To the multidimensional tame symbol

TL;DR

This paper constructs a two-dimensional tame symbol via a groupoid approach linked to third cohomology classes, proposing a method for proving Parshin reciprocity laws on algebraic surfaces.

Contribution

It introduces a novel groupoid-based construction of the two-dimensional tame symbol and suggests a new approach to proving Parshin reciprocity laws.

Findings

Constructed the two-dimensional tame symbol as a commutator in a groupoid.

Linked the construction to third cohomology classes of groups acting on local fields.

Proposed a hypothetical method for proving Parshin reciprocity laws.

Abstract

We give a construction of the two-dimensional tame symbol as the commutator of a group-like monoidal groupoid which is obtained from some group of k-linear operators acting in a two-dimensional local field and corresponds to some third cohomology class of this group. We give also the hypothetical method for the proof of the two-dimensional Parshin reciprocity laws. This text was written in 2003 as preprint 03-13 of the Humboldt University of Berlin and was available at http://edoc.hu-berlin.de/docviews/abstract.php?id=26204 (only evident misprints are corrected now). Later E. Frenkel and X. Zhu obtained in arXiv:0810.1487 [math.RT] more general results concerning the third cohomology classes of groups acting on two-dimensional local fields, and the author and X. Zhu obtained in arXiv:1002.4848 [math.AG] the proof of the Parshin reciprocity laws on an algebraic surface similar to the Tate proof of the residue formula on an algebraic curve.