The Stacks project

Lemma 59.73.4. Let $f : X \to Y$ be a finite étale morphism of schemes. Let $\Lambda $ be a Noetherian ring. If $\mathcal{F}$ is a constructible sheaf of sets, constructible sheaf of abelian groups, or constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$, the same is true for $f_*\mathcal{F}$ on $Y_{\acute{e}tale}$.

Proof. By Lemma 59.71.4 it suffices to check this Zariski locally on $Y$ and by Lemma 59.73.3 we may replace $Y$ by an étale cover (the construction of $f_*$ commutes with étale localization). A finite étale morphism is étale locally isomorphic to a disjoint union of isomorphisms, see Étale Morphisms, Lemma 41.18.3. Thus, in the case of sheaves of sets, the lemma says that if $\mathcal{F}_ i$, $i = 1, \ldots , n$ are constructible sheaves of sets, then $\prod _{i = 1, \ldots , n} \mathcal{F}_ i$ is too. This is clear. Similarly for sheaves of abelian groups and modules. $\square$

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