The Stacks project

Lemma 37.59.9. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $R\pi _*E$ is $m$-pseudo-coherent relative to $S$.

Proof. Translation of the result of More on Algebra, Lemma 15.81.5 into the language of schemes. Observe that $R\pi _*$ indeed maps $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Schemes, Lemma 36.4.1. To do the translation use Lemma 37.59.6. $\square$

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Your email address will not be published.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 09UK. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 37.59.9 as pdf