On the $K$-theory of smooth toric DM stacks

TL;DR

This paper computes the Grothendieck K-theory ring of smooth toric Deligne-Mumford stacks, introduces an analog of the Chern character, and analyzes K-theory pushforwards and pullbacks for weighted blowups.

Contribution

It provides explicit calculations of K-theory for smooth toric DM stacks and defines a new Chern character analog, advancing the understanding of their K-theoretic properties.

Findings

Explicit K-theory ring calculations for smooth toric DM stacks

Definition of an analog of the Chern character for these stacks

Calculation of K-theory pushforwards and pullbacks for weighted blowups

Abstract

We explicitly calculate the Grothendieck $K$-theory ring of a smooth toric Deligne-Mumford stack and define an analog of the Chern character. In addition, we calculate $K$-theory pushforwards and pullbacks for weighted blowups of reduced smooth toric DM stacks.