Conic modules, secondary fans and non-commutative resolutions

TL;DR

This paper explores the connection between conic modules and line bundles on toric stacks to establish criteria for non-commutative crepant resolutions of toric algebras, simplifying the combinatorial complexity involved.

Contribution

It links conic modules to line bundles on toric stacks and provides computational criteria for non-commutative resolutions, reducing complexity and addressing cases with torsion-free class groups.

Findings

Established conditions for incomplete sums of conic modules to yield NCCRs.

Reduced verification of NCCRs to torsion-free class group cases.

Classified NCCRs for almost simplicial Gorenstein cones.

Abstract

Faber, Muller and Smith used complete sums of conic modules to construct non-commutative crepant resolutions (NCCR) of simplicial toric algebras. We link these conic modules to the Bondal-Thomsen collection of line bundles on smooth toric DM stacks. This viewpoint allows us to establish computational results relating to conic modules, reducing the complexity of the combinatorics involved significantly. We formulate necessary and sufficient conditions for an incomplete sum of conic modules to give an NC(C)R of a toric algebra. Furthermore, we prove that to check if a toric algebra $R$ admits an NCCR in the form of $\End_R(\mathbb{B})$ for an incomplete sum of conic modules, we may reduce to a case where the class group of the affine toric variety $\spec R$ does not have torsion and verify the statement there. Finally, we treat the case of almost simplicial Gorenstein cones, i.e. cones with $|\sigma(1)|=\dim \sigma+1$, classifying when such cones admit NCCRs via endomorphism algebras of conic modules.