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Criteria for σ-ampleness

Dennis S. Keeler
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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 $σ$-ample divisor, where $σ$ is an automorphism of a projective scheme X. Many open questions regarding $σ$-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be $σ$-ample. As a consequence, we show right and left $σ$-ampleness are equivalent and any associated noncommutative homogeneous coordinate ring must be noetherian and have finite, integral GK-dimension. We also characterize which automorphisms $σ$ yield a $σ$-ample divisor.

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Dennis S. Keeler. 2000-02-29. Criteria for σ-ampleness. https://doi.org/10.1090/s0894-0347-00-00334-9

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