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Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$, and suppose we have a surjective finite étale morphism $f:X\rightarrow Y$ (actually $Y=X/G$ for a free action of a finite group $G$), and an abelian sheaf $\mathcal{F}$ on the étale site $Y_{\mathrm{ét}}$ such that $f^*\mathcal{F}$ is a constant sheaf with stalk $\mathbb{Z}^r$ for some $r\geq 1$. What can be said about the sheaf $\mathcal{F}$? Can we conclude that $\mathcal{F}$ is also a constant sheaf?

I apologize beforehand if this is obvious... I am still in learning stage in étale cohomology. Any reference/suggestions regarding this would be appreciated.

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    $\begingroup$ No, $\mathscr{F}$ can be non-constant. In your case, there is an action of $G$ on $\mathbb{Z}^r$, and $\mathscr{F}$ is constant if and only if this action is trivial. For instance if $G=\mathbb{Z}/2$ acting on $\mathbb{Z}$ by changing sign, $\mathscr{F}$ is a rank 1 non-constant sheaf. $\endgroup$
    – abx
    Commented Apr 3, 2022 at 6:37
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    $\begingroup$ Such sheaves correspond to homomorphisms $G\to \mathrm{GL}_r(\mathbb{Z})$ and so are not necessarily constant. $\endgroup$
    – Piotr Achinger
    Commented Apr 3, 2022 at 6:39
  • $\begingroup$ @abx thanks for your comment. Please suggest a reference to read this correspondence that you and Piotr Achinger are pointing to... I have only found similar statements in the context of local systems or when the stalks are finite. I would really like to know in what generality it holds. $\endgroup$
    – Hajime_Saito
    Commented Apr 3, 2022 at 15:32
  • $\begingroup$ @PiotrAchinger Please suggest a reference for this correspondence... it would really help me. $\endgroup$
    – Hajime_Saito
    Commented Apr 3, 2022 at 15:33
  • $\begingroup$ @Hajime_Saito One possible reference is 5.8.1 of Etale cohomology theory by Lei Fu. See also 3.2.12 in loc. cit. $\endgroup$
    – A.B.
    Commented Apr 3, 2022 at 16:52

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