Derived Representation Theory and the Algebraic K-theory of Fields

TL;DR

This paper introduces a new construction linking profinite groups to algebraic K-theory of fields, proving it for certain cases and conjecturing its general applicability, while exploring its properties and connections to representation theory.

Contribution

It presents a novel construction connecting profinite groups with algebraic K-theory of fields, with proven cases and a conjecture for all geometric fields.

Findings

Construction matches algebraic K-theory for certain Galois groups.

Proven for topologically finitely generated abelian Galois groups.

Discusses relationships with infinite group representation theory.

Abstract

We describe the construction which takes as input a profinite group, which when applied the the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. We prove that it agrees in the case of a topologically finitely generated abelian absolute Galois group, and conjecture that it agrees for all geometric fields. We also discuss properties of the construction, including relationships with the representation theory of infinite discrete groups.