On the Grothendieck groups of toric stacks
Zheng Hua

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Commutative Algebra and Its Applications
