Criteria for -ampleness

Author:
Dennis S. Keeler

Journal:
J. Amer. Math. Soc. **13** (2000), 517-532

MSC (2000):
Primary 14A22, 14F17, 14J50, 16P90, 16S38, 16W50

DOI:
https://doi.org/10.1090/S0894-0347-00-00334-9

Published electronically:
March 29, 2000

MathSciNet review:
1758752

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Abstract | References | Similar Articles | Additional Information

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 -ample divisor, where is an automorphism of a projective scheme . Many open questions regarding -ample divisors have remained.

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

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Additional Information

**Dennis S. Keeler**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Email:
dskeeler@umich.edu

DOI:
https://doi.org/10.1090/S0894-0347-00-00334-9

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