The Stacks project

Lemma 71.3.2. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $X$ is locally Noetherian, then we have

\[ \text{WeakAss}_ Y(f_*\mathcal{F}) = f(\text{WeakAss}_ X(\mathcal{F})) \]

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Set $U = X \times _ Y V$. Then $U \to V$ is an affine morphism of schemes and $U$ is locally Noetherian. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma 31.6.2. $\square$

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