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.

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).