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

We generalize Contou-Carrère symbols to higher dimensions. To an ( n + 1 stretchy="false">) (n+1) -tuple alttext="f 0 comma ellipsis f Subscript Baseline element-of upper A left-parenthesis t right-parenthesis midline-horizontal-ellipsis Superscript times"> f 0 , … ∈<!-- ∈ <mml:mi>A t ⋯<!-- ⋯ <mml:msup> ×<!-- × </mml:msup> encoding="application/x-tex">f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times } , where alttext="upper A"> encoding="application/x-tex">A denotes algebra over a field alttext="k"> k encoding="application/x-tex">k we associate element encoding="application/x-tex">(f_0,\dots ,f_n) A^{\times extending the tame symbol for alttext="k equals = encoding="application/x-tex">k = A and earlier constructions alttext="n 1"> encoding="application/x-tex">n 1 by Contou-Carrère, 2"> 2 2 Osipov–Zhu. It is based on concept of higher commutators central extensions spectra. Using these tools, describe as composition boundary maps in algebraic K"> K encoding="application/x-tex">K -theory, prove version Parshin–Kato reciprocity symbols.