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I'm looking for a reference for the following statement:

Let $X$ be a variety (over an algebraically closed field $k$), and let $F$ be a locally constant etale sheaf. Let $x \in X(k)$. Then $ \mathrm{H}^i(X, F) = \mathrm{H}^i(\pi_1(X,x), F_x)$.

It seems to me that it ought to follow as:

i) The abelian category of locally constant sheaves with values in a finite commutative groups on X is equivalent to the category of finitely generated torsion $\mathbb{Z}[\pi_1(X,x)]$-modules; and

ii) The $\mathrm{H^0}$'s are the same (in both cases), and the $\mathrm{H^i}$ form universal delta functors (in both cases).

But I can't find a reference to this in the literature.

Also, in case $k = \bar{\mathbb{F}}_q$, has anyone computed the action of Frobenius on $\mathrm{H}^1(\pi_1(X,x), F_x)$ in terms of 1-cocyles?

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    $\begingroup$ This is completely false. If $X$ is a simply-connected projective surface, $H^2(X,\mu _n)$ is nonzero. $\endgroup$
    – abx
    Commented Jul 28, 2016 at 6:55
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    $\begingroup$ Your (ii) won't quite work as suggested because sheaf cohomology is not erasable on the category of locally constant sheaves. Also, for $X = \mathbf{P}^1_k$ we have $\pi_1(X,x)=1$ but ${\rm{H}}^2(X, \mu_n) = \mathbf{Z}/(n)$ for $n$ not divisible by ${\rm{char}}(k)$. Might you be satisfied to restrict to affine (smooth) curves? Some discussion of your motivation may be helpful for context for what you wish to do with this. $\endgroup$
    – nfdc23
    Commented Jul 28, 2016 at 7:11
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    $\begingroup$ to expand @abx's comment : this would be fine if (and only if) $X$ is an algebraic $K(\pi,1)$, see for instance the answers to mathoverflow.net/questions/239195/… for references on this notion. $\endgroup$
    – Niels
    Commented Jul 28, 2016 at 7:38
  • $\begingroup$ Thanks. @nfdc23 Indeed I'm only interested in affine smooth curves. I suppose one can't extend to a larger category of sheaves as there's no corresponding extension of the $\mathbb{Z}[\pi_1(X,x)]$ modules? $\endgroup$
    – user51764
    Commented Jul 28, 2016 at 7:55
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    $\begingroup$ Are you also satisfied to limit yourself to such $F$ with finite fibers of order not divisible by ${\rm{char}}(k)$? (Once you have that functorially in $F$ you can then pass to inverse limits to handle lisse $\mathbf{Z}_{\ell}$-sheaves for $\ell \ne {\rm{char}}(k)$, for example, so the real content for the latter is the finite case.) $\endgroup$
    – nfdc23
    Commented Jul 28, 2016 at 15:07

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This isn't true in the same way that the cohomology of an ordinary (path-connected) topological space $X$ is not the same as the group cohomology of its $\pi_1$, despite the fact that local systems on $X$ are the same as $\pi_1$-modules and $H^0$ agrees for both. Taking the cohomology of a local system is simply not the same functor as taking the cohomology of the corresponding $\pi_1$-module: cohomology of local systems is sensitive to the entire homotopy type of $X$. Here "homotopy type" should be replaced by something like "etale homotopy type," I guess.

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My question was half-baked and perhaps too elementary for MO. But... based upon the suggestion of @Niels, I decided to answer my own question for people wondering about the precise relationship between the group cohomology of the fundamental group and etale cohomology. My reference is part 2.1.2 of Piotr Achinger's thesis but Appendix A of Jacob Stix's thesis is also very good.

Basically, under minimal assumptions on X there are morphisms $$\rho_i : \mathrm{H}^i(\pi_1(X,x), F_x) \rightarrow \mathrm{H}^i(X, F).$$ To say that $X$ is an algebraic $K(\pi,1)$-space is equivalent to the assertion that these morphisms are isomorphisms for each $i$.

Prototypical examples are $\mathrm{Spec}(K)$ (for any field $K$), a scheme of cohomological dimension $\leq 1$ (e.g. an affine curve), a smooth connected curve which is not geometrically isomorphic to $\mathbb{P}^1$, Abelian varieties.

One reason why the notion is important is that locally a smooth scheme in characteristic $0$ is covered by $K(\pi,1)$ spaces, and this was needed by Artin to prove the comparision theorem between 'etale cohomology and ordinary analytic cohomology. See section 1.1 of Achinger's thesis for further applications in arithmetic geometry.

The morphisms $\rho_i$ are constructed in several steps, roughly as follows:
i) $\pi_1$ is a profinite group and has a classifying space, denoted by $B \pi_1$. (Which is basically the category of finite $\pi_1$-sets endowed with a topology where $\{f_i : U_i \rightarrow U\}$ is a cover iff $\bigcup_i f_i(U_i) = U$).
ii) A sheaf (of finite Abelian groups) on $B\pi_1$ is just a finite Abelian group with a continuous $\pi_1$-action. Essentially because the categories of finite $\pi_1$-sets and finite etale covers of $X$ are equivalent.
iii) Sheaves on $B\pi_1$ are are equivalent to sheaves on $X$ on the finite etale site.
iv) There is a natural morphism from sheaves on the etale cite of X to sheaves on the finite etale site of X.

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