On the K(\pi,1)-property for rings of integers in the mixed case

TL;DR

This paper studies the Galois groups of maximal p-extensions unramified outside a finite set of primes in number fields, especially when primes dividing p are involved, and relates their cohomology to etale cohomology.

Contribution

It extends previous work on K(π,1) properties to the mixed case where primes dividing p are present inside and outside the set S.

Findings

G_S(p) often has cohomological dimension 2

Cohomology of G_S(p) is often isomorphic to etale cohomology

Results generalize previous tame case to mixed case

Abstract

We investigate the Galois group G_S(p) of the maximal p-extension unramified outside a finite set S of primes of a number field in the (mixed) case, when there are primes dividing p inside and outside S. We show that the cohomology of G_S(p) is "often" isomorphic to the etale cohomology of the scheme Spec(O_k S), in particular, G_S(p) is of cohomological dimension 2 then. We deduce this from the results in our previous paper "Rings of integers of type K(\pi,1)" (arXiv:0705.3372), which mainly dealt with the tame case.