Lemma 48.9.6. Let $i : Z \to X$ be a closed immersion of schemes. Assume $X$ is a locally Noetherian. Then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, -)$ maps $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^+_{\textit{Coh}}(\mathcal{O}_ Z)$.

**Proof.**
The question is local on $X$, hence we may assume that $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(B)$ with $A$ Noetherian and $A \to B$ surjective. In this case, we can apply Lemma 48.9.5 to translate the question into algebra. The corresponding algebra result is a consequence of Dualizing Complexes, Lemma 47.13.4.
$\square$

## 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Your email address will not be published.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)

There are also: