The Stacks project

Lemma 70.4.10. Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite presentation. Let $U \subset X$ be an open subspace 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. Note that since $Y$ is a scheme, it makes sense to take the fibres $X_ y = \mathop{\mathrm{Spec}}(\kappa (y)) \times _ Y X$. (Also, by our definitions, the set $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is exactly the fibre of $\text{Ass}_{X/Y}(\mathcal{F}) \to Y$ over $y$, but we won't need this.) The question is local on $Y$, indeed, we have to show that $E$ is constructible if $Y$ is affine. In this case $X$ is quasi-compact. Choose an affine scheme $W$ and a surjective ├ętale morphism $\varphi : W \to X$. Then $\text{Ass}_{X_ y}(\mathcal{F}_ y)$ is the image of $\text{Ass}_{W_ y}(\varphi ^*\mathcal{F}_ y)$ for all $y \in Y$. Hence the lemma follows from the case of schemes for the open $\varphi ^{-1}(U) \subset W$ and the morphism $W \to Y$. The case of schemes is More on Morphisms, Lemma 37.25.5. $\square$


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