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

TL;DR

This paper provides explicit formulas for cup products in Tate cohomology using inhomogeneous cochains and applies these formulas to compute explicit cocycles for tori over local fields that split over cyclic extensions.

Contribution

It introduces new formulas for cup products in Tate cohomology and applies them to explicitly compute cohomology classes for certain algebraic tori.

Findings

Derived explicit cocycles representing all classes in H^1(K,T)

Provided formulas for cup products in Tate cohomology

Applied formulas to tori over local fields with cyclic splitting extensions

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).