2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65449The 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.11 pages, LaTeX, minor corrections, to appear in J. AlgebraAlgebraic GeometryRings and Algebras14A22, 16S38 (Primary); 14F17 (Secondary)text