Test ideals via algebras of p−ep^{-e}-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 $F$-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.Comment: 29 pages, to appear in Journal of Algebraic Geometr

Lyubeznik's invariants for cohomologically isolated singularities

In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This strengthens the results of Bondu and myself in math.AG/0406265 while at the same time somewhat simplifying the proofs. Through examples I further point out the bad behavior of these invariants under reduction to positive characteristic.Comment: 5 page

The intersection homology D-module in finite characteristic

For Y a closed normal subvariety of codimension c of a smooth complex variety X, Brylinski and Kashiwara showed that the local cohomology module H^c_Y(X,O_X) contains a unique simple D_X-submodule, denoted by L(Y,X). In this paper the analogous result is shown for X and Y defined over a perfect field of finite characteristic. Moreover, a local construction of Ll(Y,X) is given, relating it to the theory of tight closure. From the construction one obtains a criterion for the D_X-simplicity of H^c_Y(X).Comment: 23 pages, streamlined exposition according to referee's suggestion

D-module generation in positive characteristic via Frobenius descent

In this short note I give an alternative proof of a generalization of the result in math.AC/0407464. Namely I show that for most regular rings R, the localization R[1/f] at an element f of R is generated as a module over the ring of differential operators of R by 1/f itself. Due to the greater generality this answers some questions raised in math.AC/0407464. Some further generalizations and a brief discussion of the main technique (Frobenius descent) are also included.Comment: 6 page

A short course on geometric motivic integration

These notes grew out of several introductory talks I gave during the years 2003--2005 on motivic integration. They give a short but thorough introduction to the flavor of motivic integration which nowadays goes by the name of geometric motivic integration. As an illustration of the theory some applications to birational geometry are also included.Comment: 41 page

Multiplier Ideals and Modules on Toric Varieties

A simple formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.Comment: 8 pages, to appear in Mathematische Zeitschrif

An informal introduction to multiplier ideals

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.Comment: 28 pages, 5 figures, minor corrections and improvements according to editors suggestion

Cartier Crystals

Building on our previous work "Cartier modules: finiteness results" we start in this manuscript an in depth study of the derived category of Cartier modules and the cohomological operations which are defined on them. After localizing at the sub-category of locally nilpotent objects we show that for a morphism essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for Cartier crystals. We show that, if $f$ is of finite type (but not necessarily proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$ has bounded cohomological dimension. In a sequel we will explain how Grothendieck-Serre Duality relates our theory of Cartier Crystals to the theory of $\tau$-crystals as developed by Pink and the second author.Comment: 53 page

Rational Singularities and Rational Points

If $X$ is a projective, geometrically irreducible variety defined over a finite field \F_q, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. CH_0(X\times_{\F_q}\bar{\F_q(X)})=\Q, then the second author's theorem asserts that its number of rational points satisfies |X(\F_q)| \equiv 1 modulo $q$. If $X$ is not smooth, this is no longer true. Indeed J. Koll\'ar constructed an example of a rationally connected surface over \F_q without any rational points. Based on the work by Berthelot-Bloch and the second author computing the slope $<1$ piece of rigid cohomology, we define a notion of Witt-rational singularities in characteristic $p>0$. The theorem is then that if X/\F_q is a projective, geometrically irreducible variety, such that it has Witt-rational singularities and its Chow group of 0-cycles fulfills base change, then |X(\F_q)| \equiv 1 modulo $q$.Comment: 12 page

The D-Module structure of R[F]-modules

Let R be a regular ring essentially of finite type over a perfect field k. An R-module M is called a unit R[F]-module if it comes equipped with an isomorphism F*M-->M where F denotes the Frobenius map on Spec R, and F* is the associated pullback functor. It is well known that M then carries a natural D-module structure. In this paper we investigate the relation between the unit R[F]-structure and the induced D-structure on M. In particular, it is shown that, if k is algebraically closed and M is a simple finitely generated unit R[F]-module, then it is also simple as a D-module. An example showing the necessity of k being algebraically closed is also given.Comment: 25 pages. Some minor changes following referee's suggestion. To appear in Trans. AM