The Eigenvalue Spectrum of the Inertia Operator

TL;DR

This paper studies the inertia operator on algebraic stacks' Grothendieck groups, proving it is locally finite and diagonalizable for certain classes, with implications for Hall algebras and Donaldson-Thomas theory.

Contribution

It establishes the local finiteness and diagonalizability of the inertia operator on Grothendieck groups of algebraic stacks in characteristic zero.

Findings

Inertia operator is locally finite and diagonalizable.

Results apply to Grothendieck groups of Deligne-Mumford and quasi-split Artin stacks.

Implications for Hall algebras and Donaldson-Thomas theory.

Abstract

We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We show that the inertia operator is locally finite and diagonalizable. This is proved for the Grothendieck group of Deligne-Mumford stacks over a base scheme and the category of quasi-split Artin stacks defined over a field of characteristic zero. Motivated by the quasi-splitness condition, in [2] we consider the inertia operator of the Hall algebra of algebroids, and applications of it in generalized Donaldson-Thomas theory.