Non-formality of Galois cohomology modulo all primes

TL;DR

This paper constructs specific field extensions demonstrating that the Galois cohomology ring's structure is not formal, thereby disproving the Strong Massey Vanishing Conjecture at a prime $p$ and answering a question posed by Positselski.

Contribution

It provides explicit counterexamples showing the non-formality of Galois cohomology modulo all primes, challenging previous assumptions about the structure of these cohomology rings.

Findings

Existence of field extensions where Massey products vanish but are not defined.

Counterexamples to the Strong Massey Vanishing Conjecture at prime $p$.

Demonstration that the cochain differential graded ring is not formal.

Abstract

Let $p$ be a prime number and let $F$ be a field of characteristic different from $p$. We prove that there exist a field extension $L/F$ and $a,b,c,d$ in $L^{\times}$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm{Br}(F)[p]$ but $\langle a,b,c,d\rangle$ is not defined over $L$. Thus the Strong Massey Vanishing Conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^*(\Gamma_L,\mathbb{Z}/p\mathbb{Z})$ of the absolute Galois group $\Gamma_L$ of $L$ is not formal. This answers a question of Positselski.