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

free group representations from vector valued multiplicative functions ii let gamma be a non commutative free group on finitely many generators in a previous work two of the authors have constructed the class of multiplicative representations of gamma and proved them irreducible as representation of gamma ltimes lambda c omega in this paper we analyze multiplicative representations as representations of gamma and we prove a criterium for irreducibility based on the growth of their matrix coefficients

monotony of certain free group representations abstract let be a free nonabelian group and let be its boundary let h be one of the unitary representations of introduced earlier by the authors in 1996 duke math j 82 381 436 by its definition h acts on l 2 d h for a certain measure h this gives a boundary realization of h in a sense we make precise we show that h does not have any other boundary realizations and simultaneously provide a new proof that h is irreducible

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

approximation by piecewise regular maps abstract a real algebraic variety w of dimension m is said to be uniformly rational if each of its points has a zariski open neighborhood which is biregularly isomorphic to a zariski open subset of r m let l be any nonnegative integer we prove that every map of class c l from a compact subset of a real algebraic variety into a uniformly rational real algebraic variety can be approximated in the c l topology by piecewise regular maps of class c k where k is an arbitrary integer satisfying k l next we derive consequences regarding algebraization of topological vector bundles

the blowup closure of a set of ideals with applications to ti closure abstract we introduce a new closure operation on sets of ideals in a commutative noetherian ring of characteristic p called the blowup closure we develop the theory of this operation and prove that given a set of ideals in a noetherian ring of characteristic p under mild conditions the tight integral closure of this set agrees with the blowup closure of certain extensions of those ideals in an extension of the original ring we then use properties of blowup closure to settle open questions on tight integral closure posed by m hochster j algebra230 2000 184 203 in particular we show that under mild conditions on the ring tight integral closure persists under ring maps and that it commutes with localization if and only if tight closure does

uniform behaviour of the frobenius closures of ideals generated by regular sequences this paper is concerned with ideals in a commutative noetherian ring r of prime characteristic the main purpose is to show that the frobenius closures of certain ideals of r generated by regular sequences exhibit a desirable type of uniform behaviour the principal technical tool used is a result proved by r hartshorne and r speiser in the case where r is local and contains its residue field which is perfect and subsequently extended to all local rings of prime characteristic by g lyubeznik about a left module over the skew polynomial ring r x f associated to r and the frobenius homomorphism f in the indeterminate x that is both x torsion and artinian over r

canonical elements in local cohomology modules and the direct summand conjecture one of the objectives of this paper is to show that the usual homological consequences of the existence of big cohen macaulay henceforth c m modules e g the new intersection conjecture of peskine szpiro and roberts and the evans griffith syzygy conjecture follow from the direct summand conjecture when the residual characteristic of the local ring is positive this gives a new and substantially more elementary proof of the standard homological conjectures in case the characteristic of the ring itself is positive and reduces the general case of all these conjectures to one rather down to earth conjecture of course this places the direct summand conjecture in a position of central importance so that it now merits an allout attack some partial results on this problem are given in section 6 the conjecture asserts that a regular noetherian ring r is a direct summand as an r module of every module finite extension ring s 3 r the other main objective of this paper is to formulate and develop a theory of certain canonical elements in the local cohomology of special modules of syzygies neither the modules of syzygies nor the induced maps between them are canonical but the identification between the canonical elements is independent of the choices in particular a canonical element qr e hz syz k is associated see section 3 for details with each ndimensional local ring r m k rings are commutative associative with identity local ring means noetherian ring with a unique maximal ideal the conjecture that qr is nonzero for all local rings r turns out to be equivalent to the direct summand conjecture for a given r an infinite family of cases of the latter implies the former

f injectivity and buchsbaum singularities let r mathfrak m k be a local ring that contains a field we show that when r has equal characteristic p 0 and when h mathfrak m i r has finite length for all i dim r then r is f injective if and only if every ideal generated by a system of parameters is frobenius closed as a corollary we show that such an r is in fact a buchsbaum ring this answers positively a question of s takagi that f injective singularities with isolated non cohen macaulay locus are buchsbaum we also study the characteristic 0 analogue of this question and we show that du bois singularities with isolated non cohen macaulay locus are buchsbaum in the graded case

big cohen macaulay algebras and seeds in this article we delve into the properties possessed by algebras which we have termed seeds that map to big cohen macaulay algebras we will show that over a complete local domain of positive characteristic any two big cohen macaulay algebras map to a common big cohen macaulay algebra we will also strengthen hochster and huneke s weakly functorial existence result for big cohen macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic we also show that every seed over a positive characteristic ring maps to a balanced big cohen macaulay algebra that is an absolutely integrally closed adically separated quasilocal domain

integral closures of ideals generated by regular sequences in particular our proposition contains s theorem 1 i as a special case in section 6 we shall make some remarks on theorem 2 in appendix 1 we shall give a proof of lemma 4 see section 1 for the sake of convenience this lemma is basic to our discussion of rees rings and the assertions in the lemma are all found in 9 if the ring a in the lemma is an integral domain in appendix 2 we shall make some remarks on the local cohomology modules of rees rings 390 0021 8693 88 83 00

algebras defined by powers of determinantal ideals let x be an m x n matrix of indeterminates over a field k a k x and i z x the ideal generated by the t minors of x the main objects of this article are the rees algebra 2 y these characteristics will be called non exceptional we show how this result generalizes to an arbitrary integral domain of coefficients the intersection of primary ideals which gives i in non exceptional characteristics always is the integral closure of i it follows immediately from the primary decomposition that the powers i are integrally closed in non exceptional characteristics therefore is a normal domain and the primary decomposition of i turns out easy giving some insight into the structure of y an interesting observation the primary decomposition of the ideals i can be computed very quickly if one knows in advance that all these ideals are integrally closed the best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group gl m k x gl n k on k x under which z is stable cf dep or bv sect 111 applying the theory of u invariants kraft kr one shows that g has rational singularities in particular and hence 9 are cohen macaulay rings we have no doubt that and 3 are cohen macaulay in arbitrary non exceptional characteristic it seems however that in exceptional characteristic they are as far as possible from this property the case t 2

cohomology of projective schemes from annihilators to vanishing we provide bounds on the castelnuovo mumford regularity in terms of defining equations by using elements that annihilates some cohomology modules inspired by works of miyazaki nagel schenzel and vogel the elements in these annihilators are provided either by liaison or by tight closure theories our results hold in any characteristic

complexes with homology of finite length and frobenius functors in this paper r is a noetherian local ring of prime characteristic p m its maximal ideal and k its residue field let f r r x h xp be the frobenius homomorphism for an r module m the r module m is a4 as an additive group with the scalar multiplication given by rom f r m rp m for rer rem f denotes the nth frobenius functor i e f m mor r for each r module m where the r module structure of f m is given by y m s m rs for r s e r m e m for basic properties of these functors see 2 5 and 91 remember that the functors n m resp f ro fz and n m resp fi are isomorphic the length of an r module m is denoted by m its krull dimension by dim m or dim m and its projective dimension by pd m the ith homology of a complex m will be written as hi m the support supp m of m is the set of all p e spec r such that m or r is not exact a complex m is called bounded if and only if mi 0 for only a finite number of ie z for a bounded complex m with homology modules of finite length the euler characteristic x m ciez l hi m is defined the following theorem based on arguments of dutta 2 and monsky 8 is the key to our results

a briancon skoda theorem for isolated singularities abstract letabe a normal cohen macaulay local ring which is a homomorphic image of a gorenstein local ring supposing thatais pseudo rational in the punctured spectrum and that there exists a proper birational morphismx spec asuch thatxis pseudo rational we show how one can find an integerk 0 such that i n k infor all idealsi aand integersn 0

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

extensions of arc analytic functions we prove that every arc analytic semialgebraic function on an arc symmetric set x in mathbb r n admits an arc analytic semialgebraic extension to the whole mathbb r n

equimultiplicity in hilbert kunz theory we study further the properties of hilbert kunz multiplicity as a measure of singularity this paper develops a theory of equimultiplicity for hilbert kunz multiplicity and applies it to study the behavior of hilbert kunz multiplicity on the brenner monsky hypersurface a number of applications follows in particular we show that hilbert kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed

a less restrictive briancon skoda theorem with coefficients the briancon skoda theorem in its many versions has been studied by algebraists for several decades in this paper under some assumptions on an f rational local ring r m and an ideal i of r of analytic spread l and height g l we improve on two theorems by aberbach and huneke let j be a reduction of i we first give results on when the integral closure of il w is contained in the product jw 1i l 1 for any integer w 0 where given any primary decomposition of i i l 1 is the intersection of the primary components of i of height at most l 1 in the case that r is also gorenstein we give results on when the integral closure of il 1 is contained in j

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

strong f regularity and generating morphisms of local cohomology modules abstract we establish a criterion for the strong f regularity of a non gorenstein cohen macaulay reduced complete local ring of dimension at least 2 and prime characteristic p we also describe an explicit generating morphism in the sense of lyubeznik for the top local cohomology module with support in certain ideals arising from an n n 1 matrix x of indeterminates for a perfect field k of characteristic p 5 these results led us to derive a simple new proof of the well known fact that the generic determinantal ring over k given by the maximal minors of x is strongly f regular

galois extensions plus closure and maps on local cohomology abstract given a local domain r m of prime characteristic that is a homomorphic image of a gorenstein ring huneke and lyubeznik proved that there exists a module finite extension domain s such that the induced map on local cohomology modules h m i r h m i s is zero for each i dim r we prove that the extension s may be chosen to be generically galois and analyze the galois groups that arise

tight closure plus closure and frobenius closure in cubical cones we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 74 3 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring k x y we show that frobenius closure is the same as tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c 1 we conclude that ir n r 1 for these ideals using injective modules over the ring r the union of all peth roots of elements of r we reduce the question of whether if i for 23 graded ideals to the case of 23 graded irreducible modules we classify the irreducible m primary z3 graded ideals we then show that jf 1 for most irreducible rn primary z3 graded ideals in k x y z x3 y3 z3 where k is a field of characteristic p and p 2 mod 3 hence 1 irmf n r for these ideals in this paper we discuss the conjecture that i ir n r where r denotes the integral closure of a domain r of characteristic p in an algebraic closure of its fraction field and i denotes the tight closure of i the ring r is characterized by the property that it is a domain integral over r and every monic polynomial with coefficients in r factors into monic linear factors this characterization can be used to prove the following property of r if w is a multiplicatively closed set of r then w lr w 1r aside from providing a much more concrete description of tight closure proving that i ir n r would solve the localization problem for tight closure it is known that i ir n r for parameter ideals sml and for rings in which every ideal of the normalization is tightly closed also for those ideals i of an excellent local domain r such that r i has finite phantom projective dimension it is known that i ir n r ab however the conjecture is open even for two dimensional normal gorenstein domains in particular the conjecture is open for the cubical cone k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 and more generally for rings of the form k x y z f x y z where f is a homogeneous cubic polynomial we consider tight closure plus closure and frobenius closure in the rings r k x y z x3 y3 z3 where k is a field of characteristic p and p 5l 3 in section 1 we use a z3 grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular rings k x y in section 2 we show that the frobenius closure of an ideal i denoted if is the same as the tight closure in certain classes of ideals when p 2 mod 3 since if c ir n r c i received by the editors august 27 1997 1991 mathematics subject classification primary 13a35 13a02 13hi10

order ideals and a generalized krull height theorem let n be a finitely generated module over a noetherian local ring r m we give criteria for the height of the order ideal n x of an element x n to be bounded by the rank of n the generalized principal ideal theorem of bruns eisenbud and evans says that this inequality always holds if x mn we show that the inequality even holds if the hypothesis becomes true after first extending scalars to some local domain and then factoring out torsion we give other conditions in terms of residual intersections and integral closures of modules we derive information about order ideals that leads to bounds on the heights of trace ideals of modules even in circumstances where we do not have the expected bounds for the heights of the order ideals

on the canonical element conjecture the canonical element conjecture is proved in the following two cases i depth a dim a 1 h a is decomposable ii depth a dim a 1 hm a v is cyclic the equivalence of the c e c and the improved new intersection theorem is also established in h m hochster has assigned a canonical element qa e h syz k with each n dimensional local ring a m k i e commutative noetherian local ring a with a unique maximal ideal m and k a m and has conjectured that 71a 0 for every such a in elementary terms this conjecture means the following h for every free resolution f a8i asi i a k 0 of k and for every system of parameters x1 x of a if 4 is any map of complexes k x a f which lifts the quotient surection a xl x k then 4 kn x a asn is nonzero hochster has shown that the above conjecture henceforth c e c is equivalent to the direct summand conjecture and implies the improved new intersection conjecture which can be stated as follows improved new intersection conjecture let a m be a local ring and f be a finite free complex of finitely generated free modules o fd fo such that hi f has finite length for i 0 and ho f has a nonzero minimal generator v such that avp has finite length then dim r d it is well known that the above conjecture henceforth i n i c implies several other homological conjectures in 1 of this paper we prove that a satisfies c e c a satisfies i n i c 1 4 in proving this we come across in 1 3 another equivalent version of c e c which shows that actually c e c boils down to understanding the relations among the resolutions of homologies of k x a and itself in 2 we study the case when depth a dim a 1 we prove 2 2 that a satisfies c e c in the following cases i when h j1 a v is decomposable this implies hi x a is decomposable ii when h 71 a v is cyclic this implies hi x a v is cyclic received by the editors november 18 1985 1980 mathematics subject classification 1985 revision primary 13h10 the author was supported in part by grants from the national science foundation 1987 american mathematical society 0002 9947 87 1 00 25 per page

approximation in compact nash manifolds let rn be a compact nash manifold a b the rings of nash analytic global functions on the main result of this paper is the following theorem 1 let be a pair of nash submanifolds of some rn rq and let us suppose is compact let f1 fq r be nash functions then every analytic solution y f x of the system f1 x y fq x y 0 can be approximated in the whitney topology by the global nash solutions y g x the main tool used to prove the above results is this version of neron s desingularisation theorem any homomorphism of a algebras c b with c finitely generated over a factorizes through a finitely generated a algebra d such that a d is regular using theorem 1 the authors are able to solve several interesting problems that have been open for many years for example they prove i every analytic factorization of a global nash function defined over is equivalent to a nash factorization ii every semialgebraic subset of which is a global analytic subset is also a global nash subset iii every prime ideal of a generates a prime ideal in b iv every coherent ideal subsheaf of the sheaf n of nash functions on is generated by its global sections the case where is noncompact is only partially studied in this paper in the reviewer s opinion this article makes crucial progress in the theory of global nash functions

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

uniform pointwise bounds for matrix coefficients of unitary representations and applications to kazhdan constants let k be a local field and g the group of k rational points of a connected reductive linear algebraic group over k with k semisimple rank g 2 let k be a good maximal compact subgroup of g for a unitary representation of g a vector v in is called kfinite if the subspace spanned bykv is finite dimensional we will use the termk matrix coefficients resp k finite matrix coefficients of to refer to its matrix coefficients with respect to k invariant resp k finite unit vectors following bt we denote by g the subgroup generated by the unipotent k split subgroups of g the main goal of the present paper is to construct a class of uniform pointwise bounds for the k finite matrix coefficients of all infinite dimensional irreducible unitary representations of g or more generally of all unitary representations of g without a non zero g invariant vector let a be a maximal k split torus and a the closed positive weyl chamber of a such that the cartan decomposition g ka k holds where is a finite subset of the

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

amenability and exactness for dynamical systems and their c algebras we study the relations between amenability resp amenability at infinity of c dynamical systems and equality or nuclearity resp exactness of the corresponding crossed products

absolute integral closures are big cohen macaulay algebras in characteristic p throughout this paper ring means commutative ring with identity and modules are unital our main interest is in local rings i e noetherian rings r m with a unique maximal ideal m in such a ring xx xn e m is a system of parameters if m has a power in the ideal x xn r and n is the krull dimension of r when r is complete and contains a field this means that r is module finite over a formal power series subring k xx xn a i is called cohen macaulay if some equivalently every system of parameters is a regular sequence in i which means that every xi x is a nonzerodivisor on r xx xt r for 0 i n 1 in the case where r is module finite over the formal power series subring a this means that r is a free 4 module for many theorems of commutative algebra and of algebraic geometry the cohen macaulay condition possibly on the local rings of a variety is just what is needed to make the theory work our main result which is given below in an algebraic form in theorem 1 and in a geometric form in theorem 2 asserts that under mild conditions on a local ring r of positive prime characteristic p one can correct the failure of the cohen macaulay condition in r itself by passing to a very large integral extension of r it is worth emphasizing that this is quite false in characteristic 0 and comes as a surprise we believe in characteristic p one can get an idea of how far these theorems are from the truth over a field of characteristic 0 from the following observation in the situation of theorem 2 when x is projectively normal the maps of cohomology are always injective by an easy trace argument despite the fact that both theorem 1 and theorem 2

generic graph construction ideals and greene s theorem let x be an m n matrix of indeterminates m n and t a new indeterminate consider the polynomial rings r0 k x and r r0 t for a given positive integer t m consider the ideal it it x generated by the t minors i e the determinants of the t t submatrices of x using all these determinantal ideals we define a new ideal j in r r0 t which we call the generic graph construction ideal as follows

uniform annihilators of local cohomology abstract in this paper we study the properties of noetherian rings containing uniform local cohomological annihilators it turns out that all such rings should be universally catenary and locally equidimensional we will prove a necessary and sufficient condition for such rings which enables us to show that if a locally equidimensional ring r is the image of a cohen macaulay ring then r has a uniform local cohomological annihilator moreover we will give a positive answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13 about excellent rings with dimension no more than 5

tight closure of finite length modules in graded rings in this article we look at how the equivalence of tight closure and plus closure or frobenius closure in the homogeneous m coprimary case implies the same closure equivalence in the nonhomogeneous m coprimary case in standard graded rings although our result does not depend upon dimension the primary application is based on results known in dimension 2 due to the recent work of h brenner we also demonstrate a connection between tight closure and the m adic closure of modules extended to r or r we finally show that unlike the noetherian case the injective hull of the residue field over r or r contains elements that are not killed by any power of the maximal ideal of r this fact presents an obstruction to one possible method of extending our main result to all modules

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

approximation on nash sets with monomial singularities this paper is devoted to the approximation of differentiable semialgebraic functions by nash functions approximation by nash functions is known for semialgebraic functions defined on an affine nash manifold m and here we extend it to functions defined on nash subsets x of m whose singularities are monomial to that end we discuss first finiteness and weak normality for such sets x namely we prove that i x is the union of finitely many open subsets each nash diffeomorphic to a finite union of coordinate linear varieties of an affine space and ii every function on x which is nash on every irreducible component of x extends to a nash function on m then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on nash sets with monomial singularities as a nice consequence we show that m dimensional affine nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic for k m 2 are also nash diffeomorphic

on nash images of euclidean spaces abstract in this work we characterize the subsets of r n that are images of nash maps f r m r n we prove shiota s conjecture and show that a subset s r n is the image of a nash map f r m r n if and only if s is semialgebraic pure dimensional of dimension d m and there exists an analytic path 0 1 s whose image meets all the connected components of the set of regular points of s two remarkable consequences are the following 1 pure dimensional irreducible semialgebraic sets of dimension d with arc symmetric closure are nash images of r d and 2 semialgebraic sets are projections of irreducible algebraic sets whose connected components are nash diffeomorphic to euclidean spaces

interpolating between hilbert samuel and hilbert kunz multiplicity abstract we define a function called s multiplicity that interpolates between hilbert samuel multiplicity and hilbert kunz multiplicity by comparing powers of ideals to the frobenius powers of ideals the function is continuous in s and its value is equal to hilbert samuel multiplicity for small values of s and is equal to hilbert kunz multiplicity for large values of s we prove that it has an associativity formula generalizing the associativity formulas for hilbert samuel and hilbert kunz multiplicity we also define a family of closures such that if two ideals have the same s closure then they have the same s multiplicity and the converse holds under mild conditions we describe the s multiplicity of monomial ideals in toric rings as a certain volume in real space

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

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

attached primes of local cohomology modules under localization and completion let r m be a noetherian local ring and m a finitely generated r module following i g macdonald 8 the set of all attached primes of the artinian local cohomology module hmi m is denoted by attr hmi m in 13 theorem 3 7 r y sharp proved that if r is a quotient of a gorenstein local ring then the shifted localization principle holds true for any local cohomology modules hmi m i e 1 attrp hprpi dim r p mp qrp q attrhmi m q p for any p spec r in this paper we improve sharp s result as follows the shifted localization principle holds true if and only if r is universally catenary and all its formal fibers are cohen macaulay if and only if the shifted completion principle 2 attr hmi m p attr hmi m assr r pr holds true for any local cohomology module hmi m this also improves the main result of the paper by t d m chau and the first author 2

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

a new family of representations of virtually free groups we construct a new family of irreducible unitary representations of a finitely generated virtually free group we prove furthermore a general result concerning representations of gromov hyperbolic groups that are weakly contained in the regular representation thus implying that all the representations in this family can be realized on the boundary of as a corollary we obtain an analogue of herz majorization principle

graded annihilators of modules over the frobenius skew polynomial ring and tight closure this paper is concerned with the tight closure of an ideal a in a commutative noetherian local ring r of prime characteristic p several authors including r fedder k i watanabe k e smith n hara and f enescu have used the natural frobenius action on the top local cohomology module of such an r to good effect in the study of tight closure and this paper uses that device the main part of the paper develops a theory of what are here called special annihilator submodules of a left module over the frobenius skew polynomial ring associated to r this theory is then applied in the later sections of the paper to the top local cohomology module of r and used to show that if r is cohen macaulay then it must have a weak parameter test element even if it is not excellent