Study of Derived Equivalences

派生等价研究

基本信息

批准号：
16540009
负责人：
HOSHINO Mitsuo
金额：
$ 1.28万
依托单位：
University of Tsukuba
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

项目摘要

We show that a partial tilting complex over a representation-finite selfinjective artin algebra appears as a direct summand of a tilting complex whenever it has a selfinjective endomorphism ring in the derived category, that for any two derived equivalent representation-finite selfinjective artin algebras there there exists a derived equivalence between them which is an iteration of derived equivalences induced by two-term tilting complexes, that the derived equivalent representation-finite selfinjective artin algebras have the same Nakayama permutation, and that every tilting complex over a selfinjective artin algebra with a cyclic Nakayama permutation and with a selfinjective endomorphism ring is isomorphic to a one-term complex, i.e., every derived equivalence between two selfinjective artin algebras with a cyclic Nakayama permutation is a Morita equivalence.We provide a generalization of the Auslander formula which induces a nice filtration on each finitely generated left module over a left and right noetherian ring satisfying the Auslander condition. Furthermore, we determine the concrete form of modules in this filtration.We introduce the notion of full matrix algebras with structure systems and show that certain factor algebras of Gorenstein tiledorders are full matrix algebras with Frobenius structure systems, that the converse is also true if the size of the matrix is smaller the or equal to seven and that the converse is not true if the size of the matrix is greater then seven.We show that the Grothendieck groups of derived categories of bounded above (resp., bounded below and unbounded) complexes of finitely generated left modules over a left artinian ring are trivial.

我们表明，每当有一个倾斜复合物的倾斜复合物在派生的类别中具有倾斜复合物的直接汇总，在派生类别中倾斜复合体的直接汇总似乎是一个直接汇总的倾斜复合物，这是派生的类别中的一个自我介绍性的戒指，对于两种衍生的同等表示，在其中散发出了两种等式的同等词，这是在同等的范围内，这是一个等式的等式。复合物，即派生的等效表示 - 限制自以为是的artin代数具有相同的nakayama置换，并且每个倾斜复合体在一个自我介绍性的artin代数上具有环状nakayama置换术的自我介绍，并且具有自我内向性的内态内态词与一个单期的构造，即单一的平等，即一个单独的平等。 Nakayama置换是一种莫里塔等效性。我们提供了Auslander公式的概括，该公式在满足Auslander条件的每个有限生成的左和右noetherian环上诱导了每个有限生成的左和右模块的过滤。此外，我们确定了该过滤中模块的混凝土形式。我们介绍具有结构系统的完整矩阵代数的概念，并表明戈伦斯坦·泰勒德（Gorenstein Tiledorders）的某些因子代数是完整的基质代数，该代数与frobenius结构系统相同，如果较小的是，如果是较小的，那么如果是较小的，那么如果是较小的，那么如果是真的，那么如果是较小的，则是差异的尺寸。七。我们表明，在左Artinian环上有限生成的左模块的上面有限类别的派生类别（分别，有限的下方和无界）复合物是微不足道的。