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A categorical proof of the Parshin reciprocity laws on algebraic surfaces

Osipov, Denis and Zhu, Xinwen (2011) A categorical proof of the Parshin reciprocity laws on algebraic surfaces. Algebra and Number Theory, 5 (3). pp. 289-337. ISSN 1937-0652. https://resolver.caltech.edu/CaltechAUTHORS:20140919-152325174

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.2140/ant.2011.5.289DOIArticle
http://msp.org/ant/2011/5-3/p01.xhtmlPublisherArticle
http://arxiv.org/abs/1002.4848arXivDiscussion Paper
Additional Information:© 2011 Mathematical Sciences Publishers. Received: 27 February 2010. Revised: 25 October 2010. Accepted: 21 November 2010. Published: 10 September 2010. Osipov was financially supported by Russian Foundation for Basic Research (grant no. 11-01-00145) and by the Programme for the Support of Leading Scientific Schools of the Russian Federation (grant no. NSh-4713.2010.1). Zhu’s research is supported by NSF grant under DMS-1001280. Osipov is grateful to L. Breen, M. Kapranov and A. N. Parshin for useful discussions on categorical approach to the two-dimensional tame symbol at various periods of time. Zhu thanks V. Kac for useful discussions.
Funders:
Funding AgencyGrant Number
Russian Foundation for Basic Research11-01-00145
Programme for the Support of Leading Scientific Schools of the Russian FederationNSh-4713.2010.1
NSFDMS-1001280
Subject Keywords:Picard groupoids, central extensions, commutator maps, two-dimensional local fields, higher adeles, reciprocity laws
Issue or Number:3
Classification Code:MSC2000: primary 19F15; secondary 18D05
Record Number:CaltechAUTHORS:20140919-152325174
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140919-152325174
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:49871
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:23 Sep 2014 03:50
Last Modified:03 Oct 2019 07:18

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