A Motivic Riemann-Roch Theorem for Deligne-Mumford Stacks

TL;DR

This paper develops a motivic cohomology theory for smooth Deligne-Mumford stacks, establishing an isomorphism with higher K-theory and generalizing the Grothendieck-Riemann-Roch theorem to this setting.

Contribution

It introduces a new motivic cohomology framework for stacks and proves a Riemann-Roch type theorem extending classical results to stacks.

Findings

Higher Chow groups are isomorphic to higher K-theory for stacks

Motivic cohomology theory is representable in Voevodsky's category

Generalization of Grothendieck-Riemann-Roch theorem to stacks

Abstract

We develop a motivic cohomology theory, representable in the Voevodsky's triangulated category of motives, for smooth separated Deligne-Mumford stacks and show that the resulting higher Chow groups are canonically isomorphic to the higher $K$-theory of such stacks. This generalises the Grothendieck-Riemann-Roch theorem to the category of smooth Deligne-Mumford stacks.