The Grothendieck and Picard groups of a complete toric DM stack

TL;DR

This paper calculates the Grothendieck and Picard groups of complete smooth toric Deligne-Mumford stacks using graded module categories over polynomial rings, providing explicit algebraic descriptions.

Contribution

It introduces a method to compute these groups for toric DM stacks via graded modules, advancing algebraic understanding of their structure.

Findings

Explicit descriptions of Grothendieck and Picard groups for the stacks.

Methodology using graded modules over polynomial rings.

Enhanced algebraic tools for studying toric stacks.

Abstract

We compute the Grothendieck and Picard groups of a complete smooth toric Deligne-Mumford stack by using a suitable category of graded modules over a polynomial ring.