The Stacks project

Lemma 32.19.4. Let $f : X \to Y$ be a morphism of schemes. Let $d \geq 0$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Assume

  1. $f$ is a proper morphism of finite presentation all of whose fibres have dimension $\leq d$,

  2. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

Then $R^ df_*\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

Proof. The proof is exactly the same as the proof of Lemma 32.19.3 except that the third paragraph can be skipped. We omit the details. $\square$

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Your email address will not be published.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 0EX5. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 32.19.4 as pdf