On the Grothendieck groups of toric stacks

TL;DR

This paper proves that the Grothendieck group of smooth complete toric Deligne-Mumford stacks is torsion free, and provides an example of torsion in the K-theory of an open toric stack, highlighting differences based on stackiness.

Contribution

It establishes torsion-freeness of K-theory for smooth complete toric stacks and presents a counterexample for open toric stacks with torsion.

Findings

Grothendieck group of smooth complete toric Deligne-Mumford stacks is torsion free

Open toric stacks can have torsion in their K-theory

Stackiness affects torsion properties in K-theory

Abstract

In this note, we prove that the Grothendieck group of a smooth complete toric Deligne-Mumford stack is torsion free. This statement holds when the generic point is stacky. We also construct an example of open toric stack with torsion in K-theory. This is a part of the author's Ph.D thesis.