The Stacks project

Lemma 52.16.10. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Let $M$ be as in ( Assume

  1. the inverse system $H^0(U, \mathcal{F}_ n)$ has Mittag-Leffler,

  2. the limit topology on $M$ agrees with the $I$-adic topology, and

  3. the image of $M \to H^0(U, \mathcal{F}_ n)$ is a finite $A$-module for all $n$.

Then $(\mathcal{F}_ n)$ extends canonically to $X$. In particular, if $A$ is $I$-adically complete, then $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ U$-module.

Proof. Since $H^0(U, \mathcal{F}_ n)$ has the Mittag-Leffler condition and since the limit topology on $M$ is the $I$-adic topology we see that $\{ M/I^ nM\} $ and $\{ H^0(U, \mathcal{F}_ n)\} $ are pro-isomorphic inverse systems of $A$-modules. Thus if we set

\[ \mathcal{G}_ n = \widetilde{M/I^ n M} \]

then we see that to verify the condition in Definition 52.16.7 it suffices to show that $M$ is a finite module over the $I$-adic completion of $A$. This follows from the fact that $M/I^ n M$ is finite by condition (c) and the above and Algebra, Lemma 10.96.12. $\square$

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