The Stacks project

Lemma 76.7.9. In Situation 76.7.8.

  1. The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology.

  2. If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_{flat}$ is limit preserving.

Proof. Part (1) follows from the following statement: If $T' \to T$ is a surjective flat morphism of algebraic spaces over $Y$, then $\mathcal{F}_{T'}$ is flat over $T'$ if and only if $\mathcal{F}_ T$ is flat over $T$, see Morphisms of Spaces, Lemma 66.31.3. Part (2) follows from Limits of Spaces, Lemma 69.6.12 if $f$ is also quasi-separated (i.e., $f$ is of finite presentation). For the general case, first reduce to the case where the base is affine and then cover $X$ by finitely many affines to reduce to the quasi-separated case. 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Your email address will not be published.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 0CWZ. Beware of the difference between the letter 'O' and the digit '0'.