Lemma 91.11.2. Let $(f, f')$ be a morphism of first order thickenings of ringed topoi as in Situation 91.9.1. Let $\mathcal{F}'$ be an $\mathcal{O}'$-module and set $\mathcal{F} = i^*\mathcal{F}'$. Assume that $\mathcal{F}$ is flat over $\mathcal{O}_\mathcal {B}$ and that $(f, f')$ is a strict morphism of thickenings (Definition 91.9.2). Then the following are equivalent
$\mathcal{F}'$ is flat over $\mathcal{O}_{\mathcal{B}'}$, and
the canonical map $f^*\mathcal{J} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{I}\mathcal{F}'$ is an isomorphism.
Moreover, in this case the maps
\[ f^*\mathcal{J} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{I}\mathcal{F}' \]
are isomorphisms.
Proof.
The map $f^*\mathcal{J} \to \mathcal{I}$ is surjective as $(f, f')$ is a strict morphism of thickenings. Hence the final statement is a consequence of (2).
Proof of the equivalence of (1) and (2). By definition flatness over $\mathcal{O}_\mathcal {B}$ means flatness over $f^{-1}\mathcal{O}_\mathcal {B}$. Similarly for flatness over $f^{-1}\mathcal{O}_{\mathcal{B}'}$. Note that the strictness of $(f, f')$ and the assumption that $\mathcal{F} = i^*\mathcal{F}'$ imply that
\[ \mathcal{F} = \mathcal{F}'/(f^{-1}\mathcal{J})\mathcal{F}' \]
as sheaves on $\mathcal{C}$. Moreover, observe that $f^*\mathcal{J} \otimes _\mathcal {O} \mathcal{F} = f^{-1}\mathcal{J} \otimes _{f^{-1}\mathcal{O}_\mathcal {B}} \mathcal{F}$. Hence the equivalence of (1) and (2) follows from Modules on Sites, Lemma 18.28.15.
$\square$
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