A higher-dimensional Contou-Carr\`ere symbol: local theory

TL;DR

This paper introduces a higher-dimensional generalization of the Contou-Carr ext{é}re symbol, establishing its fundamental properties, explicit formulas, and connections with Witt pairings, advancing local class field theory.

Contribution

It constructs a new higher-dimensional Contou-Carr ext{é}re symbol, proves its universal property, and provides explicit formulas and integrality results, extending local symbol theory.

Findings

Defined the higher-dimensional Contou-Carr ext{é}re symbol using $K$-groups boundary maps.

Proved the universal property of the new symbol.

Derived explicit formulas and established their integrality over $ ext{Q}$.

Abstract

We construct a higher-dimensional Contou-Carr\`ere symbol and we study its various fundamental properties. The higher-dimensional Contou-Carr\`ere symbol is defined by means of the boundary map for $K$-groups. We prove its universal property. We provide an explicit formula for the higher-dimensional Contou-Carr\`ere symbol over $\mathbb Q$ and we prove integrality of this formula. A relation with the higher-dimensional Witt pairing is also studied.