Ample filters of invertible sheaves

Details Ample filters of invertible sheaves Ample filters of invertible sheaves

2001-08-09

arxiv.org/abs/math/0108068v3

J. Algebra 259 (2003), no. 1, 243--283.

Mathematics

Algebraic Geometry

LaTeX; 35 pages; intro rewritten (since v1) and difference between ample families and ample filters clarified (since v2); to a

Scientific paper

10.1016/S0021-8693(02)00557-4

Let $X$ be a scheme, proper over a commutative noetherian ring $A$. We introduce the concept of an ample filter of invertible sheaves on $X$ and generalize the most important equivalent criteria for ampleness of an invertible sheaf. We also prove the Theorem of the Base for $X$ and generalize Serre's Vanishing Theorem. We then generalize results for twisted homogeneous coordinate rings which were previously known only when $X$ was projective over an algebraically closed field. Specifically, we show that the concepts of left and right $\sigma$-ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be noetherian.

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