Non-formality of Galois cohomology modulo all primes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912533452947456 |
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| author | Merkurjev, Alexander Scavia, Federico |
| author_facet | Merkurjev, Alexander Scavia, Federico |
| contents | 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 $C^*(Γ_L,\mathbb{Z}/p\mathbb{Z})$ of the absolute Galois group $Γ_L$ of $L$ is not formal. This answers a question of Positselski. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_17004 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-formality of Galois cohomology modulo all primes Merkurjev, Alexander Scavia, Federico Number Theory Algebraic Geometry 12G05 (Primary) 55S30, 16K50 (Secondary) 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 $C^*(Γ_L,\mathbb{Z}/p\mathbb{Z})$ of the absolute Galois group $Γ_L$ of $L$ is not formal. This answers a question of Positselski. |
| title | Non-formality of Galois cohomology modulo all primes |
| topic | Number Theory Algebraic Geometry 12G05 (Primary) 55S30, 16K50 (Secondary) |
| url | https://arxiv.org/abs/2309.17004 |