The Stacks project

Lemma 38.26.2. Let $S$ be a local scheme with closed point $s$. Let $f : X \to S$ be locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume that

  1. every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of the closed fibre $X_ s$1,

  2. $\mathcal{F}$ is flat over $S$ at every point of $X_ s$.

Then $\mathcal{F}$ is flat over $S$.

Proof. This is immediate from the fact that it suffices to check for flatness at points of the relative assassin of $\mathcal{F}$ over $S$ by Theorem 38.26.1. $\square$

[1] For example this holds if $f$ is finite type and $\mathcal{F}$ is pure along $X_ s$, or if $f$ is proper.

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Your email address will not be published.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 05UB. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 38.26.2 as pdf