multiplier ideals and modules on toric varieties a 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

the tight integral closure of a set of ideals abstract this paper celebrates the enormous contributions of david buchsbaum to commutative algebra homological algebra and representation theory with grateful appreciation for the inspiration he has provided for myself and so many others

new approximation schemes for general variational inequalities in this paper we suggest and consider a class of new three step approximation schemes for general variational inequalities our results include ishikawa and mann iterations as special cases we also study the convergence criteria of these schemes

length multiplicity and multiplier ideals let r m be an n dimensional regular local ring essentially of finite type over a field of characteristic zero given an m primary ideal a of r the relationship between the singularities of the scheme defined by a and those defined by the multiplier ideals j a c with c varying in q are quantified in this paper by showing that the samuel multiplicity of a satisfies e n n k n c n whenever j a c c m k 1 this formula generalizes an inequality on log canonical thresholds previously obtained by ein mustata and the author of this paper a refined inequality is also shown to hold for small dimensions and similar results valid for a generalization of test ideals in positive characteristics are presented

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

an efficient surface algorithm for random particle simulation of vorticity and heat transport abstract a new surface algorithm has been incorporated into the random vortex method for the simulation of 2 dimensional laminar flow in which vortex particles are deleted rather than reflected as they cross a solid surface this involves a modification to the strength and random walk of newly created vortex particles computations of the early stages of symmetric impulsively started flow around a circular cylinder for a wide range of reynolds numbers demonstrate that the number of vortices required for convergence is substantially reduced the method has been further extended to accommodate forced convective heat transfer where temperature particles are created at a surface to satisfy the condition of constant surface temperature vortex and temperature particles are handled together throughout each time step for long runs in which a steady state is reached comparison is made with some time averaged experimental heat transfer data for reynolds numbers up to a few hundred a karman vortex street occurs at the higher reynolds numbers

accuracy of the random vortex method for a problem with non smooth initial conditions abstract the accuracy of the random vortex method in approximating the solution of the navier stokes equation is investigated using the model problem of a circular vortex as suggested by milinazzo and saffman 13 the method consists of partitioning the vorticity into vortex blobs these blobs are moved via two actions first a blob is deterministically moved under the action of the velocity field associated with the other blobs then to simulate viscosity a random component is added to the position of the blob for this model problem the nonlinear terms of the navier stokes equation vanish thus the major error inherent in the deterministic component of the method vanishes consequently for this model problem concentration is on the interaction of the deterministic and random components of the method results show that the accuracy of the method depends heavily on the initial distribution and strength of the computational elements i e the vortex blobs with the right choice of initial conditions it is found that e1 t l t a t a t is o r 1 2 n 1 2 where l t and a t are respectively the exact and computed angular moment of vorticity distribution at time t for n vortex blobs at a reynolds number r

frobenius and cartier algebras of stanley reisner rings we study the generation of the frobenius algebra of the injective hull of a complete stanley reisner ring over a field with positive characteristic in particular by extending the ideas used by m katzman to give a counterexample to a question raised by g lyubeznik and k e smith about the finite generation of frobenius algebras we prove that the frobenius algebra of the injective hull of a complete stanley reisner ring can be only principally generated or infinitely generated also by using our explicit description of the generators of such algebra and applying the recent work by m blickle about cartier algebras and generalized test ideals we are able to show that the set of f jumping numbers of generalized test ideals associated to complete stanley reisner rings form a discrete subset inside the non negative real numbers

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

f injective singularities are du bois in this paper we prove that singularities of f injective type are du bois this extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of frobenius in positive characteristic

nonconvex policy search using variational inequalities policy search is a class of reinforcement learning algorithms for finding optimal policies in control problems with limited feedback these methods have been shown to be successful in high dimensional problems such as robotics control though successful current methods can lead to unsafe policy parameters that potentially could damage hardware units motivated by such constraints we propose projection based methods for safe policies these methods however can handle only convex policy constraints in this letter we propose the first safe policy search reinforcement learner capable of operating under nonconvex policy constraints this is achieved by observing for the first time a connection between nonconvex variational inequalities and policy search problems we provide two algorithms mann and two step iteration to solve the above problems and prove convergence in the nonconvex stochastic setting finally we demonstrate the performance of the algorithms on six benchmark dynamical systems and show that our new method is capable of outperforming previous methods under a variety of settings

an elementary computation of the f pure threshold of an elliptic curve abstract we compute the f pure threshold of a degree three homogeneous polynomial in three variables with an isolated singularity the computation uses elementary methods to prove a known result of bhatt and singh from 2

drops in joint hilbert kunz multiplicities and projective equivalence abstract given two m primary ideals a and i in a noetherian local or graded ring we introduce the level ideals of i with respect to a as a measure of the asymptotic growth of the drop in the joint hilbert kunz multiplicity e hk a t i e hk a t i x we study the properties that must hold for two ideals a b to determine the same level ideals for all i and we show that this is related to projective equivalence

f regularity does not deform we show that the property of f regularity does not deform and thereby settle a long standing open question in the theory of tight closure specifically we construct a three dimensional graded domain r which is not f regular or even f pure but has a quotient r tr which is f regular examples are constructed over fields of characteristic p 0 as well as over fields of characteristic zero 1 introduction throughout this paper all rings are commutative noethe rian and have an identity element the theory of tight closure was developed by melvin hochster and craig huneke in hh2 and draws attention to rings which have the property that all their ideals are tightly closed called weakly f regular rings the term f regular is reserved for rings all of whose localizations are weakly f regular a natural question that arose with the development of the the ory was whether the property of f regularity deforms i e if r m k is a local ring such that r tr is f regular for some nonzerodivisor t m m ustr be f regular see the epilogue of ho hochster and huneke showed that this is indeed true if the ring r is gorenstein hh3 and their work has been followed by various attempts at extending this result see akm si sm3 our pri mary goal here is to settle this question by constructing a family of examples to show that f regularity does not deform we shall throughout be considering graded rings but local examples can be obtained in all cases by localizing at the homogeneous maximal ideals our main result is theorem 1 1 there exists an graded ring r of dimension three finitely generated over a field r 0 k of characteristic p 2 which is not f pure but has an f regular quotient r tr where t m is a homogeneous nonzerodivisor specifically for positive integers m and n satisfying m m n 2 consider the ring r k a b c d t i where i is generated by the size two minors of the

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

on the discreteness and rationality of f jumping coefficients abstract we prove that the f jumping coefficients of a principal ideal of an excellent regular local ring of characteristic p 0 are all rational and form a discrete subset of r

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

weak f regularity deforms in q gorenstein rings weakly f regular inother words does the property of weak f regularity deform it has beenshown by singh that in general weak f regularity does not deform si2 wxit is known that f rationality deforms hh2 4 2 and the proof iswxstraightforward however the same difficulties that arise when trying to

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

f signature of pairs and the asymptotic behavior of frobenius splittings abstract we generalize f signature to pairs r d where d is a cartier subalgebra on r as defined by the first two authors in particular we show the existence and positivity of the f signature for any strongly f regular pair in one application we answer an open question of aberbach and enescu by showing that the f splitting ratio of an arbitrary f pure local ring is strictly positive furthermore we derive effective methods for computing the f signature and the f splitting ratio in the spirit of the work of r fedder

hydrodynamics in two dimensions and vortex theory we consider the navier stokes equation for a viscous and incompressible fluid inr2 we show that such an equation may be interpreted as a mean field equation vlasov like limit for a system of particles called vortices interacting via a logarithmic potential on which in addition a stochastic perturbation is acting more precisely we prove that the solutions of the navier stokes equation may be approximated in a suitable way by finite dimensional diffusion processes with the diffusion constant related to the viscosity as a particular case when the diffusion constant is zero the finite dimensional theory reduces to the usual deterministic vortex theory and the limiting equation reduces to the euler equation

natural actor critic algorithms we present four new reinforcement learning algorithms based on actor critic natural gradient and function approximation ideas and we provide their convergence proofs actor critic reinforcement learning methods are online approximations to policy iteration in which the value function parameters are estimated using temporal difference learning and the policy parameters are updated by stochastic gradient descent methods based on policy gradients in this way are of special interest because of their compatibility with function approximation methods which are needed to handle large or infinite state spaces the use of temporal difference learning in this way is of special interest because in many applications it dramatically reduces the variance of the gradient estimates the use of the natural gradient is of interest because it can produce better conditioned parameterizations and has been shown to further reduce variance in some cases our results extend prior two timescale convergence results for actor critic methods by konda and tsitsiklis by using temporal difference learning in the actor and by incorporating natural gradients our results extend prior empirical studies of natural actor critic methods by peters vijayakumar and schaal by providing the first convergence proofs and the first fully incremental algorithms

adjoint ideals along closed subvarieties of higher codimension in this paper we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using char acteristic p methods when x is a normal gorenstein closed subvariety of a smooth complex variety a we formulate a restriction property of the adjoint ideal sheaf adj x a of a along x involving the l c i ideal sheaf dx of x the proof relies on a modification of generalized test ideals of hara and yoshida 11

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

castelnuovo mumford regularity and the discreteness of f jumping coefficients in graded rings in this paper we show that the sets of f jumping coefficients of ideals form discrete sets in certain graded f finite rings we do so by giving a criterion based on linear bounds for the growth of the castelnuovo mumford regularity of certain ideals we further show that these linear bounds exists for one dimensional rings and for ideals of most two dimensional domains we conclude by applying our technique to prove that all sets of f jumping coefficients of all ideals in the determinantal ring given as t he quotient by 2 2 minors in a 2 3 matrix of indeterminates form discrete sets

grid free simulation of diffusion using random walk methods abstract the simulation of diffusion of a continuum field by the random walk displacement of a set of particles is investigated in detail computational particles are used to transport elements of the gradients of the diffusive concentration one dimensional and quasi one dimensional cases are treated for a generalized diffusion variable different types of boundary conditions of the diffusion equation are considered as well as the extensions to a system of coupled diffusion equations the reaction diffusion equations and the convection diffusion set of equations it is shown that by using concentration gradients in the random walk process statistical errors are reduced and each realization of the numerical solution is a representation of the exact solution moreover transport of higher order derivatives can be utilized to improve the smoothness the algorithm is grid free and the computational elements move to follow the gradients thus it is self adaptive and uniform resolution is attained for all times the method is particularly suitable for the simulation of diffusion in systems which involve more than one transport mechanism when the diffusivity is small and when lagrangian elements are used to model the other mechanisms

on the constancy regions for mixed test ideals abstract in this note we study the partition of r 0 n given by the regions where the mixed test ideals a 1 t 1 a n t n are constant we show that each region can be described as the preimage of a natural number under a p fractal function r 0 n n in addition we give some examples illustrating that these regions do not need to be composed of finitely many rational polytopes

the f pure threshold of a calabi yau hypersurface we compute the f pure threshold of the affine cone over a calabi yau hypersurface and relate it to the order of vanishing of the hasse invariant on the versal deformation space of the hypersurface

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

globalizing f invariants abstract in this paper we define and study the global hilbert kunz multiplicity and the global f signature of prime characteristic rings which are not necessarily local our techniques are made meaningful by extending many known theorems about hilbert kunz multiplicity and f signature to the non local case

adaptive subgradient methods for online learning and stochastic optimization we present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient based learning metaphorically the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm we describe and analyze an apparatus for adaptively modifying the proximal function which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight we give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints we experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state of the art yet non adaptive subgradient algorithms

a characteristic p analogue of plt singularities and adjoint ideals we introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure as a corollary we prove that a log pair x is plt if and only if the modulo p reduction of x is divisorially f regular for all large p 0 here divisorially f regular pairs are a class of singularities in positive characteristic introduced by hara and watanabe j algebra geom 11 363 392 2002 in terms of frobenius splitting

letter to the editor extragradient methods for solving nonconvex variational inequalities in this paper we introduce and consider a new class of variational inequalities which are called the nonconvex variational inequalities using the projection technique we suggest and analyze an extragradient method for solving the nonconvex variational inequalities we show that the extragradient method is equivalent to an implicit iterative method the convergence of which requires only pseudo monotonicity a weaker condition than monotonicity this clearly improves on the previously known result our method of proof is very simple as compared with other techniques

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

when does the subadditivity theorem for multiplier ideals hold demailly ein and lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals we prove that in the two dimensional case the subadditivity theorem holds on log terminal singularities however in the higher dimensional case we have several counterexamples we consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three dimensional toric ring

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

non commutative resolutions of toric varieties abstract let r be the coordinate ring of an affine toric variety we prove using direct elementary methods that the endomorphism ring end r a where a is the finite direct sum of all isomorphism classes of conic r modules has finite global dimension equal to the dimension of r this gives a precise version and an elementary proof of a theorem of spenko and van den bergh implying that end r a has finite global dimension furthermore we show that end r a is a non commutative crepant resolution if and only if the toric variety is simplicial for toric varieties over a perfect field k of prime characteristic we show that the ring of differential operators d k r has finite global dimension

depths of multiplier ideals and integral closure in this note we study how the depths of multiplier ideals behave under restriction we also study possible values of the depths of multiplier ideals in the filtrations induced from maximal ideal sheaves we then use it to give a sufficient condition for the integral closedness of the product of a multiplier ideal and a power of maximal ideal sheaf in the spirit of huneke

the jumping coefficients of non q gorenstein multiplier ideals abstract let a o x be a coherent ideal sheaf on a normal complex variety x and let c 0 be a real number de fernex and hacon associated a multiplier ideal sheaf to the pair x a c which coincides with the usual notion whenever the canonical divisor k x is q cartier we investigate the properties of the jumping numbers associated to these multiplier ideals we show that the set of jumping numbers of a pair is unbounded countable and satisfies a certain periodicity property we then prove that the jumping numbers form a discrete set of real numbers if the locus where k x fails to be q cartier is zero dimensional it follows that discreteness holds whenever x is a threefold with rational singularities furthermore we show that the jumping numbers are rational and discrete if one removes from x a closed subset w x of codimension at least three which does not depend on a we also obtain that outside of w the multiplier ideal reduces to the test ideal modulo sufficiently large primes p 0

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

f signature exists suppose r is a d dimensional reduced f finite noetherian local ring with prime characteristic p 0 and perfect residue field let r 1 p e be the ring of p e th roots of elements of r for e and let a e denote the maximal rank of a free r module appearing in a direct sum decomposition of r 1 p e we show the existence of the limit s r lim e to infty frac a e p ed called the f signature of r this invariant which can be extended to all local rings in prime characteristic was first formally defined by c huneke and g leuschke in math ann 324 2 391 404 2002 and has previously been shown to exist only in special cases the proof of our main result is based on the development of certain uniform hilbert kunz estimates of independent interest additionally we analyze the behavior of the f signature under finite ring extensions and recover explicit formulae for the f signatures of arbitrary finite quotient singularities

on the existence of f thresholds and related limits we show the existence of f thresholds in full generality in addition we study properties of standard graded algebras over a field for which f pure threshold and f threshold at the irrelevant maxi

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

test multiplier and invariant ideals abstract this paper gives an explicit formula for the multiplier ideals and consequently for the log canonical thresholds of any gl v gl w invariant ideal in s sym v w where v and w are vector spaces over a field of characteristic 0 this characterization is done in terms of a polytope constructed from the set of young diagrams corresponding to the schur modules generating the ideal our approach consists in computing the test ideals of some invariant ideals of s in positive characteristic namely we compute the test ideals and so the f pure thresholds of any sum of products of determinantal ideals not all the invariant ideals are as the latter not even in characteristic 0 but they are up to integral closure and this is enough to reach our goals the results concerning the test ideals are obtained as a consequence of general results holding true in a special situation within such framework fall determinantal objects of a generic matrix as well as of a symmetric matrix and of a skew symmetric one similar results are thus deduced for the gl v invariant ideals in sym sym 2 v and in sym 2 v monomial ideals also fall in this framework thus we recover howald s formula for their multiplier ideals and more generally hara yoshida s formula for their test ideals in the proof we introduce the notion of floating test ideals a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities as will be shown products of determinantal ideals and by passing to characteristic 0 ideals generated by a single schur module have this property

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

big tight closure test elements for some non reduced excellent rings abstract this paper is concerned with existence of big tight closure test elements for a commutative noetherian ring r of prime characteristic p let r denote the complement in r of the union of the minimal prime ideals of r a big test element for r is an element of r which can be used in every tight closure membership test for every r module and not just the finitely generated ones the main results of the paper are that if r is excellent and satisfies condition r 0 and c r is such that r c is gorenstein and weakly f regular then some power of c is a big test element for r if i r is a homomorphic image of an excellent regular ring of characteristic p for which the frobenius homomorphism is intersection flat or ii r is f pure or iii r is local the gamma construction is not used

f thresholds of hypersurfaces for every 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 is a nonnegative real number then the corresponding test ideal is denoted by a in this context we say that is an f jumping exponent or an f threshold if a 6 a for every 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

bernstein sato polynomials in positive characteristic abstract in characteristic zero the bernstein sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an euler operator on a suitable d module we consider analogous d modules in positive characteristic and use them to define a sequence of bernstein sato polynomials corresponding to the fact that we need to consider also divided powers euler operators we show that the information contained in these polynomials is equivalent to that given by the f jumping exponents of the hypersurface in the sense of hara and yoshida n hara k i yoshida a generalization of tight closure and multiplier ideals trans amer math soc 355 2003 3143 3174

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings