The Stacks project

Lemma 37.25.2. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U \subset X$ be an open subscheme. Assume

  1. $f$ is of finite type,

  2. $\mathcal{F}$ is of finite type,

  3. $Y$ is irreducible with generic point $\eta $, and

  4. $\text{Ass}_{X_\eta }(\mathcal{F}_\eta )$ is not contained in $U_\eta $.

Then there exists a nonempty open subscheme $V \subset Y$ such that for all $y \in V$ the set $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is not contained in $U_ y$.

Proof. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$, see Morphisms, Definition 29.5.5. Then $Z_\eta $ is the scheme theoretic support of $\mathcal{F}_\eta $ (Morphisms, Lemma 29.25.14). Hence the generic points of irreducible components of $Z_\eta $ are contained in $\text{Ass}_{X_\eta }(\mathcal{F}_\eta )$ by Divisors, Lemma 31.2.9. Hence we see that $Z_\eta \cap U_\eta = \emptyset $. Thus $T = Z \setminus U$ is a closed subset of $Z$ with $T_\eta = \emptyset $. If we endow $T$ with the induced reduced scheme structure then $T \to Y$ is a morphism of finite type. By Lemma 37.24.1 there is a nonempty open $V \subset Y$ with $T_ V = \emptyset $. Then $V$ works. $\square$

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