Realising the cup-product of local Tate duality

TL;DR

This paper provides an explicit construction of the cup-product in local Tate duality for Galois modules of prime order p, using central simple algebras to represent the product concretely.

Contribution

It introduces a minimal central simple algebra construction that explicitly realizes the cup-product in local Tate duality for prime order Galois modules.

Findings

Constructs a central simple algebra of dimension p^2 for the cup-product.

Provides an explicit algebraic description in terms of cocycles.

The algebra's class in the Brauer group corresponds to the cup-product.

Abstract

We present an explicit description, in terms of central simple algebras, of a cup-product map which occurs in the statement of local Tate duality for Galois modules of prime order p. Given cocycles f and g, we construct a central simple algebra of dimension p^2 whose class in the Brauer group gives the cup-product of f with g. This algebra is as small as possible.