1
$\begingroup$

I have a finite étale morphism $f \colon Z \rightarrow X$ and I define the sheaf of set $\mathcal{F}_Z$ on $X_{\acute{e}tale}$ as: $$ \mathcal{F}_Z(U) = Hom_X(U,Z)$$ this is a locally constant sheaf. I want to prove that the pullback of this sheaf on $Z_{\acute{e}tale}$ is constant. So basically I want to prove that for any étale map $g \colon U \rightarrow Z$ the number of map $g' \colon U \rightarrow Z$ such that $g' \circ f = g \circ f$ is $deg(f)$.

$\endgroup$
1
  • 5
    $\begingroup$ You can't prove it, because it is not true, for example, for a non-Galois connected étale cover. $\endgroup$
    – Angelo
    Dec 8, 2019 at 16:35

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.