  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.
