Chasing maximal pro-p Galois groups via 1-cyclotomicity

Abstract

Let p be a prime. A cohomologically Kummerian oriented pro-p group is a pair consisting of a pro-p group G together with a continuous G-module Zp(θ) isomorphic to Zp as an abelian pro-p group, such that the natural map in cohomology) is surjective for every n ≥ 1. One has a 1cyclotomic oriented pro-p group if cohomological Kummerianity holds for every closed subgroup. By Kummer theory, the maximal pro-p Galois group of a field containing a root of 1 of order p, together with the 1st Tate twist of Zp, is 1cyclotomic.We prove that cohomological Kummerianity is preserved by certain quotients of pro-p groups, and we extend the group-theoretic characterization of cohomologically Kummerian oriented pro-p groups, established by I. Efrat and the author, to the nonfinitely generated case. We employ these results to find interesting new examples of pro-p groups which do not occur as absolute Galois groups, which other methods fail to detect.