Degenerate fourfold Massey products over arbitrary fields

TL;DR

This paper proves that certain Massey products over fields of characteristic not 2 always vanish when defined, constructs examples where some lower products vanish but higher ones are undefined, and shows the non-formality of the Galois cochain DGA in specific fields.

Contribution

It establishes the vanishing of degenerate fourfold Massey products over arbitrary fields and constructs fields with specific Massey product properties, answering a question of Positselski.

Findings

All defined mod 2 Massey products of the form ⟨a,b,c,a⟩ vanish over fields of characteristic ≠ 2.

Existence of fields where some lower Massey products vanish but higher ones are undefined.

Examples of fields with all roots of unity where the Galois cochain DGA is not formal.

Abstract

We prove that, for all fields $F$ of characteristic different from $2$ and all $a,b,c\in F^\times$, the mod $2$ Massey product $\langle a,b,c,a \rangle$ vanishes as soon as it is defined. For every field $F_0$, we construct a field $F$ containing $F_0$ and $a,b,c,d\in F^\times$ such that $\langle a,b,c \rangle$ and $\langle b,c,d \rangle$ vanish but $\langle a,b,c,d \rangle$ is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod $2$ cochain DGA of the absolute Galois group is not formal.