Comparing the Kirwan and noncommutative resolutions of quotient varieties (Q6112402)

Let \(R\) be a normal Gorenstein domain. A noncommutative crepant resolution (NCCR) of \(R\) is an \(R\)-algebra of finite global dimension of the form \(\Lambda=\mathrm{End}_R(M)\), which in addition is Cohen-Macaulay as an \(R\)-module and where \(M\) is a nonzero finitely-generated reflexive \(R\)-module. Let a reductive group \(G\) act on a smooth variety \(X\) such that a good quotient \(X/\!\!/ G\) exists. Š. Špenko and M. Van den Bergh show that the derived category of a 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 \(X/\!\!/ G\). 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.