Lemma 71.4.10. 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 étale 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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