Noncommutative ampleness for multiple divisors

Details Noncommutative ampleness for multiple divisors Noncommutative ampleness for multiple divisors

2002-10-27

arxiv.org/abs/math/0210417v2

J. Algebra 265 (2003), no. 1, 299--311.

Mathematics

Algebraic Geometry

11 pages, LaTeX, minor corrections, to appear in J. Algebra

Scientific paper

10.1016/S0021-8693(03)00126-1

The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Thus we find under natural hypotheses that multi-homogeneous coordinate rings are noetherian and have integer GK-dimension.

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