Non-formality of Galois cohomology modulo all primes

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 $⟨a,b,c,d⟩$ is not defined over $L$. Thus the Strong Massey Vanishing Conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^{\ast }({\Gamma }_L,\mathbb{Z}/p\mathbb{Z})$ of the absolute Galois group ${\Gamma }_L$ of $L$ is not formal. This answers a question of Positselski.

27 pages