The Stacks project

Lemma 28.24.4. Let $f : X \to Y$ be an affine morphism of schemes over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is flat over $S$ if and only if $f_*\mathcal{F}$ is flat over $S$.

Proof. By Lemma 28.24.2 and the fact that $f$ is an affine morphism, this reduces us to the affine case. Say $X \to Y \to S$ corresponds to the ring maps $C \leftarrow B \leftarrow A$. Let $N$ be the $C$-module corresponding to $\mathcal{F}$. Recall that $f_*\mathcal{F}$ corresponds to $N$ viewed as a $B$-module, see Schemes, Lemma 25.7.3. Thus the result is clear. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 28.24: Flat morphisms

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Your email address will not be published.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