Geodesic
Towards a theory of Bass numbers with application to Gorenstein algebras
Shiro Goto1 ; Kenji Nishida2
1 Department of Mathematics Meiji University Kawasaki 214-8571, Japan
2 Department of Mathematical Sciences Shinshu University Matsumoto 390-8621, Japan
Colloquium Mathematicum, Tome 91 (2002) no. 2, pp. 191-253.

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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.
DOI : 10.4064/cm91-2-4
Keywords: notion gorenstein rings commutative ring theory generalized noetherian algebras which necessarily commutative faithfully follow steps commutative gorenstein algebras defined using notion cousin complexes developed sharp goals present paper characterization gorenstein algebras terms bass numbers commutative theory bass numbers turns out carry extra changes certain algebras having locally finite global dimension characterized special where algebras modules base rings explored thanks these observations clarified gorensteinness inherited under flat base changes conclusion characterization local algebras gorenstein given accounting reason why theory behaves commutative examples explored problems given see further developments

Shiro Goto 1 ; Kenji Nishida 2

1 Department of Mathematics Meiji University Kawasaki 214-8571, Japan
2 Department of Mathematical Sciences Shinshu University Matsumoto 390-8621, Japan 
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Shiro Goto; Kenji Nishida. Towards a theory of Bass numbers with
application to Gorenstein algebras. Colloquium Mathematicum, Tome 91 (2002) no. 2, pp. 191-253. doi : 10.4064/cm91-2-4. http://geodesic.mathdoc.fr/articles/10.4064/cm91-2-4/

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