Group cohomology of fundamental group of a curve
My apologies for being terse, I don't have time at the moment for a more in-depth answer. Let me know if anything is unclear.
@1: Let $X$ be a noetherian scheme. Consider the `finite etale' site $FEt(X)$ of $X$: its objects are finite etale maps to $X$, with the etale topology. The forgetful functor to the etale site defines a morphism of sites $\rho:Et(X)\to FEt(X)$. Further, a choice of a geometric point $\bar x$ of $X$ induces a map $B\pi_1(X, \bar x)\to FEt(X)$, which is an equivalence if $X$ is connected, as we shall henceforth assume. Moreover, the functor $\rho^*$ identifies the category of locally constant abelian torsion sheaves on $FEt(X)$ (which are the same as continuous $\pi_1$-representations on finite abelian groups) with the category of lcc etale sheaves on $X$, the inverse being $\rho_*$. If $\mathcal{F}$ is an lcc sheaf of $X$, we get natural maps: $$ \rho^*: H^i(\pi_1(X, \bar x), \mathcal{F}_{\bar x}) = H^i(FEt(X), \rho_* \mathcal{F}) \to H^i(X, \mathcal{F}). $$ One can check without much hassle that these are always isomorphisms for $i\leq 1$. The case $i=1$ follows easily from the torsor interpretation of $H^1$. We call $X$ a $K(\pi, 1)$ scheme if these are isomorphisms for all $\mathcal{F}$ and all $i\geq 0$.
@2: In fact this map is an isomorphism for all lcc sheaves $\mathcal{F}$ on any curve $X$ which is not $\mathbb{P}^1$, in other words, curves $\neq \mathbb{P}^1$ are $K(\pi, 1)$'s. This of course agrees with the fact that an orientable surface which is not the 2-sphere has a contractible universal cover. To show this, we need to prove that $R^i\rho_* \mathcal{F}=0$ for $i>0$. The higher direct image $R^i \rho_* \mathcal{F}$ is the sheaf associated to the presheaf (on $FEt(X)$) $$ (f:Y\to X) \mapsto H^i(Y, f^*\mathcal{F}).$$ So it suffices to show that given a cohomology class $\zeta\in H^i(X, \mathcal{F})$, there exists a finite etale surjective $f:Y\to X$ such that $f^* \zeta = 0 \in H^i(Y, f^*\mathcal{F})$. In doing so, we can assume that $\mathcal{F}$ is constant, and reduce further to the case $\mathcal{F}=\mathbb{F}_\ell$ for some prime $\ell$ (maybe $=p$). If $i>2$, cohomology vanishes for dimensional reasons, and $i\leq 1$ has been considered in @1, so the interesting case is $i=2$. Again if $X$ is not complete, it is affine so $H^2$ vanishes for dimensional reasons, so we can assume $X$ projective. In this case we see that it is enough to find a finite etale cover $f:Y\to X$ whose degree is divisible by $\ell$. If $\ell\neq p$ (or if the Jacobian of $X$ is ordinary), this is always possible, because the prime-to-$p$ part of $\pi_1$ is the same as in char. 0. The case $\ell=p$ follows from Artin-Schreier theory. The Artin-Schreier exact sequence $$ 0\to \mathbb{F}_p \to \mathcal{O}_X \to \mathcal{O}_X\to 0 $$ where the map $\mathcal{O}_X\to\mathcal{O}_X$ is $1-F$, $F$ being the absolute Frobenius, shows that $H^2(X, \mathbb{F}_p)$ is the cokernel of $1-F:H^1(X, \mathcal{O}_X)\to H^1(X, \mathcal{O}_X)$. But $1$ (that is, the identity) is $k$-linear, while $F$ is $p$-linear, and it follows from easy $p$-linear algebra that $1-F$ is always surjective.
EDIT. References:
- M. Olsson *On Faltings' method of almost etale extensions" (MR 2483956), section 5, p. 30.
- Ahmed Abbes and Michel Gros, Topos co-évanescents et généralisations, arXiv:1107.2380, section 9 (to appear in "The p-adic Simpson Correspondence", Annals of Mathematics Studies, Vol. 193).
- M. Artin, SGA4.3, Exp. XI, section 4, Variante 4.6.
- G. Faltings p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255– 299 (MR 924705): chapter II, section 2.
- section 2.1 of my PhD thesis, available here.
The best thing you could do in my opinion is to have a look at
Appendix A
Algebraic $K(\pi, 1)$ Spaces
in
Stix, Jakob Projective anabelian curves in positive characteristic and descent theory for log-étale covers. Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2002.
http://www.ams.org/mathscinet-getitem?mr=2012864
http://www.math.uni-frankfurt.de/~stix/research/preprints/STIXdissB.pdf
where this issue (for the whole fundamental group and for the pro-$\ell$ quotient) is treated with great care.