The Stacks project

Lemma 85.28.1. Let $f : V \to U$ be a morphism of simplicial schemes. Given a quasi-coherent module $\mathcal{F}$ on $U_{Zar}$ the pullback $f^*\mathcal{F}$ is a quasi-coherent module on $V_{Zar}$.

Proof. Recall that $\mathcal{F}$ is cartesian with $\mathcal{F}_ n$ quasi-coherent, see Lemma 85.12.10. By Lemma 85.2.4 we see that $(f^*\mathcal{F})_ n = f_ n^*\mathcal{F}_ n$ (some details omitted). Hence $(f^*\mathcal{F})_ n$ is quasi-coherent. The same fact and the cartesian property for $\mathcal{F}$ imply the cartesian property for $f^*\mathcal{F}$. Thus $\mathcal{F}$ is quasi-coherent by Lemma 85.12.10 again. $\square$

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Your email address will not be published.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 07TI. Beware of the difference between the letter 'O' and the digit '0'.