What are cases when Galois cohomology groups are given by étale cohomology?

Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$.

What if $G = \pi_1(X)$ and $F \in Sh(X)$? Under what conditions do we have $H^p(X, F) = H^p(G, [F])$, where $[F]$ denotes a suitable $\pi_1(X)$-module associated with $F$? (Example for this: $X = Spec(O_K)\setminus S$)

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    $\begingroup$ Did you take a look at mathoverflow.net/questions/28749/…? $\endgroup$ – Cam McLeman Apr 1 '11 at 19:56
  • $\begingroup$ One is using etale topology and the other is using "finite etale topology", so in general there is a (Grothendieck) spectral sequence converging from one to the other. They agree for $H^0$ and $H^1$ (I assume your $F$ is locally constant or lisse), but not for higher degrees in general (unless, of course, the spectral sequence degenerates, which seems unusual). $\endgroup$ – shenghao Apr 1 '11 at 21:42
  • $\begingroup$ And for the example $X=\text{Spec }O_K\backslash S$ you mentioned, maybe you want to restrict to the case where $F$ is lisse of rank 1, so that one can apply abelian class field theory to get an explicit description of $\pi_1^{ab}$ at least? $\endgroup$ – shenghao Apr 1 '11 at 21:49

All Galois cohomology groups are given by etale cohomology, but you seem to be asking the opposite. For an etale sheaf $F$ on $X$, there is a Hochschild-Serre spectral sequence $H^{p}(\pi_{1}(X),H^{q}(\tilde{X},F))\implies H^{p+q}(X,F)$ where $\tilde{X}$ is the "universal covering scheme" of $X$. Don't expect this to collapse to give isomorphisms $H^{p}(\pi _{1}(X),F)\approx H^{p}(X,F)$ except when $X$ is spec of a field or a henselian local ring. There are a few fragmentary results. For example, $H^{1}(\pi _{1}(X),A)\approx H^{1}(X,A)$ when $X$ is an open subscheme of the spec of the ring of integers in a number field and $A$ is an abelian scheme on $X$.

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