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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
Published electronically: June 15, 2009
MathSciNet review: 2529919
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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

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.

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