Noncommutative graded domains with quadratic growth Article DOI:
Cite this article as: Artin, M. & Stafford, J.T. Invent Math (1995) 122: 231. doi:10.1007/BF01231444 Abstract
k be an algebraically closed field, and let R be a finitely generated, connected graded k-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ring Q(R) of R as D[z,z −1; δ], for some automorphism δ of the division ring D. We prove that D is a finitely generated field extension of k of transcendence degree one. Moreover, we describe R in terms of geometric data. If R 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 curve Y. This implies, in particular, that R is Noetherian, that R is primitive when |δ|=∞ and that R is a finite module over its centre when |δ|<∞. If R is not generated in degree one, then R will still be Noetherian and primitive if δ has infinite order, but R need not be Noetherian when δ has finite order.
Dedicated to the memory of Shimshon Amitsur
Oblatum 5-XI-1994 & 28-III-1995
This research was supported in part by NSF grants
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