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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

GK-dimension of birationally commutative surfaces


Author: D. Rogalski
Journal: Trans. Amer. Math. Soc. 361 (2009), 5921-5945
MSC (2000): Primary 14A22, 14E05, 16P90, 16S38, 16W50
DOI: https://doi.org/10.1090/S0002-9947-09-04885-5
Published electronically: June 15, 2009
MathSciNet review: 2529919
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $k$ be an algebraically closed field, let $K/k$ be a finitely generated field extension of transcendence degree $2$, let $\sigma \in \operatorname {Aut}_k(K)$, and let $A \subseteq Q = K[t; \sigma ]$ be an $\mathbb {N}$-graded subalgebra with $\dim _k A_n < \infty$ for all $n \geq 0$. Then if $A$ is big enough in $Q$ in an appropriate sense, we prove that $\operatorname {GK} A = 3,4,5,$ or $\infty$, with the exact value depending only on the geometric properties of $\sigma$. The proof uses techniques in the birational geometry of surfaces which are of independent interest.


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Additional Information

D. Rogalski
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
MR Author ID: 734142
Email: drogalsk@math.ucsd.edu

Keywords: GK-dimension, graded rings, noncommutative projective geometry, noncommutative surfaces, birational geometry
Received by editor(s): September 19, 2007
Published electronically: June 15, 2009
Additional Notes: The author was partially supported by the NSF through grants DMS-0202479 and DMS-0600834.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.