Reciprocity laws and K-theory

Evgeny Musicantov, Alexander Yom Din

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We associate to a full flag F in an n-dimensional variety X over a field k, a “symbol map” μF: K(FX)→∑nK(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).

Original languageAmerican English
Pages (from-to)27-46
Number of pages20
JournalAnnals of K-Theory
Volume2
Issue number1
DOIs
StatePublished - 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 Mathematical Sciences Publishers.

Keywords

  • Contou-Carrère symbol
  • K-theory
  • Parshin reciprocity
  • Parshin symbol
  • Reciprocity laws
  • Symbols in arithmetic
  • Tate vector spaces

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