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 (52.16.6.1). Assume
the inverse system H^0(U, \mathcal{F}_ n) has Mittag-Leffler,
the limit topology on M agrees with the I-adic topology, and
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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