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.



pullback of a local system
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.


