Vol. 5, No. 3, 2011

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
A categorical proof of the Parshin reciprocity laws on algebraic surfaces

Denis Osipov and Xinwen Zhu

Vol. 5 (2011), No. 3, 289–337
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