The Stacks project

38.26 Flat finite type modules, Part III

The following result is one of the main results of this chapter.

Theorem 38.26.1. Let $f : X \to S$ be locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $x \in X$ with image $s \in S$. The following are equivalent

  1. $\mathcal{F}$ is flat at $x$ over $S$, and

  2. for every $x' \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ which specializes to $x$ we have that $\mathcal{F}$ is flat at $x'$ over $S$.

Proof. It is clear that (1) implies (2) as $\mathcal{F}_{x'}$ is a localization of $\mathcal{F}_ x$ for every point which specializes to $x$. Set $A = \mathcal{O}_{S, s}$, $B = \mathcal{O}_{X, x}$ and $N = \mathcal{F}_ x$. Let $\Sigma \subset B$ be the multiplicative subset of $B$ of elements which act as nonzerodivisors on $N/\mathfrak m_ AN$. Assumption (2) implies that $\Sigma ^{-1}N$ is $A$-flat by the description of $\mathop{\mathrm{Spec}}(\Sigma ^{-1}N)$ in Lemma 38.7.1. On the other hand, the map $N \to \Sigma ^{-1}N$ is injective modulo $\mathfrak m_ A$ by construction. Hence applying Lemma 38.25.5 we win. $\square$

Now we apply this directly to obtain the following useful results.

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.

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