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The Stacks project

Lemma 71.3.3. Let S be a scheme. Let f : X \to Y be a finite morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Then \text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\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 a finite morphism of schemes. 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 31.6.3. \square


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