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Massey products in Galois cohomology and the Elementary Type Conjecture

Claudio Quadrelli

Abstract

Let $p$ be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to the strengthened version of Minač--Tân's Massey Vanishing Conjecture in the case of finitely generated maximal pro-$p$ Galois groups whose pro-$p$ cyclotomic character has torsion-free image. Consequently, the maximal pro-$p$ Galois group of a field $\mathbb{K}$ containing a root of 1 of order $p$ (and also \sqrt{-1} if $p=2$) satisfies the strong $n$-Massey vanishing property for every $n>2$ (which is equivalent to the cup-defining $n$-Massey product property for every $n>2$, as defined by Minač--Tân) in several relevant cases.

Massey products in Galois cohomology and the Elementary Type Conjecture

Abstract

Let be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to the strengthened version of Minač--Tân's Massey Vanishing Conjecture in the case of finitely generated maximal pro- Galois groups whose pro- cyclotomic character has torsion-free image. Consequently, the maximal pro- Galois group of a field containing a root of 1 of order (and also \sqrt{-1} if ) satisfies the strong -Massey vanishing property for every (which is equivalent to the cup-defining -Massey product property for every , as defined by Minač--Tân) in several relevant cases.
Paper Structure (12 sections, 13 theorems, 68 equations)

This paper contains 12 sections, 13 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Massey products and pro-$p$ groups
  3. Massey products in Galois cohomology
  4. Massey products and upper unitriangular matrices
  5. Commutators of upper unitriangular matrices
  6. Oriented pro-$p$ groups
  7. Orientations
  8. Oriented pro-$p$ groups of elementary type
  9. Kummerian oriented pro-$p$ groups
  10. Oriented pro-$p$ groups and vanishing of Massey products
  11. Semidirect products
  12. Proof of Theorem 1.2

Key Result

Theorem 1.2

Let $G$ be a pro-$p$ group of elementary type. If $p=2$ assume further that the image of the orientation associated to $G$ is a subgroup of $1+4\mathbb{Z}_2$. Then $G$ satisfies the strong $n$-Massey vanishing property for every $n>2$.

Theorems & Definitions (36)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 26 more