Towards a theory of Bass numbers with application to Gorenstein algebras

Shiro Goto, and Kenji Nishida. "Towards a theory of Bass numbers with application to Gorenstein algebras." Colloquium Mathematicae 91.2 (2002): 191-253. <http://eudml.org/doc/284184>.

@article{ShiroGoto2002,
abstract = {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.},
author = {Shiro Goto, Kenji Nishida},
journal = {Colloquium Mathematicae},
keywords = {Gorenstein algebras; non-commutative rings; Cousin complexes; injective resolutions; Bass numbers; homological dimensions; module-finite algebras; Cohen-Macaulay rings; local rings; Noetherian algebras},
language = {eng},
number = {2},
pages = {191-253},
title = {Towards a theory of Bass numbers with application to Gorenstein algebras},
url = {http://eudml.org/doc/284184},
volume = {91},
year = {2002},
}

TY - JOUR
AU - Shiro Goto
AU - Kenji Nishida
TI - Towards a theory of Bass numbers with application to Gorenstein algebras
JO - Colloquium Mathematicae
PY - 2002
VL - 91
IS - 2
SP - 191
EP - 253
AB - 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.
LA - eng
KW - Gorenstein algebras; non-commutative rings; Cousin complexes; injective resolutions; Bass numbers; homological dimensions; module-finite algebras; Cohen-Macaulay rings; local rings; Noetherian algebras
UR - http://eudml.org/doc/284184
ER -