The Stacks project

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

  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 $o(\mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F}, 1) \in \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}( \mathcal{F}, \mathcal{I} \otimes _\mathcal {O} \mathcal{F})$ 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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