The Stacks project

Lemma 91.5.7. Let $(f, f')$ be a morphism of first order thickenings as in Situation 91.3.1. 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 $o(\mathcal{F}, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F}, 1) \in \mathop{\mathrm{Ext}}\nolimits ^2_{\mathcal{O}_ X}( \mathcal{F}, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F})$ of Lemma 91.4.4 is zero.

Proof. This follows immediately from the characterization of $\mathcal{O}_{X'}$-modules flat over $S'$ of Lemma 91.5.2 and Lemma 91.4.4. $\square$

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