Cellular resolutions of noncommutative toric algebras from superpotentials

TL;DR

This paper develops cellular resolutions for noncommutative toric algebras using superpotentials, generalizing dimer models to higher dimensions and providing explicit constructions of moduli spaces and resolutions.

Contribution

It introduces a new framework for cellular resolutions of noncommutative toric algebras via superpotentials, extending dimer model techniques beyond three dimensions.

Findings

Constructed explicit cellular resolutions for classes of noncommutative algebras.

Associated a cell complex in a real torus to describe projective bimodule resolutions.

Provided an example in four dimensions illustrating the construction.

Abstract

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer-Sturmfels in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of toric algebras in dimension three by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For consistent algebras $A$, the coherent component of the fine moduli space of $A$-modules is constructed explicitly by GIT and provides a partial resolution of $\Spec Z(A)$. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex $\Delta$ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of $A$. We illustrate the general construction of $\Delta$ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on $\Delta$.