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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.00886 |
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| _version_ | 1866912964134567936 |
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| author | Galet, Antoine |
| author_facet | Galet, Antoine |
| contents | A field $K$ is $d$-local if there exist fields $K=k_d,...,k_0$ with $k_{i+1}$ complete discrete valuation with residue field $k_i$, and $k_0$ finite of characteristic $p$. By work of Deninger and Wingberg, the Galois cohomology of such fields with finite coefficients satisfies a duality generalizing Tate duality when either $d=0$, $\mathrm{char} k_1=0$ or the coefficients have no $p$-torsion. Reviewing and synthesizing results of Suzuki and Kato, we obtain $p$-torsion duality statements under the weaker assumption that either $d\leq 1$ or $\mathrm{char} k_2=0$, as well as for varieties over $K$, where duality is stated in terms of locally compact Hausdorff topologies on the étale cohomology groups. More generally we obtain results for any perfect $k_0$, endowing the totally unramified cohomology groups of $K$ with the structure of ind-pro-quasi-algebraic $k_0$-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Duality for higher local fields after Kato and Suzuki Galet, Antoine Number Theory 14F20 (Primary) 20G15, 20K45 (Secondary) A field $K$ is $d$-local if there exist fields $K=k_d,...,k_0$ with $k_{i+1}$ complete discrete valuation with residue field $k_i$, and $k_0$ finite of characteristic $p$. By work of Deninger and Wingberg, the Galois cohomology of such fields with finite coefficients satisfies a duality generalizing Tate duality when either $d=0$, $\mathrm{char} k_1=0$ or the coefficients have no $p$-torsion. Reviewing and synthesizing results of Suzuki and Kato, we obtain $p$-torsion duality statements under the weaker assumption that either $d\leq 1$ or $\mathrm{char} k_2=0$, as well as for varieties over $K$, where duality is stated in terms of locally compact Hausdorff topologies on the étale cohomology groups. More generally we obtain results for any perfect $k_0$, endowing the totally unramified cohomology groups of $K$ with the structure of ind-pro-quasi-algebraic $k_0$-groups. |
| title | Duality for higher local fields after Kato and Suzuki |
| topic | Number Theory 14F20 (Primary) 20G15, 20K45 (Secondary) |
| url | https://arxiv.org/abs/2512.00886 |