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.

💡 Research Summary

The paper addresses the Bogomolov‑Positselski conjecture, which predicts that for a field K containing a primitive p‑th root of unity (and √‑1 when p = 2), the subgroup KθK(GK(p)) of the maximal pro‑p Galois group is a free pro‑p group. In the language of oriented pro‑p groups, this is equivalent to requiring that a Kummerian oriented pro‑p group (G, θ) have the Bogomolov‑Positselski property, i.e. that its “Kummer kernel” Kθ(G) be free.

Two previously known criteria for this property are:

1. Positselski’s Koszulity criterion (Theorem 2 in