Profinite Groups with a Cyclotomic $p$-Orientation

Abstract

Let $p$ be a prime. A continuous representation $\theta :G\to {\text{GL}}_1({\mathbb{Z}}_p)$ of a profinite group $G$ is called a cyclotomic $p$-orientation if for all open subgroups $U\subseteq G$ and for all $k,n\ge 1$ the natural maps $H^k(U,{\mathbb{Z}}_p(k)/p^n)\to H^k(U,{\mathbb{Z}}_p(k)/p)$ are surjective. Here ${\mathbb{Z}}_p(k)$ denotes the ${\mathbb{Z}}_p$-module of rank 1 with $U$-action induced by $\theta {|}_U^k$. By the Rost-Voevodsky theorem, the cyclotomic character of the absolute Galois group $G_{\mathbb{K}}$ of a field $\mathbb{K}$ is, indeed, a cyclotomic $p$-orientation of $G_{\mathbb{K}}$. We study profinite groups with a cyclotomic $p$-orientation. In particular, we show that cyclotomicity is preserved by several operations on profinite groups, and that Bloch-Kato pro-$p$ groups with a cyclotomic $p$-orientation satisfy a strong form of Tits' alternative and decompose as semi-direct product over a canonical abelian closed normal subgroup.