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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #994 by Johan Commelin on