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Presentation of critical modules
of GK-dimension 2 over elliptic algebras


Authors: K. Ajitabh and M. Van den Bergh
Journal: Proc. Amer. Math. Soc. 127 (1999), 1633-1639
MSC (1991): Primary 16D50, 16E10, 18G10
DOI: https://doi.org/10.1090/S0002-9939-99-04717-6
Published electronically: February 18, 1999
MathSciNet review: 1485455
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that critical modules of Gelfand-Kirillov dimension 2 and multiplicity $d$ over an elliptic algebra have (up to modules of lower GK-dimension and shifting) a presentation by $d\times d$-matrices of linear forms. In the language of non-commutative algebraic geometry this amounts to a generic description of ``curves'' of degree $d$ in a projective quantum plane.


References [Enhancements On Off] (What's this?)

  • 1. K. Ajitabh, Modules over regular algebras and quantum planes, Ph.D. thesis, MIT, 1994.
  • 2. -, Residue complex for three-dimensional Sklyanin algebras, to appear in Adv. in Math.
  • 3. -, Modules over elliptic algebras and quantum planes, Proc. London Math. Soc. (3) 72 (1996), 567-587. MR 97a:16049
  • 4. M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. in Math. 109 (1994), no. 2, 228-287. MR 96a:14004
  • 5. M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), 171-216. MR 88k:16003
  • 6. M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, vol. 1, Birkhäuser, 1990, pp. 33-85. MR 92e:14002
  • 7. -, Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), 335-388. MR 93e:16055
  • 8. M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), 249-271. MR 91k:14003
  • 9. T. Levasseur, Some properties of non-commutative regular rings, Glasgow Math. J. 34 (1992), 277-300. MR 93k:16045
  • 10. A. Yekutieli, The residue complex of a non-commutative graded algebra, J. Algebra 186 (1996), 522-543. CMP 97:05

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Additional Information

K. Ajitabh
Affiliation: Department of Mathematics, Florida International University, University Park, Miami, Florida 33199
Email: ajitabhk@zeus.fiu.edu

M. Van den Bergh
Affiliation: Departement WNI, Limburgs Universitair Centrum, Universitaire Campus, 3590 Diepenbeek, Belgium
Email: vdbergh@luc.ac.be

DOI: https://doi.org/10.1090/S0002-9939-99-04717-6
Keywords: Elliptic algebras, quantum planes, regular algebra, critical modules, Cohen-Macaulay modules
Received by editor(s): September 23, 1997
Published electronically: February 18, 1999
Additional Notes: The second author is a senior researcher at the FWO
Communicated by: Lance W. Small
Article copyright: © Copyright 1999 American Mathematical Society

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