Cartier modules: Finiteness results

By Manuel Blickle, Gebhard Bckle. 2009

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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 pe-linear map, i.e. an additive map C : X X satisfying rC(m) C(rpe m) for all m M, 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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/154763195ce5801b8e6c0cbb8db19f6610a79958

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Citations in the corpus

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “Theorem 5.4 [Gabber 2004, Lemma 13.1; Blickle and Böckle 2011, Proposition 2.14].”
Test ideals via algebras of $p^{-e}$-linear maps (2009, methodology): “We slightly generalize here some basic constructions of [BB09] from the case of a single operator to that of an algebra of operators.”
Discreteness and rationality of F-jumping numbers on singular varieties (2009, result): “This result is also interesting in other related contexts, see for example [4, 5 ].”