Abstract
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.
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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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Artin, M., Stafford, J.T. Noncommutative graded domains with quadratic growth. Invent Math 122, 231–276 (1995). https://doi.org/10.1007/BF01231444
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DOI: https://doi.org/10.1007/BF01231444