Lemma 30.24.1. Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{I} = I\mathcal{O}_ X$. Then the functor (30.23.3.1) is fully faithful.

Proof. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Then $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$ is a coherent $\mathcal{O}_ X$-module, see Modules, Lemma 17.21.5. By Lemma 30.23.5 the map

$\mathop{\mathrm{lim}}\nolimits _ n H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) \to \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge )$

is bijective. Hence fully faithfulness of (30.23.3.1) follows from the theorem on formal functions (Lemma 30.20.6) for the coherent sheaf $\mathcal{H}$. $\square$

## 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Your email address will not be published.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 0883. Beware of the difference between the letter 'O' and the digit '0'.