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 ├ętale 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 ├ętale 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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