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Duality for condensed cohomology of the Weil group of a p-adic field

Marco Artusa

TL;DR

The paper develops a condensed cohomology theory for the Weil group of a $p$-adic field by introducing the pro-condensed Weil group $\hat{W}_F$ and its classifying topos $B_{\hat{W}_F}$, along with a dualising complex $\mathbb{R}/\mathbb{Z}(1)$. It proves a perfect cup-product pairing between $\mathbb{H}^q(B_{\hat{W}_F},M)$ and $\mathbb{H}^{2-q}(B_{\hat{W}_F},M^D)$ into $\mathbb{R}/\mathbb{Z}$ for locally compact abelian groups $M$ of finite rank with finite quotient Galois action, after endowing coefficients and cohomology with appropriate topologies. The framework combines condensed mathematics, pro-condensed groups, and Hochschild–Serre spectral sequences to extend local Tate duality beyond discrete coefficients, recovering Weil-style class field theory reciprocity in a topological setting. The results provide a robust, topologically enriched duality theory with explicit structure theorems, offering a principled way to handle profinite completion issues and to study Weil-group cohomology in a broader categorical context.

Abstract

We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a $p$-adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate Duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups.

Duality for condensed cohomology of the Weil group of a p-adic field

TL;DR

The paper develops a condensed cohomology theory for the Weil group of a -adic field by introducing the pro-condensed Weil group and its classifying topos , along with a dualising complex . It proves a perfect cup-product pairing between and into for locally compact abelian groups of finite rank with finite quotient Galois action, after endowing coefficients and cohomology with appropriate topologies. The framework combines condensed mathematics, pro-condensed groups, and Hochschild–Serre spectral sequences to extend local Tate duality beyond discrete coefficients, recovering Weil-style class field theory reciprocity in a topological setting. The results provide a robust, topologically enriched duality theory with explicit structure theorems, offering a principled way to handle profinite completion issues and to study Weil-group cohomology in a broader categorical context.

Abstract

We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a -adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate Duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups.
Paper Structure (22 sections, 69 theorems, 255 equations)

This paper contains 22 sections, 69 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Outline of this article
  3. Relation to previous work
  4. Set-theoretical conventions and notation
  5. Acknowledgements
  6. Cohomology of condensed groups
  7. Topoi over condensed sets
  8. The classifying topos of a condensed group
  9. Comparison with continuous cohomology
  10. Discrete coefficients
  11. Cohomology of discrete groups
  12. The classifying topos of a pro-condensed group
  13. The category DBhatG
  14. Cohomology of WF
  15. The complex RGammaBIL
  16. ...and 7 more sections

Key Result

Theorem A

Let $M$ be a finitely generated $G_F$-module, and consider the map Then $\psi^q(M)$ is an isomorphism for all $q\ge 1$, and $\psi^0(M^D)$ defines an isomorphism (of profinite groups)

Theorems & Definitions (178)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Definition 2.2
  • Remark 2.4
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Lemma 2.9
  • proof
  • ...and 168 more