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
$A$ has a dualizing complex and $\text{cd}(A, I) = 1$,
$(\mathcal{F}_ n)$ 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. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)