Semi-steady non-commutative crepant resolutions via regular dimer models

Yusuke Nakajima

doi:10.5802/alco.39

Abstract

A consistent dimer model gives a non-commutative crepant resolution (= NCCR) of a 3-dimensional Gorenstein toric singularity. In particular, it is known that a consistent dimer model gives a class of NCCRs called steady if and only if it is homotopy equivalent to a regular hexagonal dimer model. Inspired by this result, we detect another nice property on NCCRs that characterizes square dimer models. We call such NCCRs semi-steady NCCRs, and study their properties.

Highlights

An NCCR of a quotient singularity is given by the skew group algebra, and if a given quotient singularity is d-dimensional Gorenstein with d 3, the skew group algebra is derived equivalent to crepant resolutions of such a singularity [8, 32]
We study basic properties of semi-steady NCCRs, and as a result we show that the semi-steadiness characterizes NCCRs arising from square dimer models
We will show that a splitting NCCR of a 3-dimensional Gorenstein toric singularity is obtained from a consistent dimer model

Summary

Introduction

The notion of non-commutative crepant resolution (= NCCR) was introduced by Van den Bergh [41] (see [42]). It is an algebra derived equivalent to crepant resolutions for some singularities, and it gives another perspective on Bondal–Orlov conjecture [6] and Bridgeland’s theorem [7]. Let R be a Cohen–Macaulay normal domain, and M be a non-zero reflexive R-module. We say that Λ is a non-commutative crepant resolution (= NCCR) of R or M gives an NCCR of R if Λ is a non-singular R-order, that is, gl. We refer to [30, Proposition 2.17] for several conditions that are equivalent to Λ is a non-singular R-order. Non-commutative crepant resolutions, Dimer models, Regular tilings, Toric singularities

Basic properties of semi-steady NCCRs

NCCRs arising from dimer models

Semi-steady NCCRs arising from dimer models