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Reciprocity laws and $K\mkern-2mu$-theory

Evgeny Musicantov and Alexander Yom Din

Vol. 2 (2017), No. 1, 27–46
DOI: 10.2140/akt.2017.2.27

We associate to a full flag in an n-dimensional variety X over a field k, a “symbol map” μ : K(FX) ΣnK(k). Here, FX is the field of rational functions on X, and K( ) is the K-theory spectrum. We prove a “reciprocity law” for these symbols: given a partial flag, the sum of all symbols of full flags refining it is 0. Examining this result on the level of K-groups, we derive the following known reciprocity laws: the degree of a principal divisor is zero, the Weil reciprocity law, the residue theorem, the Contou-Carrère reciprocity law (when X is a smooth complete curve), as well as the Parshin reciprocity law and the higher residue reciprocity law (when X is higher-dimensional).

reciprocity laws, $K\mkern-2mu$-theory, symbols in arithmetic, Parshin symbol, Parshin reciprocity, Contou-Carrère symbol, Tate vector spaces
Mathematical Subject Classification 2010
Primary: 19F15
Received: 11 May 2015
Revised: 28 August 2015
Accepted: 17 September 2015
Published: 3 September 2016
Evgeny Musicantov
School of Mathematical Sciences
Tel Aviv University
6997801 Tel Aviv
Alexander Yom Din
School of Mathematical Sciences
Tel Aviv University
6997801 Tel Aviv