# Cartier Modules: finiteness results

Manuel Blickle, Gebhard Böckle

Number | 31 |
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Authors |
Manuel Blickle
Gebhard Böckle |

Year | 2009 |

On a locally Noetherian scheme X over a field of positive characteristic p we study the category of coherent OX-modules M equipped with a p−e-linear map, i.e. an additive map C:OX→OX satisfying rC(m)=C(rpem) for all m in M, r in OX. 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.