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Cellular resolutions of noncommutative toric algebras from superpotentials

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 $Δ$ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of $A$. We illustrate the general construction of $Δ$ 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 $Δ$.