  Let $X$ be a projective scheme over a field. We show that the vanishing
cohomology of any sequence of coherent sheaves is closely related to vanishing
under pullbacks by the Frobenius morphism. We also compare various definitions
of ample locally free sheaf and show that the vanishing given by the Frobenius
morphism is, in a certain sense, the strongest possible. Our work can be viewed
as various generalizations of the Serre Vanishing Theorem.
