# Atiyah-Segal theorem for Deligne-Mumford stacks and applications

## TL;DR

This paper proves an Atiyah-Segal isomorphism for higher K-theory of quotient Deligne-Mumford stacks over complex numbers and applies it to establish a Grothendieck-Riemann-Roch theorem for these stacks.

## Contribution

It introduces an Atiyah-Segal isomorphism for higher K-theory of stacks and proves a Grothendieck-Riemann-Roch theorem in this context, extending classical results.

## Key findings

- Establishes an isomorphism between higher K-theory and higher Chow groups for stacks.
- Proves the Grothendieck-Riemann-Roch theorem for quotient Deligne-Mumford stacks.
- Demonstrates the covariant nature of the isomorphism for proper maps.

## Abstract

We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-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.