On steady non-commutative crepant resolutions. (English) Zbl 1419.16012

Iyama, Osamu; Nakajima, Yusuke

J. Noncommut. Geom. 12, No. 2, 457-471 (2018).

Summary: We introduce special classes of non-commutative crepant resolutions (= NCCR) which we call steady and splitting. We show that a singularity has a steady splitting NCCR if and only if it is a quotient singularity by a finite abelian group. We apply our results to toric singularities and dimer models.

MSC:

16G50	Cohen-Macaulay modules in associative algebras
14A22	Noncommutative algebraic geometry
14E15	Global theory and resolution of singularities (algebro-geometric aspects)

Keywords:

non-commutative crepant resolutions; class groups; quotient singularities