Lemma 18.28.15. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a flat homorphism of sheaves of rings. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals such that the induced map $\mathcal{O}/\mathcal{I} \to \mathcal{O}'/\mathcal{I}\mathcal{O}'$ is an isomorphism. For any $\mathcal{O}$-module $\mathcal{F}$ annihilated by $\mathcal{I}^ n$ for some $n \geq 0$ the map $\text{id} \otimes 1 : \mathcal{F} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{O}'$ is an isomorphism.

Proof. Omitted. Hint: See More on Algebra, Lemma 15.88.2. $\square$

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