The Stacks project

The local case of this result is [IV Corollaire 2.9, MRaynaud-book].

Proposition 52.22.2 (Algebraization in cohomological dimension 1). In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Assume

  1. $A$ has a dualizing complex and $\text{cd}(A, I) = 1$,

  2. $(\mathcal{F}_ n)$ satisfies the $(2, 3)$-inequalities, see Definition 52.19.1.

Then $(\mathcal{F}_ n)$ extends 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. By Lemma 52.17.1 we may replace $(\mathcal{F}_ n)$ by the object $(\mathcal{H}_ n)$ of $\textit{Coh}(U, I\mathcal{O}_ U)$ found in Lemma 52.21.3. Thus we may assume that $(\mathcal{F}_ n)$ is pro-isomorphic to a inverse system $(\mathcal{F}_ n'')$ with the properties mentioned in Lemma 52.21.3. In Lemma 52.22.1 we proved that $(\mathcal{F}_ n)$ canonically extends to $X$. The final statement follows from Lemma 52.16.8. $\square$

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