Lemma 91.11.1. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ 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 $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{I}\mathcal{G}')$.
91.11 Infinitesimal deformations of modules on ringed topoi
Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a first order thickening of ringed topoi. We freely use the notation introduced in Section 91.9. 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 (91.10.0.1). 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
\[ 0 \to \mathcal{I} \to \mathcal{O}' \to \mathcal{O} \to 0 \]
Proof. This is a special case of Lemma 91.10.1 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
\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{I}\mathcal{G}') \quad \text{and}\quad \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]
The lemma now follows from the exact sequence
\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{G}') \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) \]
see Homology, Lemma 12.5.8. $\square$
Lemma 91.11.2. Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation 91.9.1. 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 91.9.2). Then the following are equivalent
$\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$, and
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 18.28.15. $\square$
Lemma 91.11.3. Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation 91.9.1. 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
$\mathcal{F}'$ is an $\mathcal{O}'$-module of finite presentation, and
$\mathcal{F}$ is an $\mathcal{O}$-module of finite presentation.
Proof. The implication (1) $\Rightarrow $ (2) follows from Modules on Sites, Lemma 18.23.4. For the converse, assume $\mathcal{F}$ of finite presentation. We may and do assume that $\mathcal{C} = \mathcal{C}'$. By Lemma 91.11.2 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. $\square$
Lemma 91.11.4. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.9.1. 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
\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G} \otimes _\mathcal {O} f^*\mathcal{J}) \]
Lemma 91.11.5. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ 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 whose vanishing is a necessary and sufficient condition for the existence of a lift of $\varphi $ to an $\mathcal{O}'$-linear map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$.
\[ o(\varphi ) \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') \]
Proof. It is clear from the proof of Lemma 91.11.1 that the vanishing of the boundary of $\varphi $ via the map
\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') \]
is a necessary and sufficient condition for the existence of a lift. We conclude as
\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') \]
the adjointness of $i_* = Ri_*$ and $Li^*$ on the derived category (Cohomology on Sites, Lemma 21.19.1). $\square$
Lemma 91.11.6. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.9.1. 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 whose vanishing is a necessary and sufficient condition for the existence of a lift of $\varphi $ to an $\mathcal{O}'$-linear map $\varphi ' : \mathcal{F}' \to \mathcal{G}'$.
\[ o(\varphi ) \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{G} \otimes _\mathcal {O} f^*\mathcal{J}) \]
First proof. This follows from Lemma 91.11.5 as we claim that under the assumptions of the lemma we have
\[ \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{G} \otimes _\mathcal {O} f^*\mathcal{J}) \]
Namely, we have $\mathcal{I}\mathcal{G}' = \mathcal{G} \otimes _\mathcal {O} f^*\mathcal{J}$ by Lemma 91.11.2. 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 91.11.2. Thus $\tau _{\geq -1}Li^*\mathcal{F}' = \mathcal{F}$. On the other hand, we have
\[ \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\tau _{\geq -1}Li^*\mathcal{F}', \mathcal{I}\mathcal{G}') \]
by the dual of Derived Categories, Lemma 13.16.1. $\square$
Second proof. We can apply Lemma 91.10.2 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 91.11.2, 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. $\square$
Lemma 91.11.7. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.9.1. 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 $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}( \mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F})$.
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 91.11.2. Apply Lemma 91.10.3 with $\mathcal{K} = \mathcal{I} \otimes _\mathcal {O} \mathcal{F}$ and $c = 1$. By Lemma 91.11.2 the corresponding extensions $\mathcal{F}'$ are all flat over $\mathcal{O}_{\mathcal{B}'}$. $\square$
Lemma 91.11.8. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.9.1. 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
the canonical map $f^*\mathcal{J} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{I} \otimes _\mathcal {O} \mathcal{F}$ is an isomorphism, and
the class $o(\mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F}, 1) \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}( \mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F})$ of Lemma 91.10.4 is zero.
Proof. This follows immediately from the characterization of $\mathcal{O}'$-modules flat over $\mathcal{O}_{\mathcal{B}'}$ of Lemma 91.11.2 and Lemma 91.10.4. $\square$
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