The Stacks project

Theorem 77.4.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume

  1. $X \to Y$ is locally of finite presentation,

  2. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type, and

  3. the set of weakly associated points of $Y$ is locally finite in $Y$.

Then $U = \{ x \in |X| : \mathcal{F}\text{ flat at }x\text{ over }Y\} $ is open in $X$ and $\mathcal{F}|_ U$ is an $\mathcal{O}_ U$-module of finite presentation and flat over $Y$.

Proof. Condition (3) means that if $V \to Y$ is a surjective étale morphism where $V$ is a scheme, then the weakly associated points of $V$ are locally finite on the scheme $V$. (Recall that the weakly associated points of $V$ are exactly the inverse image of the weakly associated points of $Y$ by Divisors on Spaces, Definition 71.2.2.) Having said this the question is étale local on $X$ and $Y$, hence we may assume $X$ and $Y$ are schemes. Thus the result follows from More on Flatness, Theorem 38.13.6. $\square$


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Your email address will not be published.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