The Stacks project

Lemma 52.19.6. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume

  1. $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,

  2. the completion $\mathcal{F}^\wedge $ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities, and

  3. for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{F}_ x) \geq 2$.

Then the map

\[ \mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{G}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)}(\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]

is bijective for every coherent $\mathcal{O}_ U$-module $\mathcal{G}$.

Proof. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathcal{F})$. Using Cohomology of Schemes, Lemma 30.11.2 or More on Algebra, Lemma 15.23.10 we see that the completion of $\mathcal{H}$ satisfies the strict $(1, 1 + d)$-inequalities and that for $x \in U$ with $\overline{\{ x\} } \cap Y \subset Z$ we have $\text{depth}(\mathcal{H}_ x) \geq 2$. Details omitted. Thus by Lemma 52.19.5 we have

\[ \mathop{\mathrm{Hom}}\nolimits _ U(\mathcal{G}, \mathcal{F}) = H^0(U, \mathcal{H}) = \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Coh}(U, I\mathcal{O}_ U)} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]

See Cohomology of Schemes, Lemma 30.23.5 for the final equality. $\square$

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Your email address will not be published.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 0EJ7. Beware of the difference between the letter 'O' and the digit '0'.