Lemma 52.16.3. In Situation 52.16.1 let (\mathcal{F}_ n) be an object of \textit{Coh}(U, I\mathcal{O}_ U). Consider the following conditions:
(\mathcal{F}_ n) is in the essential image of the functor (52.16.2.1),
(\mathcal{F}_ n) is the completion of a coherent \mathcal{O}_ U-module,
(\mathcal{F}_ n) is the completion of a coherent \mathcal{O}_ V-module for U \cap Y \subset V \subset U open,
(\mathcal{F}_ n) is the completion of the restriction to U of a coherent \mathcal{O}_ X-module,
(\mathcal{F}_ n) is the restriction to U of the completion of a coherent \mathcal{O}_ X-module,
there exists an object (\mathcal{G}_ n) of \textit{Coh}(X, I\mathcal{O}_ X) whose restriction to U is (\mathcal{F}_ n).
Then conditions (1), (2), (3), (4), and (5) are equivalent and imply (6). If A is I-adically complete then condition (6) implies the others.
Proof.
Parts (1) and (2) are equivalent, because the completion of a coherent \mathcal{O}_ U-module \mathcal{F} is by definition the image of \mathcal{F} under the functor (52.16.2.1). If V \subset U is an open subscheme containing U \cap Y, then we have
\textit{Coh}(V, I\mathcal{O}_ V) = \textit{Coh}(U, I\mathcal{O}_ U)
since the category of coherent \mathcal{O}_ V-modules supported on V \cap Y is the same as the category of coherent \mathcal{O}_ U-modules supported on U \cap Y. Thus the completion of a coherent \mathcal{O}_ V-module is an object of \textit{Coh}(U, I\mathcal{O}_ U). Having said this the equivalence of (2), (3), (4), and (5) holds because the functors \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ U) \to \textit{Coh}(\mathcal{O}_ V) are essentially surjective. See Properties, Lemma 28.22.5.
It is always the case that (5) implies (6). Assume A is I-adically complete. Then any object of \textit{Coh}(X, I\mathcal{O}_ X) corresponds to a finite A-module by Cohomology of Schemes, Lemma 30.23.1. Thus we see that (6) implies (5) in this case.
\square
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