Lemma 66.30.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module with scheme theoretic support $Z \subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.

Proof. Using the characterization of the scheme theoretic support as given in Lemma 66.15.3 and using the characterization of flat morphisms in terms of étale coverings in Lemma 66.30.5 we reduce to the case of schemes which is Morphisms, Lemma 29.25.14. $\square$

There are also:

• 2 comment(s) on Section 66.30: Flat morphisms

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.

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 089C. Beware of the difference between the letter 'O' and the digit '0'.