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
\mathcal{F} is flat at x over S, and
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
every point of \text{Ass}_{X/S}(\mathcal{F}) specializes to a point of the closed fibre X_ s1,
\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
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