The Stacks project

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 $\mathcal{O}_{\mathcal{B}'}$. $\square$

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