Rings of integers of type $K(\pi,1)$

TL;DR

This paper studies the Galois groups of maximal p-extensions unramified outside a finite set of primes in number fields, showing their cohomology often matches étale cohomology of certain schemes, especially when cohomological dimension is 2.

Contribution

It demonstrates conditions under which the Galois group’s cohomology aligns with étale cohomology, revealing new insights into the structure of these Galois groups.

Findings

G_S(p) often has cohomological dimension 2

Cohomology of G_S(p) is isomorphic to étale cohomology of Spec(O_k ackslash S)

Provides conditions for the isomorphism of cohomologies

Abstract

We investigate the Galois group $G_S(p)$ of the maximal $p$-extension unramified outside a finite $S$ of primes of a number field in the (tame) case, when no prime dividing $p$ is in $S$. We show that the cohomology of $G_S(p)$ is 'often' isomorphic to the etale cohomology of the scheme $\Spec(\O_k \sm S)$, in particular, $G_S(p)$ is of cohomological dimension~2 then.