Twisted Multi-Homogeneous Coordinate Rings

Artin and Van den Bergh have constructed noncommutative analogues of the homogeneous coordinate ring of a projective scheme. We generalize their results to the multiprojective setting. Such multi-homogeneous coordinate rings include tensor products of twisted homogeneous coordinate rings as well as their Rees algebras. Finally, we show that these last two examples are noetherian (assuming ampleness conditions) though the coordinate ring in general is not.