Lemma 35.7.9 (05B3)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 35: Descent
Section 35.7: Descent of finiteness properties of modules
Lemma 35.7.9 (cite)

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Lemma 35.7.9. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Assume $f$ is a finite morphism. Then $\mathcal{F}$ is an $\mathcal{O}_ X$ -module of finite type if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_ Y$ -module of finite type.

Proof. As $f$ is finite it is affine. This reduces us to the case where $f$ is the morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ given by a finite ring map $A \to B$ . Moreover, then $\mathcal{F} = \widetilde{M}$ is the sheaf of modules associated to the $B$ -module $M$ . Note that $M$ is finite as a $B$ -module if and only if $M$ is finite as an $A$ -module, see Algebra, Lemma 10.7.2. Combined with Properties, Lemma 28.16.1 this proves the lemma. $\square$

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