Inventiones mathematicae

, Volume 122, Issue 1, pp 231–276

Noncommutative graded domains with quadratic growth

  • M. Artin
  • J. T. Stafford

DOI: 10.1007/BF01231444

Cite this article as:
Artin, M. & Stafford, J.T. Invent Math (1995) 122: 231. doi:10.1007/BF01231444


Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ|<∞. IfR is not generated in degree one, thenR will still be Noetherian and primitive if δ has infinite order, butR need not be Noetherian when δ has finite order.

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • M. Artin
    • 1
  • J. T. Stafford
    • 2
  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA