Algebra & Number Theory

A categorical proof of the Parshin reciprocity laws on algebraic surfaces

Denis Osipov and Xinwen Zhu

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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.

Article information

Algebra Number Theory, Volume 5, Number 3 (2011), 289-337.

Received: 27 February 2010
Revised: 25 October 2010
Accepted: 21 November 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19F15: Symbols and arithmetic [See also 11R37]
Secondary: 18D05: Double categories, 2-categories, bicategories and generalizations

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


Osipov, Denis; Zhu, Xinwen. A categorical proof of the Parshin reciprocity laws on algebraic surfaces. Algebra Number Theory 5 (2011), no. 3, 289--337. doi:10.2140/ant.2011.5.289.

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