Algebraic cycles and completions of equivariant K-theory

Abstract

Let G be a complex, linear algebraic group acting on an algebraic space X. The purpose of this article is to prove a Riemann-Roch theorem (Theorem 6.5) that gives a description of the completion of the equivariant Grothendieck group G0(G,X)⊗C at any maximal ideal of the representation ring R(G)⊗C in terms of equivariant cycles. The main new technique for proving this theorem is our nonabelian completion theorem (Theorem 5.3) for equivariant K-theory. Theorem 5.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups