On the Bogomolov–Positselski Conjecture

Let p be a prime. An oriented pro-p group $(G,θ)$ is said to have the Bogomolov--Positselski property if it is Kummerian and if I_θ(G) is a free pro-p group. In this paper, we provide a new criterion for an oriented pro-p group to satisfy the Bogomolov--Positselski property. This criterion builds on earlier work of Positselski (arXiv:1405.0965) and Quadrelli--Weigel (arXiv:2103.12438), relates their approaches, and answers a question raised in (arXiv:2103.12438). Under additional assumptions, we obtain two further sufficient criteria. The first is analogous to a Merkurjev--Suslin type statement. The second allows one to weaken the hypotheses appearing in Positselski's criterion (arXiv:1405.0965 Theorem 2). Finally, we show that the stronger conditions are satisfied by pro-p groups of elementary type. As a consequence, the Elementary Type Conjecture implies Positselski's ``Module Koszulity Conjecture 1'' (arXiv:1008.0095) for fields with finitely generated maximal pro-p Galois group.