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Cup product of inhomogeneous Tate cochains, and application to tori over local fields that split over cyclic extensions

Mikhail Borovoi

TL;DR

This work develops explicit formulas for cup products in Tate cohomology using inhomogeneous cocycles, enabling concrete representatives of cohomology classes for tori over local fields that split over cyclic extensions. It provides a cyclic-case explicit fundamental class $u_{L/K}$ by expressing the associated 2-cocycle $a$ as $a=b\cup e_\sigma$, and then applies this to give explicit cocycles $z_x$ in $H^{1}(K,T)$ for tori $T$ split by a cyclic extension $L/K$. The results yield concrete, computable representatives of all $H^{1}(K,T)$ classes via $z_x(\sigma^g)=\sum_{t=1}^{g} \sigma^{t}\cdot x \otimes e_\sigma$, facilitating explicit twisting and local-class-field-theory computations. Overall, the paper connects the algebraic framework of Tate cohomology with practical, cycle-by-cycle cocycle constructions for tori over local fields.

Abstract

In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T).

Cup product of inhomogeneous Tate cochains, and application to tori over local fields that split over cyclic extensions

TL;DR

This work develops explicit formulas for cup products in Tate cohomology using inhomogeneous cocycles, enabling concrete representatives of cohomology classes for tori over local fields that split over cyclic extensions. It provides a cyclic-case explicit fundamental class by expressing the associated 2-cocycle as , and then applies this to give explicit cocycles in for tori split by a cyclic extension . The results yield concrete, computable representatives of all classes via , facilitating explicit twisting and local-class-field-theory computations. Overall, the paper connects the algebraic framework of Tate cohomology with practical, cycle-by-cycle cocycle constructions for tori over local fields.

Abstract

In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T).

Paper Structure

This paper contains 7 sections, 2 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The homogeneous and inhomogeneous complexes
  3. Tori over local fields
  4. Cup product in terms of inhomogeneous cocycles
  5. The fundamental class of a cyclic extension
  6. Explicit cocycles for a torus
  7. Cup product: calculations

Key Result

Theorem 1.1

Let $L/K$ be a cyclic extension of degree $n$ of non-archimedean local fields. Choose a generator $\sigma$ of the cyclic group (of order $n$) $G={\rm Gal}(L/K)$. Let $e_\sigma\in K^\times$ be an element such that where is the local reciprocity homomorphism; see CF. Then the fundamental class $u_{L/K}\in H^2(G,L^\times)$ is represented by the following cocycle $a\colon G\times G\to L^\times$:

Theorems & Definitions (2)

  • Theorem 1.1: Sawin Sawin
  • Theorem 1.2