Cartier Modules: finiteness results

TL;DR

This paper introduces Cartier crystals on schemes over fields of positive characteristic, proving they have finite length when the scheme is F-finite, and generalizes several existing finiteness results in Frobenius module theory.

Contribution

It establishes the finiteness of Cartier crystals' length on F-finite schemes and generalizes prior finiteness results for modules with Frobenius actions.

Findings

Cartier crystals have finite length on F-finite schemes.

Generalization of finiteness results for Frobenius modules.

Recovery of known results by Hartshorne-Speiser, Lyubeznik, and others.

Abstract

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.