Theorem 37.15.1. Let $S$ be a scheme. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module which is locally of finite presentation. Then

is open in $X$.

[IV Theorem 11.3.1, EGA]

Theorem 37.15.1. Let $S$ be a scheme. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module which is locally of finite presentation. Then

\[ U = \{ x \in X \mid \mathcal{F}\text{ is flat over }S\text{ at }x\} \]

is open in $X$.

**Proof.**
We may test for openness locally on $X$ hence we may assume that $f$ is a morphism of affine schemes. In this case the theorem is exactly Algebra, Theorem 10.129.4.
$\square$

Your email address will not be published.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.

## Comments (2)

Comment #2691 by Johan on

Comment #2719 by Takumi Murayama on