The forgetful map in rational K-theory

TL;DR

This paper investigates the relationship between equivariant and non-equivariant K-theory for algebraic groups, proving that a certain natural map becomes an isomorphism over the rationals even when the fundamental group has torsion.

Contribution

It extends Merkurjev's result by showing the forgetful map in rational K-theory is an isomorphism without the torsion-free fundamental group assumption.

Findings

The map becomes an isomorphism after tensoring with the rationals.

The result holds even when the fundamental group has torsion.

Provides a broader understanding of the structure of rational K-theory.

Abstract

Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and let I be the ideal in R(G) of virtual representations of rank 0. Let G(X) (resp. G(G,X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. Merkurjev proved that if the fundamental group of G is torsion-free, then the map of G(G,X)/IG(G,X) to G(X) is an isomorphism. Although this map need not be an isomorphism if the fundamental group of G has torsion, we prove that without the assumption on the fundamental group of G, this map is an isomorphism after tensoring with the rational numbers.