Higher local duality in Galois cohomology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916594491326464 |
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| author | Galet, Antoine |
| author_facet | Galet, Antoine |
| contents | A field $K$ is quasi-classical $d$-local if there exist fields $K=k_d,\dots,k_0$ with $k_{i+1}$ Henselian admissible discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of such $K$ with many coefficients, including finite coefficients of any order. Previously, such duality was only known in few cases : as a perfect pairing of finite groups for finite coefficients prime to $\mathrm{char} k_0$ in general, or for any finite coefficients when $k_1$ is $p$-adic ; or as a perfect pairing of locally compact Hausdorff groups for the $\mathrm{fppf}$ cohomology of finite group schemes when $K$ is local. With no obvious reasonable topology available, we abandon perfectness altogether and instead obtain nondegenerate pairings of abstract abelian groups. This is done with new diagram-chasing results for pairings of torsion groups, allowing a dévissage approach which reduces our results to the study of $K^M_r(K)/p\times H^{d+1-r}_p(K)\to\mathbb{Z}/p$ using results of Kato. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16047 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher local duality in Galois cohomology Galet, Antoine Number Theory K-Theory and Homology 11S25 (Primary) 11S70, 20K45 (Secondary) A field $K$ is quasi-classical $d$-local if there exist fields $K=k_d,\dots,k_0$ with $k_{i+1}$ Henselian admissible discretely valued with residue field $k_i$, and $k_0$ quasi-finite. We prove a duality theorem for the Galois cohomology of such $K$ with many coefficients, including finite coefficients of any order. Previously, such duality was only known in few cases : as a perfect pairing of finite groups for finite coefficients prime to $\mathrm{char} k_0$ in general, or for any finite coefficients when $k_1$ is $p$-adic ; or as a perfect pairing of locally compact Hausdorff groups for the $\mathrm{fppf}$ cohomology of finite group schemes when $K$ is local. With no obvious reasonable topology available, we abandon perfectness altogether and instead obtain nondegenerate pairings of abstract abelian groups. This is done with new diagram-chasing results for pairings of torsion groups, allowing a dévissage approach which reduces our results to the study of $K^M_r(K)/p\times H^{d+1-r}_p(K)\to\mathbb{Z}/p$ using results of Kato. |
| title | Higher local duality in Galois cohomology |
| topic | Number Theory K-Theory and Homology 11S25 (Primary) 11S70, 20K45 (Secondary) |
| url | https://arxiv.org/abs/2410.16047 |