Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T02:01:22.914Z Has data issue: false hasContentIssue false
  • English
  • Français

Some projective surfaces of GK-dimension 4

Part of: Rings and algebras arising under various constructions Rings and algebras with additional structure Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  15 May 2012

D. Rogalski  and
Susan J. Sierra
D. Rogalski
Affiliation:
UCSD Department of Mathematics, 9500 Gilman Dr. #0112, La Jolla, CA 92093-0112, USA (email: [email protected])
Susan J. Sierra
Affiliation:
School of Mathematics, James Clerk Maxwell Building, King’s Buildings, University of Edinburgh, Edinburgh EH9 3JZ, UK (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

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.

We construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ2, and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ1×ℙ1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.

Keywords

noncommutative projective geometrynoncommutative surfacesnoetherian graded ringsample sequence

MSC classification

Secondary: 16S38: Rings arising from non-commutative algebraic geometry 16P40: Noetherian rings and modules 16S80: Deformations of rings 16W50: Graded rings and modules
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1195 - 1237
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AV90]Artin, M. and Van den Bergh, M., Twisted homogeneous coordinate rings, J. Algebra 133 (1990), 249–271.CrossRefGoogle Scholar
[Ber78]Bergman, G. M., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218.CrossRefGoogle Scholar
[BGT08]Bell, J., Ghioca, D. and Tucker, T. J., The dynamical Mordell-Lang problem for étale maps (2008), arXiv:0808.3266.Google Scholar
[DF01]Diller, J. and Favre, C., Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135–1169.CrossRefGoogle Scholar
[Har77]Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, NY, 1977).CrossRefGoogle Scholar
[J{\OT1\o }r98]Jørgensen, P., Non-commutative graded homological identities, J. Lond. Math. Soc. (2) 57 (1998), 336–350.CrossRefGoogle Scholar
[KRS05]Keeler, D. S., Rogalski, D. and Stafford, J. T., Naïve noncommutative blowing up, Duke Math. J. 126 (2005), 491–546.CrossRefGoogle Scholar
[Li96]Li, H., Global dimension of graded local rings, Comm. Algebra 24 (1996), 2399–2405.Google Scholar
[Rog04]Rogalski, D., Generic noncommutative surfaces, Adv. Math. 184 (2004), 289–341.CrossRefGoogle Scholar
[Rog09]Rogalski, D., GK-dimension of birationally commutative surfaces, Trans. Amer. Math. Soc. 361 (2009), 5921–5945.CrossRefGoogle Scholar
[RS07]Rogalski, D. and Stafford, J. T., Naïve noncommutative blowups at zero-dimensional schemes, J. Algebra 318 (2007), 794–833.CrossRefGoogle Scholar
[RS09]Rogalski, D. and Stafford, J. T., A class of noncommutative projective surfaces, Proc. Lond. Math. Soc. (3) 99 (2009), 100–144.CrossRefGoogle Scholar
[Sie08]Sierra, S. J., Geometric idealizer rings, Trans. Amer. Math. Soc. 363 (2008), 457–500.CrossRefGoogle Scholar
[Sie10]Sierra, S. J., Geometric algebras on projective surfaces, J. Algebra 324 (2010), 1687–1730.CrossRefGoogle Scholar
[Sie11]Sierra, S. J., Classifying birationally commutative projective surfaces, Proc. Lond. Math. Soc. (3) 103 (2011), 139–196.CrossRefGoogle Scholar
[Sie]Sierra, S. J., Algebras of GK-dimension five birational to commutative surfaces, in preparation.Google Scholar
[SV01]Stafford, J. T. and Van den Bergh, M., Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc. 38 (2001), 171–216.CrossRefGoogle Scholar
[SV06]Sidman, J. and Van Tuyl, A., Multigraded regularity: syzygies and fat points, Beiträge Algebra Geom. 47 (2006), 67–87.Google Scholar
[SZ00]Stephenson, D. R. and Zhang, J. J., Noetherian connected graded algebras of global dimension 3, J. Algebra 230 (2000), 474–495.CrossRefGoogle Scholar
[Van96]Van den Bergh, M., A translation principle for the four-dimensional Sklyanin algebras, J. Algebra 184 (1996), 435–490.CrossRefGoogle Scholar
[Van97]Van den Bergh, M., Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), 662–679.CrossRefGoogle Scholar
[Wei94]Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar
[YZ06]Yekutieli, A. and Zhang, J. J., Homological transcendence degree, Proc. Lond. Math. Soc. (3) 93 (2006), 105–137.CrossRefGoogle Scholar