Arithmetic duality for finite Galois modules over two-dimensional local fields of mixed characteristic

Abstract

A field $K$ is $d$-local if there exist fields $K=k_d,\dots,k_0$ where each $k_{i+1}$ is a complete discrete valuation field with residue field $k_i$, and $k_0$ is a finite field of characteristic $p$. By work of Deninger and Wingberg, the Galois cohomology of such fields with coefficients in finite Galois modules satisfies a duality generalizing Tate duality when either $d=0$, $\mathrm{char} k_1=0$ or the coefficients have no $p$-torsion. Based on recent progress by Kato and Suzuki, we obtain duality statements for arbitrary finite Galois modules, under the weaker assumption that either $d\leq 1$ or $\mathrm{char} k_2=0$. We also get duality for the \'etale cohomology of $K$-varieties with coefficients in finite \'etale groups, in the style of Artin-Verdier duality. These dualities are stated in terms of condensed structures (in fact, locally compact Hausdorff topologies) on the 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.