Lemma 20.34.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. If $\mathcal{I}^\bullet $ is a K-injective complex of $\mathcal{O}_ X$-modules, then $\mathcal{H}_ Z(\mathcal{I}^\bullet )$ is K-injective complex of $\mathcal{O}_ X|_ Z$-modules.

**Proof.**
Since $i_* : \textit{Mod}(\mathcal{O}_ X|_ Z) \to \textit{Mod}(\mathcal{O}_ X)$ is exact and left adjoint to $\mathcal{H}_ Z$ (Modules, Lemma 17.13.6) this follows from Derived Categories, Lemma 13.31.9.
$\square$

## 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Your email address will not be published.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.

## Comments (0)