Completions of Higher Equivariant K-theory

TL;DR

This paper extends the non-abelian localization theorem to higher equivariant K-theory for smooth varieties with group actions and establishes a Riemann-Roch theorem relating completed K-theory to equivariant higher Chow groups.

Contribution

It proves a non-abelian localization theorem for higher equivariant K-theory and generalizes the Riemann-Roch theorem to this setting with rational coefficients.

Findings

Established a non-abelian localization theorem for higher equivariant K-theory.

Proved a Riemann-Roch theorem relating completed K-theory to equivariant higher Chow groups.

Generalized previous results of Edidin and Graham to higher K-theory.

Abstract

The goal of this paper is to prove a version of the non-abelian localization theorem for the rational equivariant K-theory of a smooth variety $X$ with the action of a linear algebraic group $G$. We then use this to prove a Riemann-Roch theorem which represents the completion of the higher equivariant K-theory of $X$ at various maximal ideals of the representation ring, in terms the equivariant higher Chow groups. This generalizes a result of Edidin and Graham to higher $K$-theory with rational coefficients.