above complex of flat \(\mathcal{O}\)-modules (by picking a qis from such to \(\mathcal{G}^\bullet\) and replacing). Then we see that \(\xi\) is represented by a map of complexes \(t : \mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I}[1]\) and the quasi-isomorphism \(\mathcal{G}_0^\bullet \to \mathcal{F}_0^\bullet\). Set \[\mathcal{H}^n = \mathcal{F}^n \times_{\mathcal{F}_0^n} \mathcal{G}_0^n\] with differentials \[\mathcal{H}^n \to \mathcal{H}^{n + 1},\quad (f^n, g_0^n) \mapsto (d(f^n) + t(g_0^n), d(g_0^n))\] This makes sense as \(\mathcal{F}_0^{n + 1} \otimes_{\mathcal{O}_0} \mathcal{I} = \mathcal{F}^{n + 1} \otimes_\mathcal{O} \mathcal{I} = \mathcal{I}\mathcal{F}^{n + 1} \subset \mathcal{F}^{n + 1}\). We omit the computation that shows that \(\mathcal{H}^\bullet\) is a complex of \(\mathcal{O}\)-modules. By construction there is a short exact sequence \[0 \to \mathcal{F}_0^\bullet \otimes_{\mathcal{O}_0} \mathcal{I} \to \mathcal{H}^\bullet \to \mathcal{G}_0^\bullet \to 0\] of complexes of \(\mathcal{O}\)-modules. Exactly as in the proof of Lemma 0DIW one shows that this sequence induces an isomorphism \(\alpha_0 : \mathcal{H}^\bullet \otimes_\mathcal{O}^\mathbf{L} \mathcal{O}_0 \to \mathcal{G}_0^\bullet\) in \(D(\mathcal{O}_0)\). In other words, we have produced a pair \((\mathcal{H}^\bullet, \alpha_0)\). We omit the verification that \(o(\alpha_0) = \xi\); hint: \(o(\alpha_0)\) can be computed explicitly in this case as we have maps \(\mathcal{H}^n \to \mathcal{F}^n\) (not compatible with differentials) lifting the components of \(\alpha_0\). This finishes the proof.