Cellular resolutions of noncommutative toric algebras from superpotentials

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 $\Delta$.

34 pages, 10 figures; v2 simplified proof of Theorem 5.9 and expanded Section 6; v3 added Lemma 2.1, final version

References in corpus (7)

• Calabi-Yau algebras
• Non-commutative Donaldson-Thomas theory and the conifold
• On moduli spaces of quiver representations associated with dimer models
• Higgsing M2-brane Theories
• Crepant resolutions and brane tilings I: Toric realization
• Brane Tilings, M2-branes and Chern-Simons Theories
• Brane Tilings and Non-Commutative Geometry

Cited by in corpus (8)

• Holomorphic field theories and Calabi--Yau algebras
• Combinatorial Reid's recipe for consistent dimer models
• Dimer algebras, ghor algebras, and cyclic contractions
• Mutations of splitting maximal modifying modules: The case of reflexive polygons
• Nonlocality and the central geometry of dimer algebras
• Geometric Reid's recipe for dimer models
• Non-commutative crepant resolutions of Hibi rings with small class group
• Linear Strands Supported on Regular CW Complexes