Manuel Blickle

F-THRESHOLDS OF HYPERSURFACES

′) for every λ ′ 0 that is F-finite, i.e. such that the Frobenius morphism F : R − → R is finite. If a is an ideal in R and if λ is a nonnegative real number, then the corresponding test ideal is denoted by τ(a � ). In this context we say that λ is an F-jumping exponent (or an F-threshold) if τ(a � ) 6 τ(a � ′ ) for every λ ′ < λ. The following is our main result about F-jumping exponents in positive characteristic. Theorem 1.1. If R is an F-finite regular ring, and if a = (f) is a principal ideal, then the F-jumping exponents of a are rational, and they form a discrete set. The discreteness and the rationality of F-jumping numbers has been proved in [BMS] for every ideal when the ring R is essentially of finite type over an F-finite field. We mention also that for R = k[[x, y]], with k a finite field, the above result has been proved in [Ha] using a completely different approach. We stress that the difficulty in attacking this result does notcome from the fact that there is no available resolution of singularities in positive characteristic. Even

Test ideals via algebras of ^{-}-linear maps

Continuing ideas of a recent preprint of Schwede arXiv:0906.4313 we study test ideals by viewing them as minimal objects in a certain class of

Cartier modules: Finiteness results

F

F-singularities via alterations

-pure modules over algebras of p^{-e}-linear operators. This shift in the viewpoint leads to a simplified and generalized treatment, also allowing us to define test ideals in non-reduced settings. In combining this with an observation of Anderson on the contracting property of p^{-e}-linear operators we obtain an elementary approach to test ideals in the case of affine k-algebras, where k is an F-finite field. It also yields a short and completely elementary proof of the discreteness of their jumping numbers extending most cases where the discreteness of jumping numbers was shown in arXiv:0906.4679.

p−1-Linear Maps in Algebra and Geometry

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 Lyubezniks F-finite modules J. reine angew. Math. 491: 65130, 1997 have finite length.

An Informal Introduction to Multiplier Ideals

We give characterizations of test ideals and

Local cohomology modules of a smooth Z-algebra have finitely many associated primes

F

On rings with small Hilbert-Kunz multiplicity

-rational singularities via (regular) alterations. Formally, the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions. Lastly, we establish Nadel-type vanishing theorems (up to finite maps) for test ideals, and further demonstrate how these vanishing theorems may be used to extend sections.

Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry: A short course on geometric motivic integration

At least since Habousch’s proof of Kempf’s vanishing theorem, Frobenius splitting techniques have played a crucial role in geometric representation theory and algebraic geometry over a field of positive characteristic. In this article we survey some recent developments which grew out of the confluence of Frobenius splitting techniques and tight closure theory and which provide a framework for higher dimension geometry in positive characteristic. We focus on local properties, i.e. singularities, test ideals, and local cohomology on the one hand and global geometric applicatioms to vanishing theorems and lifting of sections on the other.

F-signature of pairs: continuity, p-fractals and minimal log discrepancies

Multiplier ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety. More recently, they have led to the discovery of some surprising uniform results in local algebra. The present notes aim to provide a gentle introduction to the algebraically-oriented local side of the theory. They follow closely a short course on multiplier ideals given in September 2002 at the Introductory Workshop of the program in commutative algebra at MSRI.