Comparing the Kirwan and noncommutative resolutions of quotient varieties

Abstract

Let a reductive group G act on a smooth variety X such that a good quotient X//G exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X//G, obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X//G. In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne-Mumford stacks.