IV - Noncommutative resolutions

Summary

The notion of a noncommutative crepant resolution (NCCR) was introduced by Van den Bergh [231], following his interpretation [230] of work of Bridgeland [42] and Bridgeland, King and Reid [44]. Since then, NCCRs have appeared prominently in both the mathematics and physics literature as a general homological structure that underpins many topics currently of interest, for example moduli spaces, dimer models, curve counting Donaldson–Thomas invariants, spherical–type twists, the minimal model program and mirror symmetry.

My purpose in writing these notes is to give an example based approach to some of the ideas and constructions for NCCRs, with latter sections focussing more on the explicit geometry and restricting mainly to dimensions two and three, rather than simply presenting results in full generality. The participants at the MSRI Summer School had a wonderful mix of diverse backgrounds, so the content and presentation of these notes reflect this. There are exercises scattered throughout the text, at various levels of sophistication, and also computer exercises that hopefully add to the intuition.

The following is a brief outline of the content of the notes. In Section 1 we begin by outlining some of the motivation and natural questions for NCCRs through the simple example of the Z3 surface singularity. We then progress to the setting of twodimensional Gorenstein quotient singularities for simplicity, although most things work much more generally. We introduce the notion of Auslander algebras, and link to the idea of finite CM type. We then introduce skew group rings and use this to show that a certain endomorphism ring in the running example has finite global dimension.

Section 2 begins with the formal definitions of Gorenstein and CM rings, depth and CM modules, before giving the definition of a noncommutative crepant resolution (NCCR). This comes in two parts, and the second part is motivated using some classical commutative algebra. We then deal with uniqueness issues, showing that in dimension two NCCRs are unique up to Morita equivalence, whereas in dimension three they are unique up to derived equivalence. We give examples to show that these results are the best possible. Along the way, the three key technical results of the depth lemma, reflexive equivalence and the Auslander–Buchsbaum formula are formulated.