Maximal modifications and Auslander–Reiten duality for non-isolated singularities
Maximal modifications and Auslander–Reiten duality for non-isolated singularities
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DOI：
10.1007/s00222-013-0491-y
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发表时间：
2010-07
影响因子：
3.1
通讯作者：
O. Iyama;M. Wemyss
中科院分区：
文献类型：
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作者：
O. Iyama;M. Wemyss
我们首先将经典的Auslander-Reiten对偶推广到具有一维奇异轨迹的孤立奇点。然后，我们定义了非孤立奇点的CT模的概念，并证明了它与非对易分解(NCCR)密切相关。当R分离奇点时，CT模块恢复了经典的集群倾斜模块的概念，但总的来说，这两个概念是不同的。然后，为了将NCCRs的概念推广到的部分分解，在本文的主体部分，我们引入了修正和极大修正模的理论。在较弱的假设下，证明了三维Gorenstein环的极大修正模的所有对应的自同态代数是导出等价的。然后，我们发展了一种修改模的突变理论，它类似于但不同于簇倾斜理论中出现的突变。我们的突变在任意维度上起作用，而在第三维，我们突变的行为很大程度上取决于某个因子代数是否为Artin代数。
We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). WhenRhas isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of, in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.