Graded, Selfinjective, and Koszul Algebras
Graded, Selfinjective, and Koszul Algebras
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DOI：
10.1006/jabr.1998.7728
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发表时间：
1999-05
影响因子：
0.9
通讯作者：
R. Martínez-Villa
中科院分区：
文献类型：
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作者：
R. Martínez-Villa
摘要在本文中，我们继续在早期研究中发起的Koszul代数的工作。关于Koszul模块的几乎分裂序列存在的考虑在我们的早期工作中出现，只有部分答案是已知的。 Koszul二元性将无限全局尺寸的有限维代数与无限维代数相关联，因此分级小组的注入尺寸是有限的。自然出现了两个问题：无限尺寸分级代数的几乎分裂序列以及分级代数的整体维度与分级小组的注射尺寸之间的关系。在本文的第一部分中，我们处理第一件事；在第二个中，我们考虑了分级的全球维度与分级小组的注射尺寸的最大关系之间的关系，证明在诺瑟里亚的情况下它们是重合的。在第三部分中，我们考虑有限的尺寸级别的自我注射颤动代数，并证明对于此类代数，所有不可分解的投影模块具有相同的lowey长度。在本文的第四部分和最后一部分中，我们专门研究自我介绍性的Koszul代数，并将其代数为代数，证明它们构成了交换性常规代数的非交通性版本。
Abstract In this paper we continue our work on Koszul algebras initiated in earlier studies. The consideration about the existence of almost split sequences for Koszul modules appeared in our early work and only a partial answer is known. Koszul duality relates finite dimensional algebras of infinite global dimension with infinite dimensional algebras such that the injective dimension of the graded simples is finite. Two questions naturally arise: the existence of almost split sequences for infinite dimensional graded algebras and the relations between the global dimension of a graded algebra and the injective dimension of the graded simples. In the first part of the paper, we deal with the first matter; in the second, we consider the relations between the graded global dimension and the maximum of the injective dimensions of the graded simples, proving that in the noetherian case they coincide. In the third part, we consider finite dimensional graded selfinjective quiver algebras and prove that for such algebras all indecomposable projective modules have the same Lowey length. In the forth and last part of the paper, we specialize to selfinjective Koszul algebras and characterize their Yoneda algebras, proving they constitute non-commutative versions of the commutative regular algebras.