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arXiv · 2309.17004

Non-formality of Galois cohomology modulo all primes

Alexander Merkurjev·Federico Scavia
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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.

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Alexander Merkurjev, Federico Scavia. 2023-09-29. Non-formality of Galois cohomology modulo all primes. https://doi.org/10.1112/s0010437x25007018

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