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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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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.
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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