2002/10/31 by Dennis S. Keeler
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #math.AG #math.RA #msc:14A22 #msc:14F17 #msc:16S38
paper · pdf · doi:10.1016/s0021-8693(03)00126-1
published as J. Algebra 265 (2003), no. 1, 299--311. · 11 pages, LaTeX, minor corrections, to appear in J. Algebra
arxiv created 2002/11/11 · openalex publication_date 2003/05/12 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31
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.