Lemma 56.63.3. Let $f : X \to Y$ be a finite étale morphism of schemes. If $\mathcal{F}$ is a (finite) locally constant sheaf of sets, (finite) locally constant sheaf of abelian groups, or (finite type) locally constant sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$, the same is true for $f_*\mathcal{F}$ on $Y_{\acute{e}tale}$.

Proof. The construction of $f_*$ commutes with étale localization. A finite étale morphism is locally isomorphic to a disjoint union of isomorphisms, see Étale Morphisms, Lemma 40.18.3. Thus the lemma says that if $\mathcal{F}_ i$, $i = 1, \ldots , n$ are (finite) locally constant 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. $\square$

Comment #994 by on

Suggested slogan: The pushforward of a locally constant sheaf along a finite étale morphism is a locally constant sheaf.

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