Reciprocity laws and K-theory

Evgeny Musicantov; Alexander Yom Din

doi:10.2140/akt.2017.2.27

Reciprocity laws and K-theory

Evgeny Musicantov, Alexander Yom Din

Research output: Contribution to journal › Article › peer-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” μ_F: K(F_X)→∑ⁿK(k). Here, F_X 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 language	American English
Pages (from-to)	27-46
Number of pages	20
Journal	Annals of K-Theory
Volume	2
Issue number	1
DOIs	https://doi.org/10.2140/akt.2017.2.27
State	Published - 2017
Externally published	Yes

Bibliographical note

Funding Information:
This research was partially supported by the ERC grant 291612 and by the ISF grant 533/14. MSC2010: 19F15. Keywords: reciprocity laws, K-theory, symbols in arithmetic, Parshin symbol, Parshin reciprocity, Contou-Carrère symbol, Tate vector spaces.

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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10.2140/akt.2017.2.27

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title = "Reciprocity laws and K-theory",

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-Carr{\`e}re 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).",

keywords = "Contou-Carr{\`e}re symbol, K-theory, Parshin reciprocity, Parshin symbol, Reciprocity laws, Symbols in arithmetic, Tate vector spaces",

author = "Evgeny Musicantov and {Yom Din}, Alexander",

note = "Funding Information: This research was partially supported by the ERC grant 291612 and by the ISF grant 533/14. MSC2010: 19F15. Keywords: reciprocity laws, K-theory, symbols in arithmetic, Parshin symbol, Parshin reciprocity, Contou-Carr{\`e}re symbol, Tate vector spaces. Publisher Copyright: {\textcopyright} 2017 Mathematical Sciences Publishers.",

year = "2017",

doi = "10.2140/akt.2017.2.27",

language = "American English",

volume = "2",

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AU - Yom Din, Alexander

N1 - Funding Information: This research was partially supported by the ERC grant 291612 and by the ISF grant 533/14. MSC2010: 19F15. Keywords: reciprocity laws, K-theory, symbols in arithmetic, Parshin symbol, Parshin reciprocity, Contou-Carrère symbol, Tate vector spaces. Publisher Copyright: © 2017 Mathematical Sciences Publishers.

PY - 2017

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N2 - 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).

AB - 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).

KW - Contou-Carrère symbol

KW - K-theory

KW - Parshin reciprocity

KW - Parshin symbol

KW - Reciprocity laws

KW - Symbols in arithmetic

KW - Tate vector spaces

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DO - 10.2140/akt.2017.2.27

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VL - 2

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EP - 46

JO - Annals of K-Theory

JF - Annals of K-Theory

IS - 1

ER -

Reciprocity laws and K-theory

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