  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.
