Mathematics – Algebraic Geometry
Scientific paper
2010-08-09
Advances in Mathematics 229 (2012), no. 3., 1516-1554
Mathematics
Algebraic Geometry
34 pages, 10 figures; v2 simplified proof of Theorem 5.9 and expanded Section 6; v3 added Lemma 2.1, final version
Scientific paper
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 $\Delta$.
Craw Alastair
Velez Alexander Quintero
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