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Towards a theory of Bass numbers with application to Gorenstein algebras

100%

Goto S., Nishida K.

Colloquium Mathematicum

2002

tom 91

nr 2

191-253

The notion of Gorenstein rings in the commutative ring theory is generalized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the notion of Cousin complexes developed by R. Y. Sharp [Sh1]. One of the goals of the present paper is the characterization of Gorenstein algebras in terms of Bass numbers. The commutative theory of Bass numbers turns out to carry over with no extra changes. Certain algebras having locally finite global dimension are also characterized. The special case where the algebras are free modules over base rings is explored. Thanks to these observations, it is clarified how the Gorensteinness is inherited under flat base changes. In conclusion, a characterization for local algebras to be Gorenstein is given, accounting for the reason why the theory behaves so well in the commutative case. Examples are explored and open problems are given. See [GN2] and [GN3] for further developments.

On stable equivalences of module subcategories over a semiperfect noetherian ring

81%

Kameyama N., Kimura Y., Nishida K.

Colloquium Mathematicum

2014

tom 137

nr 1

7-26

Given a semiperfect two-sided noetherian ring Λ, we study two subcategories $𝓐_k(Λ) = {M ∈ mod Λ | Ext_{Λ}^{j}(Tr M,Λ) = 0 (1 ≤ j ≤ k)}$ and $𝓑_k(Λ) = {N ∈ mod Λ | Ext_{Λ}^{j}(N,Λ) = 0 (1 ≤ j ≤ k)}$ of the category mod Λ of finitely generated right Λ-modules, where Tr M is Auslander's transpose of M. In particular, we give another convenient description of the categories $𝓐_{k}(Λ)$ and $𝓑_{k}(Λ)$, and we study category equivalences and stable equivalences between them. Several results proved in [J. Algebra 301 (2006), 748-780] are extended to the case when Λ is a two-sided noetherian semiperfect ring.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Nishida K.

1 Goto S.

1 Kameyama N.

1 Kimura Y.

1 2014

1 2002

Wyniki wyszukiwania

Towards a theory of Bass numbers with application to Gorenstein algebras

On stable equivalences of module subcategories over a semiperfect noetherian ring