Non-commutative crepant resolutions for (almost) simplicial toric algebras

TL;DR

This paper demonstrates that toric non-commutative crepant resolutions (NCCRs) for certain Gorenstein cones can be extended to their faces, providing new proofs for the existence of NCCRs in specific toric algebra cases.

Contribution

It introduces a method for descending NCCRs from cones to faces and offers new proofs for the existence of NCCRs in simplicial and almost simplicial affine toric Gorenstein algebras.

Findings

NCCRs descend from cones to faces of the polytope.

New proofs for the existence of toric NCCRs in specific cases.

Extension of NCCRs to almost simplicial toric algebras.

Abstract

Given a rational convex polyhedral Gorenstein cone constructed as cone over a lattice polytope P, we establish that toric non-commutative crepant resolutions (NCCRs) of its associated toric algebra descend to toric NCCRs of the algebras associated to faces of the polytope P. As consequence, we present two new, short proofs to the existence of toric NCCRs for simplicial affine toric Gorenstein algebras and for almost simplicial affine toric Gorenstein algebras, i.e. those associated to cones $\sigma$ with $\dim\sigma+1$ extremal rays.