Lemma 91.5.6. 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. If there exists a pair (\mathcal{F}', \alpha ) consisting of an \mathcal{O}_{X'}-module \mathcal{F}' flat over S' 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}_ X}( \mathcal{F}, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F}).
Proof. If we assume there exists one such module, then the canonical map
f^*\mathcal{J} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F}
is an isomorphism by Lemma 91.5.2. Apply Lemma 91.4.3 with \mathcal{K} = \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} and c = 1. By Lemma 91.5.2 the corresponding extensions \mathcal{F}' are all flat over S'. \square
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