Cellular resolutions of noncommutative toric algebras from superpotentials

Alastair Craw, Alexander Quintero Velez

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8 Citations (SciVal)


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 $\Delta$.
Original languageEnglish
Pages (from-to)1516-1554
JournalAdvances in Mathematics
Issue number3
Early online date21 Dec 2011
Publication statusPublished - 15 Feb 2012


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