Cartier Modules: finiteness results

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.