## Criteria for $\sigma$-ampleness

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- by Dennis S. Keeler PDF
- J. Amer. Math. Soc.
**13**(2000), 517-532 Request permission

## Abstract:

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.## References

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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
- Received by editor(s): December 13, 1999
- Published electronically: March 29, 2000
- Additional Notes: The author was partially supported by NSF grant DMS-9801148.
- © Copyright 2000 American Mathematical Society
- 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
- MathSciNet review: 1758752