The Stacks project

Lemma 73.6.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. 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 representable. Choose a scheme $V$ and a surjective ├ętale morphism $V \to Y$. Then $U = V \times _ Y X$ is a scheme with a surjective ├ętale morphism towards $X$ and a finite morphism $\psi : U \to V$ (the base change of $f$). Since $\psi _*(\mathcal{F}|_ U) = f_*\mathcal{F}|_ V$ the result of the lemma follows immediately from the schemes version which is Descent, Lemma 35.7.9. $\square$


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