Proposition 52.23.3. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume there is Noetherian local ring $(R, \mathfrak m)$ and a ring map $R \to A$ such that

$I = \mathfrak m A$,

for $y \in U \cap Y$ the stalk $\mathcal{F}_ y^\wedge $ is $R$-flat,

$H^0(U, \mathcal{F}_1)$ and $H^1(U, \mathcal{F}_1)$ are finite $A$-modules.

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

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