# Group cohomology of fundamental group of a curve

The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this rudimenatry question.

Let $C$ be a connected hyperbolic curve defined over an algebraically closed field of characteristic 0. Let us write $\pi_1$ for the maximal pro-$p$ quotient of the etale fundamental group of $C$ with some choice of base-point. There is a statement which claims that, for any integers $i,r$, there exists a natural morphism

$H^i(\pi_1,\mathbb{Z}_p(r))\to H^i(C,\mathbb{Z}_p(r))$

which in fact an isomoprhism.

My questions are:

1. How is the natural map defined?
2. Why is it an isomorphism?

Surely any comment would helpful, but if anyone can explain it with details, that would be greatly helpful.

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:

1. M. Olsson *On Faltings' method of almost etale extensions" (MR 2483956), section 5, p. 30.
2. 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).
3. M. Artin, SGA4.3, Exp. XI, section 4, Variante 4.6.
4. G. Faltings p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255– 299 (MR 924705): chapter II, section 2.
5. 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.