Atiyah-Segal theorem for Deligne-Mumford stacks and applications

Abstract

We prove an Atiyah-Segal isomorphism for the higher K K -theory of coherent sheaves on quotient Deligne-Mumford stacks over C \mathbb {C} . As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher K K -theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.