Duality for higher local fields after Kato and Suzuki

TL;DR

This paper explores duality theories for higher local fields, extending previous results to cases with weaker assumptions and incorporating topological structures in étale cohomology.

Contribution

It synthesizes and extends duality results for higher local fields, especially for cases with p-torsion and weaker conditions, including topological aspects of cohomology.

Findings

Established p-torsion duality under weaker assumptions (d ≤ 1 or char k_2=0).

Extended duality to varieties over K with topological structures on cohomology.

Provided a framework for cohomology groups as ind-pro-quasi-algebraic k_0-groups.

Abstract

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 \'etale 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.