A Motivic Riemann-Roch Theorem for Deligne-Mumford Stacks

TL;DR

This work develops a motivic cohomology theory for smooth Deligne-Mumford stacks by introducing the twisted motive $M_\text{χ}(F)$, built from the character stack $C^t_F$, and its associated motivic cohomology $H_{\text{χ}}^{p,q}(F,\mathbb{Q})$. It proves a motivic Grothendieck-Riemann-Roch isomorphism $K_i(F)_{\mathbb{Q}} \simeq \prod_j H_{\text{χ}}^{2j-i,j}(F,\mathbb{Q})$, with functoriality along coarse moduli and a connection to Toën's Chow motives after inverting roots of unity. The paper develops descent, Gysin triangles, and homotopy invariance for $M_\text{χ}(F)$, and provides explicit computations for quotient stacks and inertia-related decompositions, unifying several known Riemann-Roch formalisms. Together, these results extend motivic methods to the stacky setting and offer a coherent framework for comparing $K$-theory and motivic cohomology of DM 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.

Theorems & Definitions (50)

• Theorem A: see Theorem \ref{['theorem-riemann-roch']}
• Theorem B: see Theorem \ref{['comp inertia']}
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Lemma 3.1
• Lemma 3.2
• proof