A higher-dimensional Contou-Carrère symbol: local theory

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.