The Stacks project

Lemma 59.73.9. Let $f : X \to Y$ be a finite and finitely presented morphism of schemes. Let $\Lambda $ be a Noetherian ring. If $\mathcal{F}$ is a constructible sheaf of sets, abelian groups, or $\Lambda $-modules on $X_{\acute{e}tale}$, then $f_*\mathcal{F}$ is too.

Proof. It suffices to prove this when $X$ and $Y$ are affine by Lemma 59.71.4. By Lemmas 59.55.3 and 59.73.3 we may base change to any affine scheme surjective over $X$. By Lemma 59.72.3 this reduces us to the case of a finite étale morphism (because a thickening leads to an equivalence of étale topoi and even small étale sites, see Theorem 59.45.2). The finite étale case is Lemma 59.73.4. $\square$

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