Ample filters of invertible sheaves

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 Serreʹs 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 σ-ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be Noetherian