Complete Cohomologies and Some Homological Invariants

• 来源期刊: 代数集刊（英文版）
• 关键词: complete cohomology Auslander's δ-invariant Loewy length
• 年，卷（期）: 2007,（1）
• 页码范围: 155-166
• 页数: 12页
• 语种: 英文

摘要

There is a complete cohomology theory developed over a commutative noetherian ring in which injectives take the role of projectives in Vogel's construction of complete cohomology theory. We study the interaction between this complete cohomology, that is referred to as I-complete cohomology, and Vogel's one and give some sufficient conditions for their equivalence. Using I-complete functors, we assign a new homological invariant to any finitely generated module over an arbitrary commutative noetherian local ring,that would generalize Auslander's delta invariant. We generalize the results about the δ-invariant to arbitrary rings and give a sufficient condition for the vanishing of this new invariant. We also introduce an analogue of the notion of the index of a Gorenstein local ring, introduced by Auslander, for arbitrary local rings and study its behavior under flat extensions of local rings. Finally, we study the connection between the index and Loewy length of a local ring and generalize the main result of [11] to arbitrary rings.