A categorical proof of the Parshin reciprocity laws on algebraic surfaces

TL;DR

This paper introduces a categorical framework to prove Parshin reciprocity laws on algebraic surfaces, utilizing torsors over Picard groupoids and central extensions to provide a new proof of these classical results.

Contribution

It develops a novel categorical approach involving torsors and Picard groupoids to establish reciprocity laws on algebraic surfaces, offering a new proof method.

Findings

Defined the 2-category of torsors over a Picard groupoid

Constructed central extensions and commutator maps in this context

Provided a new proof of Parshin reciprocity laws

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 it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.