A categorical proof of the Parshin reciprocity laws on algebraic surfaces

Osipov, Denis; Zhu, Xinwen

doi:10.2140/ant.2011.5.289

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A categorical proof of the Parshin reciprocity laws on algebraic surfaces

Denis Osipov and Xinwen Zhu

Vol. 5 (2011), No. 3, 289–337

DOI: 10.2140/ant.2011.5.289

Abstract

We define and study the $2$ -category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using this in the context of two-dimensional local fields and two-dimensional adèle theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.

Keywords

Picard groupoids, central extensions, commutator maps, two-dimensional local fields, higher adeles, reciprocity laws

Mathematical Subject Classification 2000

Primary: 19F15

Secondary: 18D05

Milestones

Received: 27 February 2010

Revised: 25 October 2010

Accepted: 21 November 2010

Published: 10 September 2011

Authors

Denis Osipov
Steklov Mathematical Institute Gubkina str. 8 Moscow 119991 Russia

Xinwen Zhu
Department of Mathematics Harvard University 1 Oxford Street Cambridge, MA, 02138 United States

Vol. 5, No. 3, 2011