Some properties of non-commutative regular graded rings

Levasseur, Thierry

Glasgow mathematical journal, Tome 34 (1992) no. 3, pp. 277-300

Résumé

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

Détail

DOI : 10.1017/S0017089500008843

Bibliographie

[1] 1.Artin, M. and Schelter, W. F., Graded algebras of global dimension 3, Adv. in Math. 66 (1987), 171–216. Google Scholar | DOI

[2] 2.Atiyah, M. F., Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 27 (1957), 414–452. Google Scholar | DOI

[3] 3.Artin, M., Tate, J. and Van den Bergh, M., Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift (Birkhauser, 1990), 33–85. Google Scholar

[4] 4.Artin, M., Tate, J. and Van den Bergh, M., Modules over regular algebras of dimension 3, preprint (1988). Google Scholar

[5] 5.Artin, M. and Van den Bergh, M., Twisted homogeneous coordinate rings, Algebra 133 (1990), 249–271. Google Scholar | DOI

[6] 6.Björk, J. E., Rings of differential operators (North-Holland, 1979). Google Scholar

[7] 7.Björk, J. E., The Auslander condition on noetherian rings, Séminaire Dubreil-Malliavin 1987–88, Lecture Notes in Mathematics 1404 (Springer, 1989), 137–173. Google Scholar | DOI

[8] 8.Björk, J. E. and Ekström, E. K., Filtered Auslander-Gorenstein rings, Colloque en l'honneur de J. Dixmier (Birkhauser, 1990), 424–448. Google Scholar

[9] 9.Bourbaki, N., Algébre, Chapitre 10 (Masson, 1980). Google Scholar

[10] 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), 220–245. Google Scholar | DOI

[11] 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). Google Scholar | DOI

[12] 12.Ischebeck, F., Eine Dualität Zwischen den Funktoren Ext und Tor, J. Algebra 11 (1969), 510–531. Google Scholar | DOI

[13] 13.Krause, G. and Lenagan, T. H., Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics 116 (Pitman, 1985). Google Scholar

[14] 14.Levasseur, T., Complexe bidualisant en algèbre non commutative, Séminaire Dubreil-Malliavin 1983–84, Lecture Notes in Mathematics 1146 (Springer, 1985), 270–287. Google Scholar | DOI

[15] 15.Levasseur, T. and Smith, S. P., Modules over the Sklyanin algebra, preprint (1991). Google Scholar

[16] 16.Huishi, Li, Non-commutative Zariskian rings (Ph. Doc. thesis, U.I.A., Antwerp, 1989). Google Scholar

[17] 17.McConnell, J. C. and Robson, J. C., Non-commutative noetherian rings (Wiley, 1987). Google Scholar

[18] 18.Smith, S. P. and Stafford, J. T., Regularity of the four dimensional Sklyanin algebra, preprint (1989). Google Scholar

[19] 19.Stafford, J. T., Regularity of algebras related to the Sklyanin algebra, preprint (1990). Google Scholar

[20] 20.Yekutieli, A., The residue complex and duality for some non-commutative rings (Ph. Doc. thesis, MIT, 1990). Google Scholar

[21] 21.Yekutieli, A., Dualizing complexes over noncommutative graded algebras, preprint (1990). Google Scholar