The Stacks project

Lemma 75.13.5. Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then $E$ is a perfect object of $D(\mathcal{O}_ X)$ if and only if $\epsilon ^*E$ is a perfect object of $D(\mathcal{O}_{\acute{e}tale})$. Here $\epsilon $ is as in (

Proof. The easy implication follows from the general result contained in Cohomology on Sites, Lemma 21.47.5. For the converse, we can use the equivalence of Cohomology on Sites, Lemma 21.47.4 and the corresponding results for pseudo-coherent and complexes of finite tor dimension, namely Lemmas 75.13.2 and 75.13.3. Some details omitted. $\square$

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Your email address will not be published.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 08HG. Beware of the difference between the letter 'O' and the digit '0'.