Criteria for σ-ampleness
Abstract
Description
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.
16 pages, LaTeX2e, to appear in J. of the AMS, minor errors corrected (esp. in 1.4 and 3.1), proofs simplified
16 pages, LaTeX2e, to appear in J. of the AMS, minor errors corrected (esp. in 1.4 and 3.1), proofs simplified