The Stacks project

Lemma 37.25.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\xi \in \text{Ass}_{X/S}(\mathcal{F})$ and set $Z = \overline{\{ \xi \} } \subset X$. If $f$ is locally of finite type and $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, then there exists a nonempty open $V \subset Z$ such that for every $s \in f(V)$ the generic points of $V_ s$ are elements of $\text{Ass}_{X/S}(\mathcal{F})$.

Proof. We may replace $S$ by an affine open neighbourhood of $f(\xi )$ and $X$ by an affine open neighbourhood of $\xi $. Hence we may assume $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$ and that $f$ is given by the finite type ring map $A \to B$, see Morphisms, Lemma 29.15.2. Moreover, we may write $\mathcal{F} = \widetilde{M}$ for some finite $B$-module $M$, see Properties, Lemma 28.16.1. Let $\mathfrak q \subset B$ be the prime corresponding to $\xi $ and let $\mathfrak p \subset A$ be the corresponding prime of $A$. By assumption $\mathfrak q \in \text{Ass}_ B(M \otimes _ A \kappa (\mathfrak p))$, see Algebra, Remark 10.65.6 and Divisors, Lemma 31.2.2. With this notation $Z = V(\mathfrak q) \subset \mathop{\mathrm{Spec}}(B)$. In particular $f(Z) \subset V(\mathfrak p)$. Hence clearly it suffices to prove the lemma after replacing $A$, $B$, and $M$ by $A/\mathfrak pA$, $B/\mathfrak pB$, and $M/\mathfrak pM$. In other words we may assume that $A$ is a domain with fraction field $K$ and $\mathfrak q \subset B$ is an associated prime of $M \otimes _ A K$.

At this point we can use generic flatness. Namely, by Algebra, Lemma 10.118.3 there exists a nonzero $g \in A$ such that $M_ g$ is flat as an $A_ g$-module. After replacing $A$ by $A_ g$ we may assume that $M$ is flat as an $A$-module.

In this case, by Algebra, Lemma 10.65.4 we see that $\mathfrak q$ is also an associated prime of $M$. Hence we obtain an injective $B$-module map $B/\mathfrak q \to M$. Let $Q$ be the cokernel so that we obtain a short exact sequence

\[ 0 \to B/\mathfrak q \to M \to Q \to 0 \]

of finite $B$-modules. After applying generic flatness Algebra, Lemma 10.118.3 once more, this time to the $B$-module $Q$, we may assume that $Q$ is a flat $A$-module. In particular we may assume the short exact sequence above is universally injective, see Algebra, Lemma 10.39.12. In this situation $(B/\mathfrak q) \otimes _ A \kappa (\mathfrak p') \subset M \otimes _ A \kappa (\mathfrak p')$ for any prime $\mathfrak p'$ of $A$. The lemma follows as a minimal prime $\mathfrak q'$ of the support of $(B/\mathfrak q) \otimes _ A \kappa (\mathfrak p')$ is an associated prime of $(B/\mathfrak q) \otimes _ A \kappa (\mathfrak p')$ by Divisors, Lemma 31.2.9. $\square$

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