The Stacks project

Lemma 48.17.6. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Then $f^!$ maps $D_{\textit{Coh}}^+(\mathcal{O}_ Y)$ into $D_{\textit{Coh}}^+(\mathcal{O}_ X)$.

Proof. The question is local on $X$ hence we may assume that $X$ and $Y$ are affine schemes. In this case we can factor $f : X \to Y$ as

\[ X \xrightarrow {i} \mathbf{A}^ n_ Y \to \mathbf{A}^{n - 1}_ Y \to \ldots \to \mathbf{A}^1_ Y \to Y \]

where $i$ is a closed immersion. The lemma follows from By Lemmas 48.17.3 and 48.9.6 and Dualizing Complexes, Lemma 47.15.10 and induction. $\square$


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Your email address will not be published.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 0AU1. Beware of the difference between the letter 'O' and the digit '0'.