Lemma 71.4.9 (0CV4)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 71: Divisors on Algebraic Spaces
Section 71.4: Relative weak assassin
Lemma 71.4.9 (cite)

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Lemma 71.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ . Let $i : Z \to X$ be a finite morphism. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Z$ -module. Then $\text{WeakAss}_{X/Y}(i_*\mathcal{G}) = i(\text{WeakAss}_{Z/Y}(\mathcal{G}))$ .

Proof. Follows from the case of schemes (Divisors, Lemma 31.8.3) by étale localization. Details omitted. $\square$

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