Cellular resolutions of noncommutative toric algebras from superpotentials
Cellular resolutions of noncommutative toric algebras from superpotentials
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DOI：
10.1016/j.aim.2011.11.012
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发表时间：
2010-08
影响因子：
1.7
通讯作者：
Alastair Craw;A. Q. Vélez
中科院分区：
文献类型：
--
作者：
Alastair Craw;A. Q. Vélez
本文构造了非交换代数类的元胞分解，类似于Bayer和Sturmfels（1998）在交换情况下引入的分解。为了实现这一点，我们通过将超势和一致性概念与任意维的环代数联系起来，推广了三维环代数非交换渐变分辨率的二聚体模型构造。对于阿贝尔偏群代数和代数上一致的二聚体模型代数，我们在一个实环面中引入了一个细胞复合体Δ，其细胞在a的最小投影双模分辨率下均匀地描述了所有映射。我们举例说明了Δ在四维中的一般构造，该构造是由光滑环面Fano三倍上的倾斜束产生的，以突出Δ上的关联函数的重要性。
This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998) [2] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. 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.