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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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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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