A generalized Contou-Carrere symbol and its reciprocity laws in higher dimensions.

We generalize Contou-Carrère symbols to higher dimensions. To an (n + 1)-tuple ƒ₀, . . . ,ƒ_n ∈ A((t₁))⋅⋅⋅((t_n))^×, where A denotes an algebra over a field k, we associate an element (ƒ₀, . . . ,ƒ_n) ∈ A^×, extending the higher tame symbol for k = A, and earlier constructions for n = 1 by Contou-Carrère, and n = 2 by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic K-theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols. [ABSTRACT FROM AUTHOR]