The forgetful map in rational K-theory

摘要

Let G be a connected reductive algebraic group acting on a scheme X. Let R( G) denote the representation ring of G, and I subset of 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 pi(1)( G) is torsion-free, then the forgetful map G( G, X) --> G( X) induces an isomorphism G(G, X)/IG(G, X) --> G(X). Although this map need not be an isomorphism if pi(1)( G) has torsion, we prove that without the assumption on pi(1)( G), the map G(G, X)/IG(G, X)circle times Q --> G(X) circle times Q is an isomorphism.