Lemma 91.5.3 (08LJ)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 91: Deformation Theory
Section 91.5: Infinitesimal deformations of modules on ringed spaces
Lemma 91.5.3 (cite)

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Lemma 91.5.3. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.3.1. 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

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G} \otimes _{\mathcal{O}_ X} f^*\mathcal{J})$

Proof. Combine Lemmas 91.5.1 and 91.5.2. $\square$

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