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The Stacks project

Lemma 71.2.10. Let S be a scheme. Let f : X \to Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Let y \in |Y| be a point which is not in the image of |f|. Then y is not weakly associated to f_*\mathcal{F}.

Proof. By Morphisms of Spaces, Lemma 67.11.2 the \mathcal{O}_ Y-module f_*\mathcal{F} is quasi-coherent hence the lemma makes sense. Choose an affine scheme V, a point v \in V, and an étale morphism V \to Y mapping v to y. We may replace f : X \to Y, \mathcal{F}, y by X \times _ Y V \to V, \mathcal{F}|_{X \times _ Y V}, v. Thus we may assume Y is an affine scheme. In this case X is quasi-compact, hence we can choose an affine scheme U and a surjective étale morphism U \to X. Denote g : U \to Y the composition. Then f_*\mathcal{F} \subset g_*(\mathcal{F}|_ U). By Lemma 71.2.4 we reduce to the case of schemes which is Divisors, Lemma 31.5.9. \square


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