Cartier modules: Finiteness results

Abstract

On a locally Noetherian scheme X over a field of positive characteristic p, we study the category of coherent _X-modules M equipped with a p^e-linear map, i.e. an additive map C : _XX satisfying rC(m) C(r^pe m) for all mM, r_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 result 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 HartshorneSpeiser Ann. Math. 105: 4579, 1977, Lyubeznik J. reine angew. Math. 491: 65130, 1997, Sharp Trans. Amer. Math. Soc. 359: 42374258, 2007, EnescuHochster Alg. Num. Th. 2: 721754, 2008, and Hochster Contemp. Math. 448: 119127, 2007 about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F-finite scheme X Lyubeznik's F-finite modules J. reine angew. Math. 491: 65130, 1997 have finite length.