On algebras of finite Cohen-Macaulay type
On algebras of finite Cohen-Macaulay type
复制标题
DOI：
10.1016/j.aim.2010.09.006
复制
发表时间：
2011-01
影响因子：
1.7
通讯作者：
Apostolos Beligiannis
中科院分区：
文献类型：
--
作者：
Apostolos Beligiannis
我们研究了Artin代数Λ和交换Noether完备局部环R与Gorenstein投射模的以下分解性质：我们证明了代数Λ enjoy（†）的类与Bkillannis和Reiten（2007）[27]，Bkillannis（2005）[24]中引入的有限Cohen-Macaulay型虚Gorenstein代数的类重合。因此，我们解决了Chen（2008）[33]中提出的问题。这是通过刻画一个分解子范畴在滤余极限下闭包的分解性质来证明的，从而推广了Auslander（1976）[9]和Ringel and Tachikawa（1974）[63]的一个经典结果。在交换情形下，若R中存在一个非自由生成Gorenstein-投射模，则证明了R是有限Cohen-Macaulay型的当且仅当R是Gorenstein模且满足（†）.我们还推广了Yoshino（2005）[68]的一个结果，刻画了无环扩张的n-生成模何时是Gorenstein-投射的。最后，研究了有限Cohen-Macaulay型虚Gorenstein代数的（稳定）相对Auslander代数，并在簇倾斜对象存在的情况下，利用与稳定相对Auslander代数的簇相关的簇范畴给出了Gorenstein投射模的稳定范畴的刻画.在这种情况下，我们表明，集群类别是不变的派生等价。
We study Artin algebras Λ and commutative Noetherian complete local rings R in connection with the following decomposition property of Gorenstein-projective modules: We show that the class of algebras Λ enjoying (†) coincides with the class of virtually Gorenstein algebras of finite Cohen–Macaulay type, introduced in Beligiannis and Reiten (2007) [27], Beligiannis (2005) [24]. Thus we solve the problem stated in Chen (2008) [33]. This is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits, thus generalizing a classical result of Auslander (1976) [9] and Ringel and Tachikawa (1974) [63]. In the commutative case, if R admits a non-free finitely generated Gorenstein-projective module, then we show that R is of finite Cohen–Macaulay type iff R is Gorenstein and satisfies (†). We also generalize a result of Yoshino (2005) [68] by characterizing when finitely generated modules without extensions with the ring are Gorenstein-projective. Finally we study the (stable) relative Auslander algebra of a virtually Gorenstein algebra of finite Cohen–Macaulay type and, under the presence of a cluster tilting object, we give descriptions of the stable category of Gorenstein-projective modules in terms of the cluster category associated to the quiver of the stable relative Auslander algebra. In this setting we show that the cluster category is invariant under derived equivalences.