The Stacks project

Lemma 37.24.5. Let $f : X \to Y$ be a morphism of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation. Let $U \subset X$ be an open subscheme such that $U \to Y$ is quasi-compact. Then the set

\[ E = \{ y \in Y \mid \text{Ass}_{X_ y}(\mathcal{F}_ y) \subset U_ y\} \]

is locally constructible in $Y$.

Proof. Let $y \in Y$. We have to show that there exists an open neighbourhood $V$ of $y$ in $Y$ such that $E \cap V$ is constructible in $V$. Thus we may assume that $Y$ is affine. Write $Y = \mathop{\mathrm{Spec}}(A)$ and $A = \mathop{\mathrm{colim}}\nolimits A_ i$ as a directed limit of finite type $\mathbf{Z}$-algebras. By Limits, Lemma 32.10.1 we can find an $i$ and a morphism $f_ i : X_ i \to \mathop{\mathrm{Spec}}(A_ i)$ of finite presentation whose base change to $Y$ recovers $f$. After possibly increasing $i$ we may assume there exists a quasi-coherent $\mathcal{O}_{X_ i}$-module $\mathcal{F}_ i$ of finite presentation whose pullback to $X$ is isomorphic to $\mathcal{F}$, see Limits, Lemma 32.10.2. After possibly increasing $i$ one more time we may assume there exists an open subscheme $U_ i \subset X_ i$ whose inverse image in $X$ is $U$, see Limits, Lemma 32.4.11. By Lemma 37.24.4 it suffices to prove the lemma for $f_ i$. Thus we reduce to the case where $Y$ is the spectrum of a Noetherian ring.

We will use the criterion of Topology, Lemma 5.16.3 to prove that $E$ is constructible in case $Y$ is a Noetherian scheme. To see this let $Z \subset Y$ be an irreducible closed subscheme. We have to show that $E \cap Z$ either contains a nonempty open subset or is not dense in $Z$. This follows from Lemmas 37.24.2 and 37.24.3 applied to the base change $(X, \mathcal{F}, U) \times _ Y Z$ over $Z$. $\square$


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