The Stacks project

Lemma 30.9.9. Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is finite and $Y$ locally Noetherian. Then $R^ pf_*\mathcal{F} = 0$ for $p > 0$ and $f_*\mathcal{F}$ is coherent if $\mathcal{F}$ is coherent.

Proof. The higher direct images vanish by Lemma 30.2.3 and because a finite morphism is affine (by definition). Note that the assumptions imply that also $X$ is locally Noetherian (see Morphisms, Lemma 29.15.6) and hence the statement makes sense. Let $\mathop{\mathrm{Spec}}(A) = V \subset Y$ be an affine open subset. By Morphisms, Definition 29.44.1 we see that $f^{-1}(V) = \mathop{\mathrm{Spec}}(B)$ with $A \to B$ finite. Lemma 30.9.1 turns the statement of the lemma into the following algebra fact: If $M$ is a finite $B$-module, then $M$ is also finite viewed as a $A$-module, see Algebra, Lemma 10.7.2. $\square$

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