The Stacks project

Lemma 52.17.1. In Situation 52.16.1. Let $(\mathcal{F}_ n) \to (\mathcal{F}'_ n)$ be a morphism of $\textit{Coh}(U, I\mathcal{O}_ U)$ whose kernel and cokernel are annihilated by a power of $I$. Then

  1. $(\mathcal{F}_ n)$ extends to $X$ if and only if $(\mathcal{F}'_ n)$ extends to $X$, and

  2. $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ U$-module if and only if $(\mathcal{F}'_ n)$ is.

Proof. Part (2) follows immediately from Cohomology of Schemes, Lemma 30.23.6. To see part (1), we first use Lemma 52.16.6 to reduce to the case where $A$ is $I$-adically complete. However, in that case (1) reduces to (2) by Lemma 52.16.3. $\square$

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