MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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: $X = Spec(O_K)\setminus S$)

share|cite|improve this question
Did you take a look at…? – Cam McLeman Apr 1 '11 at 19:56
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). – shenghao Apr 1 '11 at 21:42
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? – 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 $X$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.