good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

test ideals in diagonal hypersurface rings ii abstract let r k x 1 x n x 1 d x n d where k is a field of characteristic p p does not divide d and n 3 we describe a method for computing the test ideal for these diagonal hypersurface rings this method involves using a characterization of test ideals in gorenstein rings as well as developing a way to compute tight closures of certain ideals despite the lack of a general algorithm in addition we compute examples of test ideals in diagonal hypersurface rings of small characteristic relative to d including several that are not integrally closed these examples provide a negative answer to smith s question k e smith the multiplier ideal is universal test ideal comm algebra 28 12 2000 5912 5929 of whether the test ideal in general is always integrally closed

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

on the behavior of f rational rings under flat base change abstract we examine the behavior of f rationality under flat homomorphisms with geometrically f rational fibres the goal is to prove that if r s is a homomorphism with geometrically f rational fibres and r is an f rational ring then s is an f rational ring we prove a stronger version of this under a mild condition and introduce a new perspective for dealing with flat base change problems in tight closure theory

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

cyclic covers of rings with rational singularities we examine some recent work of phillip griffith on etale covers and fibered products from the point of view of tight closure theory while it is known that cyclic covers of gorenstein rings with rational singularities are cohen macaulay we show this is not true in general in the absence of the gorenstein hypothesis specifically we show that the canonical cover of a q gorenstein ring with rational singularities need not be cohen macaulay

theoretical analysis and numerical verification of the consistency of viscous smoothed particle hydrodynamics formulations in simulating free surface flows the theoretical formulation of the smoothed particle hydrodynamics sph method deserves great care because of some inconsistencies occurring when considering free surface inviscid flows actually in sph formulations one usually assumes that i surface integral terms on the boundary of the interpolation kernel support are neglected ii free surface conditions are implicitly verified these assumptions are studied in detail in the present work for free surface newtonian viscous flow the consistency of classical viscous weakly compressible sph formulations is investigated in particular the principle of virtual work is used to study the verification of the free surface boundary conditions in a weak sense the latter can be related to the global energy dissipation induced by the viscous term formulations and their consistency numerical verification of this theoretical analysis is provided on three free surface test cases including a standing wave with the three viscous term formulations investigated

strongly liftable schemes and the kawamata viehweg vanishing in positive characteristic iii abstract a smooth scheme x over a field k of positive characteristic is said to be strongly liftable over w 2 k if x and all prime divisors on x can be lifted simultaneously over w 2 k in this paper we first deduce the kummer covering trick over w 2 k which can be used to construct a large class of smooth projective varieties liftable over w 2 k and to give a direct proof of the kawamata viehweg vanishing theorem on strongly liftable schemes secondly we generalize almost all of the results in 18 19 to the case where everything is considered over w k the ring of witt vectors of k

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

vanishing theorems on toric varieties in positive characteristic we use the liftability of the relative frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the bott vanishing theorem the degeneration of the hodge to de rham spectral sequence and the kawamata viehweg vanishing theorem for log pairs on toric varieties in positive characteristic these results generalize those results of danilov buch thomsen lauritzen mehta musta and fujino to the case where concerned weil divisors are not necessarily torus invariant

frobenius amplitude and strong vanishing theorems for vector bundles the primary goal of this paper is to systematically exploit the method of deligne illusie to obtain kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties the key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the frobenius or f amplitude the f amplitude enters into the statement of the basic vanishing theorem and this leads to the problem of calculating or at least estimating this number most of the work in this paper is devoted to doing this various situations

computing the tight closure in dimension two we study computational aspects of the tight closure of a homogeneous primary ideal in a two dimensional normal standard graded domain we show how to use slope criteria for the sheaf of relations for generators of the ideal to compute its tight closure in particular our method gives an algorithm to compute the tight closure of three elements under the condition that we are able to compute the harder narasimhan filtration we apply this to the computation of x a y a z a in k x y z f where f is a homogeneous polynomial

behavior of test ideals under smooth and etale homomorphisms we investigate the behavior of the test ideal of an excellent reduced ring of prime characteristic under base change it is shown that if h a d is a smooth homomorphism then ad d assuming that all residue fields of a at maximal ideals are perfect and that formation of the test ideal commutes with localization it is also shown that if h a m d is a finite flat homomorphism of gorenstein normal rings etale in codimension 1 then ad d more generally this last result holds under the assumption that the closed fiber of h a m d is gorenstein provided one knows that the tight closure of zero and the finitistic tight closure of zero in the injective hulls of the residue fields of a and s are equal

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

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

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 purity and rational singularity in graded complete intersection rings a simple criterion is given for determining almost completely whether the positively graded complete intersection ring r k x1 xn t g of dimension n has an f pure type singularity at m x1 x specifically if deg x a 0 for 1 0 for 1 0 the characterization given by this theorem is particularly effective if the singularity of r at m is isolated in that case 8 0 so that only the condition t 1d y lta 0 is not solved by the above result in particular it follows from work of kei ichi watanabe that if r has an isolated rational singularity then r has f pure type the converse is also almost true with the only exception being the case where t 1d e i ta 0 in proving this criterion a weak but more stable form of f purity called f contractedness is defined and explored r is f contracted in characteristic p 0 if every system of parameters for m is contracted with respect to the frobenius map f r r just as for f purity the notion of f contracted type is defined in characteristic 0 by reduction to characteristic p the two notions of f pure type and f contracted type coincide when r is gorenstein whence in particular when r is a complete intersection ring 0 introduction let r be a ring of characteristic p and let r denote the ring r viewed as an r module via the frobenius map f r rp r is f pure if for every r module m 0 r m r m is exact a notion of f pure type can then be defined in characteristic 0 by reduction to characteristic p the concept of f purity was suggested originally by the hochster roberts proof that the ring of invariants of a linearly reductive affine linear group acting on a regular ring is cohen macaulay 4 the same two authors later demonstrated that f purity measures nice singularities in the sense that it implies much simplification and vanishing in the computation of local cohomology 5 the problem of identifying those rings which are f pure type is quite difficult the following special cases are known 1 the invariant ring of a reductive linear group acting on a regular ring is f pure type 5 received by the editors october 8 1985 1980 mathematics subject classification 1985 revision primary 13h99 the author wishes to thank the research council of the university of missouri for a research grant which partially supported this project 1987 american mathematical society 0002 9947 87 1 00 25 per page

on the connection between hamiltonian many particle systems and the hydrodynamical equations we consider certain hamiltonian systems with many particles interacting through a potential whose range is large in comparison with the typical distance between neighbouring particles it is shown that the empirical processes of the positions and the velocities respectively converge to solutions of the continuity equation and the euler equation in the limit as the particle number tends to infinity

a tight closure proof of fujita s freeness conjecture for very ample line bundles the proof here actually proves a stronger statement than theorem 1 above the variety x need not be smooth f rationality is sufficient the definition is recalled in the next section in characteristic zero it is equivalent to rational singularities also the line bundle l need not be very ample it is sufficient if l is globally generated and the dimension the complete linear system l is greater than d these generalizations are summarized in theorem 2 furthermore with a little more work the same ideas prove even stronger statements which are interesting algebraically but difficult to interpret geometrically see theorem 3 fujita s freeness conjecture predicts the same conclusion under the much weaker hypothesis that l is only ample while open in general for varieties defined over a field of characteristic zero it is known in dimension four or less r el ka see also as for important progress on the conjecture and ko for a good survey about it in characteristic zero it is not hard to give a geometric argument of the special case above using the kodaira vanishing theorem the goal here is to give a simple quite different proof that is purely algebraic and valid in any characteristic this argument offers a nice illustration of how tight closure can be used to prove geometric theorems in arbitrary characteristic without the use of the usual tools of desingularization or vanishing theorems

convergence of the hamiltonian particle mesh method for barotropic fluid flow we prove convergence of the hamiltonian particle mesh hpm method initially proposed by j frank g gottwald and s reich on a pe riodic domain when applied to the irrotational shallow water equations as a prototypical model for barotropic compressible uid ow under appropriate assumptions most notably suciently fast decay in fourier space of the global smoothing operator and a strang fix condition of order 3 for the local par tition of unity kernel the hpm method converges as the number of particles tends to innity and the global interaction scale tends to zero in such a way that the average number of particles per computational mesh cell remains con stant and the number of particles within the global interaction scale tends to innity the classical sph method emerges as a particular limiting case of the hpm algorithm and we nd that the respective rates of convergence are comparable under suitable assumptions since the computational complexity of bare sph is algebraically superlinear and the complexity of hpm is logarithmically su perlinear in the number of particles we can interpret the hpm method as a fast sph algorithm

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

generic bounds for frobenius closure and tight closure we use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring p k x 0 ldots x d one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard graded k algebra r of dimension d 1 this indicates that the tight closure of an ideal behaves more uniformly than the ideal itself moreover if r is normal one obtains a generic bound for membership in the frobenius closure if d leq 2 then the bound for ideal membership in p can be computed from the known cases of the fr oberg conjecture and yields explicit generic tight closure bounds

pure subrings of regular rings are pseudo rational we prove a generalization conjectured by aschenbrenner and schoutens 2003 of the hochster roberts boutot kawamata theorem let r s be a pure homomorphism of equicharacteristic zero noetherian local rings if s is regular then r is pseudo rational and if r is moreover q gorenstein then it is pseudo log terminal

nilpotence of frobenius action and the hodge filtration on local cohomology abstract an f nilpotent local ring is a local ring r m of prime characteristic defined by the nilpotence of the frobenius action on its local cohomology modules h m i r a singularity in characteristic zero is said to be of f nilpotent type if its modulo p reduction is f nilpotent for almost all p in this paper we give a hodge theoretic interpretation of three dimensional normal isolated singularities of f nilpotent type in the graded case this yields a characterization of these singularities in terms of divisor class groups and brauer groups

kawamata viehweg vanishing on rational surfaces in positive characteristic we prove that the kawamata viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to w2 k of certain log pairs on smooth rational surfaces as a corollary the kawamata viehweg vanishing theorem holds on log del pezzo surfaces in positive characteristic

a new insight into the consistency of the sph interpolation formula abstract in this paper the consistency of the smoothed particle hydrodynamics sph interpolation formula is investigated by analytical means a novel error analysis is developed in n dimensional space using the poisson summation formula which enables the simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions new consistency integral relations are derived for the particle approximation which correspond to the cosine fourier transform of the kernel consistency conditions the functional dependence of the error bounds on the sph interpolation parameters namely the smoothing length h and the number of particles within the kernel support n is demonstrated explicitly from which consistency conditions arise as n the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m h with n where n is the spatial dimension this implies that as h 0 the joint limit m 0 n and n is necessary for complete convergence to the continuum where n is the total number of particles the analysis also reveals a dominant error term of the form ln n n n for finite n as it has long been conjectured based on the similarity between the sph and the quasi monte carlo estimates when n 1 the error of the sph interpolant decays as n 1 independently of the dimension this ensures approximate partition of unity of the kernel volume

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

good filtrations of symmetric algebras and strong f regularity of invariant subrings in this paper we prove that a linear action of a reductive group on a polynomial ring with good filtrations over a field of characteristic p 0 yields a strongly f regular in particular cohen macaulay invariant subring the strongly f regular property of some known examples of invariant subrings such as the coordinate rings of schubert varieties in grassmannians are recovered a similar result over a field of characteristic zero is also proved

multiple solutions of two point boundary value problems of ordinary differential equations in banach spaces soit p un cone normal de l espace de banach reel e on note par n la constante normale de p on considere le probleme aux deux points limites x f t x t i x 0 x 1 ou i 0 1 et f une application continue de i p dans p on etudie l existence de solutions positives multiples

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by 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 y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

test ideals in quotients of f finite regular local rings let s be an f finite regular local ring and i an ideal contained in s let r s i fedder proved that r is f pure if and only if i p i m p we have noted a new proof for his criterion along with showing that i q i q where is the pullback of the test ideal for r combining the the f purity criterion and the above result we see that if r s i is f pure then r is also f pure in fact we can form a filtration of r i 0 1 i that stabilizes such that each r i is f pure and its test ideal is i 1 to find examples of these filtrations we have made explicit calculations of test ideals in the following setting let r t i where t is either a polynomial or a power series ring and i p1 pn is generated by monomials and the r pi are regular set j p1 pi pn then j par this paper concerns the study of the test ideal in f finite quotients of regular local rings test elements play a key role in tight closure theory once known they make computing tight closures of ideals and modules easier in fact in an excellent gorenstein local ring with an isolated singularity r hom i i e where i is generated by a system of parameters that are test elements and e is the injective hull of r see hu1 and s1 we also know for parameter ideals i that i i thus knowing is basically equivalent to knowing the tight closure of a system of parameters which is contained in the test ideal a recent paper of huneke and smith hs links tight closure to kodaira vanishing for graded rings r with characteristic either 0 or p where p 0 recall that the a invariant for a graded ring r denoted a is equal to min i r i 6 0 where r is the canonical module for r if r is gorenstein then r r a huneke and smith prove that the test ideal is exactly the ideal generated by elements of degree greater than the a invariant of r if and only if a strong kodaira vanishing holds on r a recent paper of hara ha confirms that this strong kodaira vanishing holds in finitely generated algebras over a field of characteristic zero in this paper we study test ideals of f finite rings which are reduced quotients of a regular local ring reduced quotients of f finite regular local rings have been studied by both fedder fe and glassbrenner gl fedder s work concerns f purity aspects of these rings and glassbrenner s results use fedder s techniques to examine strong f regularity the object that plays a key role in their work is received by the editors november 4 1996 1991 mathematics subject classification primary 13a35

cartier modules finiteness results on a locally noetherian scheme x over a field of positive characteristic p we study the category of coherent ox modules m equipped with a pe linear map i e an additive map c ox ox satisfying rc m c r p e m for all m a m r a ox the notion of nilpotence meaning that some power of the map c is zero is used to rigidify this cate gory 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 hartshorne speiser 19 lyubeznik 27 sharp 30 enescu hochster 14 and hochster 20 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 27 have finite length

on rational connectedness of globally f regular threefolds in this paper we show that projective globally f regular threefolds defined over an algebraically closed field of characteristic p 11 are rationally chain connected

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

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

on the arithmetic of tight closure we provide a negative answer to an old question in tight closure theory by showing that the containment x 3y 3 in x 4 y 4 z 4 in k x y z x 7 y 7 z 7 holds for infinitely many but not for almost all prime characteristics of the field k this proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic the ideal x y z subset k x y z u v w x 7 y 7 z 7 ux 4 vy 4 wz 4 x 3y 3 has then the property that the cohomological dimension fluctuates arithmetically between 0 and 1

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

a frobenius characterization of rational singularity in 2 dimensional graded rings a ring r is said to be f rational if for every prime p in r the local ring r p has the property that every system of parameters ideal is tightly closed as defined by hochster huneke it is proved that if r is a 2 dimensional graded ring with an isolated singularity at the irrelevant maximal ideal m then the following are equivalent 1 r has a rational singularity at m 2 r is f rational 3 a r 0 here a r as defined by goto watanabe denotes the least nonvanishing graded piece of the local cohomology module h m r the proof of this result relies heavily on the properties of derivations of r and suggests further questions in that direction paradigmatically if one knows that d a satisfies a certain property for every derivation d what can one conclude about the original ring element a

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

multigraded rings diagonal subalgebras and rational singularities we study f rationality and f regularity in diagonal subalgebras of multigraded rings and use this to construct large families of rings that are f rational but not f regular we also use diagonal subalgebras to construct rings with divisor class groups that are finitely generated but not discrete in the sense of danilov

a canonical linear system associated to adjoint divisors in characteristic p 0 suppose that x is a projective variety over an algebraically closed field of characteristic p 0 further suppose that l is an ample or more generally in some sense positive divisor we study a natural linear system in h 0 x ox kx l we further generalize this to incorporate a boundary divisor we show that these subsystems behave like the global sections associated to multiplier ideals h 0 x j x l in characteristic zero in particular we show that these systems are in many cases base point free while the original proof utilized kawamata viehweg vanishing and variants of multiplier ideals our proof uses test ideals

the convergence of the sph method abstract in this paper we prove the convergence of a smooth particle flow to the solution of a regularized version of the euler equations describing a generic polytropic fluid this result combined with a stability property of the euler equations with respect to suitable regularizations result of the same authors of the present paper which will be published elsewhere allows to achieve the proof of the convergence of the so called smoothed particle hydrodynamics sph method

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