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A higher-dimensional Contou-Carrère symbol: local theory

We construct a higher-dimensional Contou-Carrère symbol and we study its various fundamental properties. The higher-dimensional Contou-Carrère 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-Carrère symbol over $\mathbb Q$ and we prove integrality of this formula. A relation with the higher-dimensional Witt pairing is also studied.