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Examples in non-commutative projective geometry

Published online by Cambridge University Press:  24 October 2008

J. T. Stafford  and
J. J. Zhang
J. T. Stafford
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.
J. J. Zhang
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.
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Extract

Let A = k ⊕ ⊕n ≥ 1An connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 3 , November 1994 , pp. 415 - 433
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[Ar]Abtin, M.. Lecture at the Ring Theory Conference, Wake Forest (11, 1991).Google Scholar
[AS]Artin, M. and Schelter, W.. Graded algebras of dimension 3. Adv. Math. 66 (1987), 172–216.CrossRefGoogle Scholar
[ATV1]Artin, M., Tate, J. and van den Bergh, M.. Some algebras associated to automorphisms of curves, in The Grothendieck Festschrift (Eds. Cartier, P. et al. ,). (Birkhauser, 1990).Google Scholar
[ATV2]Artin, M., Tate, J. and van den Bergh, M.. Modules over regular algebras of dimension 3. Inventories Math. 106 (1991), 335–389.CrossRefGoogle Scholar
[AV]Artin, M. and Bergh, M. van den. Twisted homogeneous coordinate rings. J. Algebra, 186 (1990), 249–271.CrossRefGoogle Scholar
[AZ]Artin, M. and Zhang, J. J.. Noncommutative projective schemes, Adv. Math., to appear.Google Scholar
[Ha]Hartshorne, R.. Algebraic Geometry (Springer-Verlag, 1977).CrossRefGoogle Scholar
[LS]Levasseur, T. and Stafford, J. T.. The quantum coordinate ring of the special linear group. J. Pure and Appl. Algebra 86 (1993), 181–186.CrossRefGoogle Scholar
[MR]McConnell, J. C. and Robson, J. C.. Non-commutative Noetherian Rings. (Wiley-Inter-science, 1987).Google Scholar
[Mu]Mumford, D.. The Red Book on Varieties and Schemes. Lecture Notes in Math. No. 1358. (Springer-Verlag, 1988).CrossRefGoogle Scholar
[Po]Popescu, N.. Abelian Categories with Applications to Rings and Modules. (Academic Press, 1973).Google Scholar
[Ro]Rotman, J. J.. An Introduction to Homological Algebra. (Academic Press, 1979).Google Scholar
[Se]Serre, J.-P.. Faisceaux Algebriques Coherents. Annals of Math. 61 (1955), 197–278.CrossRefGoogle Scholar
[Sm]Smith, S. P.. The four dimensional Sklyanin algebra at points of finite order, to appear.Google Scholar
[SS]Smith, S. P. and Stafford, J. T.. Regularity of the four dimensional Sklyanin algebra. Comp. Math. 83 (1992), 259–289.Google Scholar
[St1]Stafford, J. T.. On the ideals of a Noetherian ring. Trans. Amer. Math. Soc. 289 (1985), 381–392.CrossRefGoogle Scholar
[St2]Stafford, J. T.. Global dimension of semiprime, Noetherian rings, in Séminaire Malliavin (Lecture Notes in Math. No. 1296). (Springer-Verlag, 1987).Google Scholar
[SZ]Stafford, J. T. and Zhang, J. J.. Homological properties of (graded) Noetherian PI rings. J. Algebra, to appear.Google Scholar
[TV]Tate, J. and Behgh, M. van den. Homological properties of Sklyanin algebras, to appear.Google Scholar
[Zh]Zhang, J. J.. Twisted graded algebras and equivalences of graded categories, to appear.Google Scholar