The Stacks project

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

  1. $I = \mathfrak m A$,

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

  3. $H^0(U, \mathcal{F}_1)$ 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.

Proof. The proof is exactly the same as the proof of Proposition 52.23.1. Namely, if $\kappa = R/\mathfrak m$ then for $n \geq 0$ there is an isomorphism

\[ I^ n \mathcal{F}_{n + 1} \cong \mathcal{F}_1 \otimes _\kappa \mathfrak m^ n/\mathfrak m^{n + 1} \]

and the right hand side is a finite direct sum of copies of $\mathcal{F}_1$. This can be checked by looking at stalks. Everything else is exactly the same. $\square$

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