Lemma 31.8.3 (0CUD)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.8: Relative weak assassin
Lemma 31.8.3 (cite)

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Lemma 31.8.3. Let $f : X \to S$ be a morphism of schemes. Let $i : Z \to X$ be a finite morphism. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ Z$ -module. Then $\text{WeakAss}_{X/S}(i_*\mathcal{F}) = i(\text{WeakAss}_{Z/S}(\mathcal{F}))$ .

Proof. Let $i_ s : Z_ s \to X_ s$ be the induced morphism between fibres. Then $(i_*\mathcal{F})_ s = i_{s, *}(\mathcal{F}_ s)$ by Cohomology of Schemes, Lemma 30.5.1 and the fact that $i$ is affine. Hence we may apply Lemma 31.6.3 to conclude. $\square$

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