  On a locally Noetherian scheme X over a field of positive characteristic p we
study the category of coherent O_X-modules M equipped with a p^{-e}-linear map,
i.e. an additive map C: O_X \to O_X satisfying rC(m)=C(r^{p^e}m) for all m in
M, r in O_X. The notion of nilpotence, meaning that some power of the map C is
zero, is used to rigidify this category. The resulting quotient category,
called Cartier crystals, satisfies some strong finiteness conditions. The main
reasult in this paper states that, if the Frobenius morphism on X is a finite
map, i.e. if X is F-finite, then all Cartier crystals have finite length. We
further show how this and related results can be used to recover and generalize
other finiteness results of Hartshorne-Speiser, Lyubeznik, Sharp,
Enescu-Hochster, and Hochster about the structure of modules with a left action
of the Frobenius. For example, we show that over any regular F-finite scheme X
Lyubeznkik's F-finite modules have finite length.
