The Stacks project

Lemma 85.28.2. Let $f : V \to U$ be a cartesian morphism of simplicial schemes. Assume the morphisms $d^ n_ j : U_ n \to U_{n - 1}$ are flat and the morphisms $V_ n \to U_ n$ are quasi-compact and quasi-separated. For a quasi-coherent module $\mathcal{G}$ on $V_{Zar}$ the pushforward $f_*\mathcal{G}$ is a quasi-coherent module on $U_{Zar}$.

Proof. If $\mathcal{F} = f_* \mathcal{G}$, then $\mathcal{F}_ n = f_{n , *}\mathcal{G}_ n$ by Lemma 85.2.4. The maps $\mathcal{F}(\varphi )$ are defined using the base change maps, see Cohomology, Section 20.17. The sheaves $\mathcal{F}_ n$ are quasi-coherent by Schemes, Lemma 26.24.1 and the fact that $\mathcal{G}_ n$ is quasi-coherent by Lemma 85.12.10. The base change maps along the degeneracies $d^ n_ j$ are isomorphisms by Cohomology of Schemes, Lemma 30.5.2 and the fact that $\mathcal{G}$ is cartesian by Lemma 85.12.10. Hence $\mathcal{F}$ is cartesian by Lemma 85.12.2. Thus $\mathcal{F}$ is quasi-coherent by Lemma 85.12.10. $\square$

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Your email address will not be published.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