On the Bogomolov-Positselski Conjecture

Let $p$ be a prime. An oriented pro-$p$ group $(G,\theta)$ is said to have the Bogomolov--Positselski property if it is Kummerian and if $I_\theta(G)$ 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.