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I have a smooth projective $k$-scheme $X$ with a local system $F$ (locally constant sheaf) of finite dimensional $k$-vector spaces (on étale topology). My question is whether there exists a finite étale morphism $f \colon Y \rightarrow X$ such that pullback of $F$ is a constant sheaf on $Y$.

I am still not familiar with the theory of étale fundamental groups, but if we work in traditional topology then there is a correspondence between local systems of $k$-vector spaces and $k$-vector space representations of $\pi_1(X,x)$. Moreover if $F$ correspond to $\phi \colon \pi_1(X, x) \rightarrow V$, then $f^{-1}F$ correspond to $\phi' \colon \pi_1(Y,y) \rightarrow \pi_1(X,x) \rightarrow V$.

Is there a similar correpondence for étale fundamental group?

From that $f^{-1}F$ is trivial if $Im(\phi')$ is trivial, i.e. $Im(\pi_1(Y,y)) \subset ker(\phi)$. In topology we can find a cover such that the image is exactly $ker(\phi)$.

Can we do the same for étale fundamental group? I suppose that since I am looking for finite étale morphism, I need the condition that $ker(\phi)$ has finite index.

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  • $\begingroup$ In your definition of local system, you mean locally constant for the étale topology (for the Zariski topology a locally constant sheaf is constant). $\endgroup$ – abx Dec 5 '19 at 14:45
  • $\begingroup$ Oh, that's true. Thank you I will fix that. This is actually a topology question that I want to answer algebraically. $\endgroup$ – Anh Dũng Lê Dec 5 '19 at 14:51
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Yes, there is such a correspondence. This is mentioned in Milne's étale cohomology and in other places. You actually want the kernel to be an open subgroup and not just finite index, as only these correspond to finite étale covers

But the correspondence covers only continuous representations, where $k$ is normally given the discrete topology, so you're fine because a stabilizer of a continuous representation on a discrete space is open and because the vector space is finite dimensional.

The only exception is $\ell$-adic local systems (and some similar constructs with more general coefficients), which are defined in a more complicated way than just locally constant sheaves (as certain inverse systems of sheaves up to equivalence) which correspond to representations of $\pi_1$ continuous with respect to the $\ell$-adic topology. In this case it is not true that there is a finite etale pullback that trivializes the local system.

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    $\begingroup$ @AnhDũngLê In the online version jmilne.org/math/CourseNotes/LEC.pdf Proposition 6.15 - it of course works over an arbitrary base field. $\endgroup$ – Will Sawin Dec 7 '19 at 19:45
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    $\begingroup$ @AnhDũngLê Oh I see what you mean - sorry I misinterpreted. I think it should work as long as you take continuous representations with the discrete topology on $k$. But I don't know why would want to do this.. $\endgroup$ – Will Sawin Dec 7 '19 at 22:51
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    $\begingroup$ @AnhDũngLê This is probably extreme overkill but you can get this out of Lemmas 7.4.5 and 7.4.10 of arxiv.org/pdf/1309.1198.pdf $\endgroup$ – Will Sawin Dec 7 '19 at 23:02
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    $\begingroup$ @AnhDũngLê Lemma 7.4.5 there works for sheaves of sets, which includes arbitrary representations. The local field lemma is needed because the topology on the local field may be relevant. But the topology on $\mathbb C$ is never relevant. $\endgroup$ – Will Sawin Dec 8 '19 at 1:38
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    $\begingroup$ @AnhDũngLê Yeah, that's what I'm suggesting. The point is because it is continuous on a discrete set, the stabilizer is open, and because it has a finite basis, the kernel is a finite intersection of stabilizers and is open. $\endgroup$ – Will Sawin Dec 9 '19 at 13:26

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