The Stacks project

Lemma 85.28.3. 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. Then $f^*$ and $f_*$ form an adjoint pair of functors between the categories of quasi-coherent modules on $U_{Zar}$ and $V_{Zar}$.

Proof. We have seen in Lemmas 85.28.1 and 85.28.2 that the statement makes sense. The adjointness property follows immediately from the fact that each $f_ n^*$ is adjoint to $f_{n, *}$. $\square$

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Your email address will not be published.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