Weak factorization and the Grothendieck group of Deligne-Mumford stacks

TL;DR

This paper develops a presentation for the Grothendieck group of Deligne-Mumford stacks over characteristic zero fields, utilizing stacky blow-ups and establishing a weak factorization theorem for these stacks.

Contribution

It introduces a new presentation for the Grothendieck group of Deligne-Mumford stacks and proves a weak factorization theorem specific to these stacks.

Findings

Presentation of the Grothendieck group using smooth, proper stacks

Relations expressed via stacky blow-ups

Proof of a weak factorization theorem for Deligne-Mumford stacks

Abstract

We construct a presentation for the Grothendieck group of Deligne-Mumford stacks over a field of characteristic zero. The generators for this presentation are smooth, proper Deligne-Mumford stacks and the relations are expressed in terms of stacky blow-ups. In the process we prove a version of the weak factorization theorem for Deligne-Mumford stacks.