The forgetful map in rational K-theory

William Graham

Vol. 236 (2008), No. 1, 45–55

DOI: 10.2140/pjm.2008.236.45

Abstract

Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and I ⊂ R(G) the ideal of virtual representations of rank 0. Let G(X) (respectively, G(G,X)) denote the Grothendieck group of coherent sheaves (respectively, G-equivariant coherent sheaves) on X. Merkurjev proved that if π₁(G) is torsion-free, then the forgetful map G(G,X) → G(X) induces an isomorphism

Although this map need not be an isomorphism if π₁(G) has torsion, we prove that without the assumption on π₁(G), the map G(G,X)∕IG(G,X) ⊗ ℚ → G(X) ⊗ ℚ is an isomorphism.

Keywords

K-theory, equivariant, equivariant K-theory, Riemann–Roch

Mathematical Subject Classification 2000

Primary: 19E08, 18F30

Milestones

Received: 29 October 2007

Accepted: 4 December 2007

Published: 1 May 2008