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Criteria for $\sigma$-ampleness

Author: Dennis S. Keeler
Journal: J. Amer. Math. Soc. 13 (2000), 517-532
MSC (2000): Primary 14A22, 14F17, 14J50, 16P90, 16S38, 16W50
Published electronically: March 29, 2000
MathSciNet review: 1758752
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In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a $\sigma$-ample divisor, where $\sigma$ is an automorphism of a projective scheme $X$. Many open questions regarding $\sigma$-ample divisors have remained.

We derive a relatively simple necessary and sufficient condition for a divisor on $X$ to be $\sigma$-ample. As a consequence, we show right and left $\sigma$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $\sigma$ yield a $\sigma$-ample divisor.

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Additional Information

Dennis S. Keeler
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Keywords: Noetherian graded rings, noncommutative projective geometry, automorphisms, vanishing theorems
Received by editor(s): December 13, 1999
Published electronically: March 29, 2000
Additional Notes: The author was partially supported by NSF grant DMS-9801148.
Article copyright: © Copyright 2000 American Mathematical Society

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