Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-27T22:09:48.830Z Has data issue: false hasContentIssue false

Some properties of non-commutative regular graded rings

Published online by Cambridge University Press:  18 May 2009

Thierry Levasseur
Affiliation:
Département de Mathématiques, Université de Bretagne Occidentale, 6, Avenue Victor Le Gorgeu, 29287 Brest Cedex, France
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a supplementary condition. Before stating it, we recall that the grade of a finitely generated (left or right) module is defined by

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Artin, M. and Schelter, W. F., Graded algebras of global dimension 3, Adv. in Math. 66 (1987), 171216.CrossRefGoogle Scholar
2.Atiyah, M. F., Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 27 (1957), 414452.CrossRefGoogle Scholar
3.Artin, M., Tate, J. and Van den Bergh, M., Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift (Birkhauser, 1990), 3385.Google Scholar
4.Artin, M., Tate, J. and Van den Bergh, M., Modules over regular algebras of dimension 3, preprint (1988).Google Scholar
5.Artin, M. and Van den Bergh, M., Twisted homogeneous coordinate rings, Algebra 133 (1990), 249271.CrossRefGoogle Scholar
6.Björk, J. E., Rings of differential operators (North-Holland, 1979).Google Scholar
7.Björk, J. E., The Auslander condition on noetherian rings, Séminaire Dubreil-Malliavin 1987–88, Lecture Notes in Mathematics 1404 (Springer, 1989), 137173.CrossRefGoogle Scholar
8.Björk, J. E. and Ekström, E. K., Filtered Auslander-Gorenstein rings, Colloque en l'honneur de J. Dixmier (Birkhauser, 1990), 424448.Google Scholar
9.Bourbaki, N., Algébre, Chapitre 10 (Masson, 1980).Google Scholar
10.Ekström, E. K., The Auslander condition on graded and filtered noetherian rings, Séminaire Dubreil-Malliavin 1987–1988, Lecture Notes in Mathematics 1404 (Springer, 1989), 220245.CrossRefGoogle Scholar
11.Fossum, R. M., Griffith, P. A. and Reiten, I., Trivial extensions of abelian categories with applications to ring theory, Lecture Notes in Mathematics 456 (Springer, 1975).CrossRefGoogle Scholar
12.Ischebeck, F., Eine Dualität Zwischen den Funktoren Ext und Tor, J. Algebra 11 (1969), 510531.CrossRefGoogle Scholar
13.Krause, G. and Lenagan, T. H., Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics 116 (Pitman, 1985).Google Scholar
14.Levasseur, T., Complexe bidualisant en algèbre non commutative, Séminaire Dubreil-Malliavin 1983–84, Lecture Notes in Mathematics 1146 (Springer, 1985), 270287.CrossRefGoogle Scholar
15.Levasseur, T. and Smith, S. P., Modules over the Sklyanin algebra, preprint (1991).Google Scholar
16.Huishi, Li, Non-commutative Zariskian rings (Ph. Doc. thesis, U.I.A., Antwerp, 1989).Google Scholar
17.McConnell, J. C. and Robson, J. C., Non-commutative noetherian rings (Wiley, 1987).Google Scholar
18.Smith, S. P. and Stafford, J. T., Regularity of the four dimensional Sklyanin algebra, preprint (1989).Google Scholar
19.Stafford, J. T., Regularity of algebras related to the Sklyanin algebra, preprint (1990).Google Scholar
20.Yekutieli, A., The residue complex and duality for some non-commutative rings (Ph. Doc. thesis, MIT, 1990).Google Scholar
21.Yekutieli, A., Dualizing complexes over noncommutative graded algebras, preprint (1990).Google Scholar