Algebraic analogue of Atiyah's theorem

TL;DR

This paper establishes an algebraic analogue of Atiyah's theorem, demonstrating an isomorphism between the K-theory of the etale classifying space of a split reductive algebraic group and a completion of its equivariant K-theory over a field.

Contribution

It introduces an algebraic version of Atiyah's theorem, linking etale classifying space K-theory with equivariant K-theory for algebraic groups.

Findings

Proves an isomorphism between etale classifying space K-theory and completed equivariant K-theory.

Extends topological Atiyah's theorem to algebraic setting.

Provides new tools for algebraic K-theory of algebraic groups.

Abstract

In topology there is a theorem of Atiyah, concerning K-theory of classifying space of connected compact Lie group. We consider an algebraic analogue of this theorem. We prove that for a split reductive algebraic group G over a field there is an isomorphism between K-theory of etale classifying space of group G and a completion of the G-equivariant K-theory of the base field.