The Stacks project

Lemma 52.17.1. In Situation 52.16.1. Let $(\mathcal{F}_ n) \to (\mathcal{F}'_ n)$ be a morphism of $\textit{Coh}(U, I\mathcal{O}_ U)$ whose kernel and cokernel are annihilated by a power of $I$. Then

  1. $(\mathcal{F}_ n)$ extends to $X$ if and only if $(\mathcal{F}'_ n)$ extends to $X$, and

  2. $(\mathcal{F}_ n)$ is the completion of a coherent $\mathcal{O}_ U$-module if and only if $(\mathcal{F}'_ n)$ is.

Proof. Part (2) follows immediately from Cohomology of Schemes, Lemma 30.23.6. To see part (1), we first use Lemma 52.16.6 to reduce to the case where $A$ is $I$-adically complete. However, in that case (1) reduces to (2) by Lemma 52.16.3. $\square$

Comments (0)

Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0EIU. Beware of the difference between the letter 'O' and the digit '0'.