Duality for higher local fields after Kato and Suzuki
Antoine Galet
TL;DR
The paper develops a unified duality framework for higher local fields by integrating relatively perfect unipotent group schemes (RPAU) with ind-pro and pro-finite extensions, all within various étale sites (RP, RPS, proet) and their condensed-geometry realizations. It constructs a robust derived-category setup, proves Serre and Cartier dualities for RPAU and related ind-pro-finite objects, and shows how Weil restrictions and trace maps preserve the RP-structure under field extensions. The key contributions include a detailed dévissage for RPAU groups, a derived Yoneda framework linking RP-objects to sheaves, and the assembly of positive equal-characteristic, mixed-characteristic, and prime-to-p steps into comprehensive higher-local-duality theorems for both cohomology and varieties, with explicit topologies on cohomology groups. These results extend Tate-type dualities to higher local fields in greater generality, enabling refined duality statements for p-torsion and prime-to-p coefficients and providing a coherent interpretation via ind-pro-sheaves and condensed groups. An appendix on Kato lifting underpins the equal-characteristic and mixed-characteristic arguments, tying the local theory to global cohomological dualities.
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 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.