Conic modules, secondary fans and non-commutative resolutions

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 $|σ(1)|=\dim σ+1$, classifying when such cones admit NCCRs via endomorphism algebras of conic modules.