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2017 Reciprocity laws and $K\mkern-2mu$-theory
Evgeny Musicantov, Alexander Yom Din
Ann. K-Theory 2(1): 27-46 (2017). 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).


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Evgeny Musicantov. Alexander Yom Din. "Reciprocity laws and $K\mkern-2mu$-theory." Ann. K-Theory 2 (1) 27 - 46, 2017.


Received: 11 May 2015; Revised: 28 August 2015; Accepted: 17 September 2015; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1354.19003
MathSciNet: MR3599515
Digital Object Identifier: 10.2140/akt.2017.2.27

Primary: 19F15

Keywords: $K\mkern-2mu$-theory , Contou-Carrère symbol , Parshin reciprocity , Parshin symbol , reciprocity laws , symbols in arithmetic , Tate vector spaces

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.2 • No. 1 • 2017
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