generalized serre conditions and perverse coherent sheaves abstract in algebraic geometry one often encounters the following problem given a scheme x find a proper birational morphism y x where the geometry of y is nicer than that of x one version of this problem first studied by faltings requires y to be cohen macaulay in this case y x is called a macaulayfication of x in another variant one requires y to satisfy the serre condition s r in this paper the authors introduce generalized serre conditions these are local cohomology conditions which include s r and the cohen macaulay condition as special cases to any generalized serre condition s there exists an associated perverse t structure on the derived category of coherent sheaves on a suitable scheme x under appropriate hypotheses the authors characterize those schemes for which a canonical finite s ification exists in terms of the intermediate extension functor for the associated perversity similar results including a universal property are obtained for a more general morphism extension problem called s extension

on cohomologically complete intersections abstract an ideal i of a local gorenstein ring r m is called cohomologically complete intersection whenever h i i r 0 for all i height i here h i i r i z denotes the local cohomology of r with respect to i for instance a set theoretic complete intersection is a cohomologically complete intersection here we study cohomologically complete intersections from various homological points of view in particular in terms of their bass numbers of h i c r c height i as a main result it is shown that the vanishing h i i r 0 for all i c is completely encoded in homological properties of h i c r in particular in its bass numbers

on birational macaulayfications and cohen macaulay canonical modules abstract the aim of the paper is twofold at first there is a characterization of those local domains admitting a birational macaulayfication it turns out that a local domain a quotient of a gorenstein ring possesses a cohen macaulay birational extension b if and only if its canonical module k a is a cohen macaulay module in this situation it follows that b is isomorphic to the endomorphism ring of k a the second part of the paper is devoted to the question when the canonical module of a local ring is a cohen macaulay module there are applications on sequentially cohen macaulay rings a large class of rings whose canonical module are cohen macaulay is given by the simplicial affine semigroup rings as a technical tool there is a spectral sequence related to the duality with respect to the dualizing complex

on invariants and endomorphism rings of certain local cohomology modules abstract let r m denote an n dimensional gorenstein ring for an ideal i r with grade i c we define new numerical invariants i j i as the socle dimensions of h m i h i n j r in case of a regular local ring containing a field these numbers coincide with the lyubeznik numbers i j r i we use d d i d dim r i to characterize the surjectivity of the natural homomorphism f r hom r h i r c r h i r c r as a technical tool we study several natural homomorphisms moreover we prove a few results on i j i

finiteness properties of local cohomology modules an application of d modules to commutative algebra the main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by c huneke and r sharp in characteristicp 0 our method based on the theory of algebraicd modules seems to be the first application ofd modules to commutative algebra

lyubeznik s invariants for cohomologically isolated singularities abstract in this note i give a description of lyubeznik s local cohomology invariants for a certain natural class of local rings namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity this strengthens our results of manuel blickle raphael bondu local cohomology multiplicities in terms of etale cohomology ann inst fourier 55 7 2005 while at the same time somewhat simplifying the proofs through examples i further point out the bad behavior of these invariants under reduction mod p

associated primes of local cohomology of flat extensions with regular fibers and σ finite d modules abstract in this manuscript we study the following question raised by mel hochster let r m k be a local ring and s be a flat extension with regular closed fiber is v m s ass s h i i s finite for every ideal i s and i n we prove that the answer is positive when s is either a polynomial or a power series ring over r and dim r i r 1 in addition we analyze when this question can be reduced to the case where s is a power series ring over r an important tool for our proof is the use of finite d modules which are not necessarily finitely generated as d modules but whose associated primes are finite we give examples of this class of d modules and applications to local cohomology

lyubeznik numbers of monomial ideals let r k x1 xn be the polynomial ring in n independent variables where k is a field in this work we will study bass numbers of local cohomology modules h r i r supported on a squarefree monomial ideal i r among them we are mainly interested in lyubeznik numbers we build a dictionary between the modules h r i r and the minimal free resolution of the alexander dual ideal i that allow us to interpret lyubeznik numbers as the obstruction to the acyclicity of the linear strands of i the methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support

on formal local cohomology and connectedness let a denote an ideal of a local ring r m let m be a finitely generated r module there is a systematic study of the formal cohomology modules lim hmi m anm i z we analyze their r module structure the upper and lower vanishing and non vanishing in terms of intrinsic data of m and its functorial behavior these cohomology modules occur in relation to the formal completion of the punctured spectrum specr v m as a new cohomological data there is a description on the formal grade fgrade a m defined as the minimal non vanishing of the formal cohomology modules there are various exact sequences concerning the formal cohomology modules among them a mayer vietoris sequence for two ideals it applies to new connectedness results there are also relations to local cohomological dimensions

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

castelnuovo mumford regularity of deficiency modules abstract let d n and let m be a finitely generated graded module of dimension d over a noetherian homogeneous ring r with local artinian base ring r 0 let beg m gendeg m and reg m respectively denote the beginning the generating degree and the castelnuovo mumford regularity of m if i n 0 and n z let d m i n denote the r 0 length of the n th graded component of the i th r transform module d r i m of m and let k i m denote the i th deficiency module of m our main result says that reg k i m is bounded in terms of beg m and the diagonal values d m j j with j 0 d 1 as an application of this we get a number of further bounding results for reg k i m

finite dimensional invariant subspaces for algebras of linear operators and amenable banach algebras abstract we study a finite dimensional invariant subspace property similar to fan s theorem on semigroups for arbitrary banach algebras a in terms of amenability of x a the closed subalgebra of a generated by the set of all maximal elements in a with respect to a character as a consequence we offer some applications to the measure algebra m g and the generalized fourier algebra a p g of a locally compact group g

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

common fixed point properties and amenability of a class of banach algebras abstract let a be a lau algebra and let s a be the semigroup of all positive functionals in a with norm one we obtain some equivalent conditions for left amenability of a in terms of a common fixed point property and the hahn banach property we then apply these results for certain lau algebras on a locally compact group g to give characterizations for amenability of g

the isoperimetric problem for pinwheel tilings in aperiodic pinwheel tilings of the plane there exist unions of tiles with ratio area perimeter 2 arbitrarily close to that of a circle such approximate circles can be constructed with arbitrary center and any sufficiently large radius the existence of such circles follows from the metric on pinwheel space being almost euclidean at large distances ifp andq are points separated by large euclidean distancer then the shortest path along tile edges fromp toq has lengthr o r

on arithmetic macaulayfication of noetherian rings the rees algebra is the homogeneous coordinate ring of a blowing up the present paper gives a necessary and sufficient condition for a noetherian local ring to have a cohen macaulay rees algebra a noetherian local ring has a cohen macaulay rees algebra if and only if it is unmixed and all the formal fibers of it are cohen macaulay as a consequence of it we characterize a homomorphic image of a cohen macaulay local ring for non local rings this paper gives only a sufficient condition by using it however we obtain the affirmative answer to sharp s conjecture that is a noetherian ring having a dualizing complex is a homomorphic image of a finite dimensional gorenstein ring

asymptotically non expansive actions of strongly amenable semigroups and fixed points abstract in this paper we deal with some fixed point properties for a semi topological semigroup s acting on a compact convex subset k of a banach space we first focus on the space l m c s of left multiplicatively continuous functions on s and its strong left amenability the existence of a compact left ideal group in the lmc compactification of s we then study the relation between left amenability and strong left amenability of l m c s with a common fixed point property for non expansive and asymptotically non expansive actions of s our results improve a result of t mitchell in 1970 and answer an open problem of a t m lau in 2010 for the class of strongly left amenable semi topological semigroups

dualizing complexes of affine semigroup rings for an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup as a consequence there are characterizations of locally cohen macaulay rings buchsbaum rings and cohen macaulay rings as well as serre s condition 9 introduction let s denote a finitely generated submonoid of the additive monoid n n being a positive integer by the affine semigroup ring i s of s over a field x let us denote the subring of x xl xn generated by all monomials xs s e s so one has up to isomorphisms a one to one correspondence between affine semigroup rings and affine varieties given parametrically by monomials now it is of some interest to characterize ring theoretic properties as cohenmacaulay gorenstein etc in terms of the semigroup s a first breakthrough in this direction was done by kempf et al kkms and hochster hol who showed that x s is a cohen macaulay ring provided it is normal moreover the normality of x s is described in terms of s see hol another particular case if s is a simplicial semigroup is treated by goto suzuki and watanabe gsw resp stanley st2 who showed that x s is a cohen macaulay ring if it satisfies the 22 condition of serre note that normality is equivalent to 5y2 and r1 moreover there is a description of the 5 condition in terms of s see theorem 6 3 in general the 52 condition is not sufficient for i s to be a cohen macaulay ring as follows by examples in hol resp th 1 2 on the other side grobner g posed the problem to classify cohen macaulay affine semigroup rings the solution of this problem was done by hoa and trung th related to structural properties of graded rings is a technical tool of the duality theory this is the dualizing complex introduced by grothendieck see ha in recent years it became a helpful tool in commutative algebra in particular its knowledge allows characterizations of cohen macaulay rings locally cohenmacaulay rings buchsbaum rings gorenstein rings and the 5 conditions of received by the editors december 18 1988 1980 mathematics subject classification 1985 revision primary 13h10 secondary 14m05 06a99 1990 american mathematical society 0002 9947 90 1 q0 25 per page

invariant means and semigroups of nonexpansive mappings on uniformly convex banach spaces let s be a semitopological semigroup i e s is a semigroup with hausdorff topology such that for each a s the mappings s sa and s as from s into s are continuous let ruc s denote the space of bounded right uniformly continuous real valued functions on s in 141 t mitchell generalizing a well known fixed point theorem of day s showed that ruc s has a right invariant mean if and only if whenever s x c 4 c is an affine jointly continuous anti action of s on a compact convex subset of a separated locally convex space then c contains a common fixed point for s furthermore an argument similar to that for the proof of theorem 1 in 14 shows that if s has the following fixed point property for dual banach space then ruc s has a right invariant mean

non cohen macaulay locus of canonical modules abstract let r m be a noetherian local ring which is a quotient of a gorenstein local ring let m be a finitely generated r module and k m the canonical module of m denote by ncm m the non cohen macaulay locus of m in this paper the relationship between the two loci ncm k m and ncm m is given the structure of m is clarified in case where dim ncm k m 0 i e k m is generalized cohen macaulay

castelnuovo mumford regularity of canonical and deficiency modules abstract we give two kinds of bounds for the castelnuovo mumford regularity of the canonical module and the deficiency modules of a ring respectively in terms of the homological degree and the castelnuovo mumford regularity of the original ring

on canonical cohen macaulay modules abstract let r m be a noetherian local ring which is a homomorphic image of a local gorenstein ring and let m be a finitely generated r module of dimension d 0 according to schenzel 2004 sc3 m is called a canonical cohen macaulay module ccm module for short if the canonical module k m of m is cohen macaulay we give another characterization of ccm modules we describe the non canonical cohen macaulay locus nccm m of m if d 4 then nccm m is closed in spec r for each d 5 there are reduced geometric local rings r of dimension d such that nccm r is not stable under specialization

on some local cohomology invariants of local rings let a be a commutative noetherian local ring containing a field of characteristic p 0 the integer invariants i j a have been introduced in an old paper of ours in this paper we completely describe d d a where d dima in terms of the topology of speca

pseudo cohen macaulay and pseudo generalized cohen macaulay modules abstract in this paper we study the structure of two classes of modules called pseudo cohen macaulay and pseudo generalized cohen macaulay modules we also give a characterization for these modules in term of the cohen macaulayness and generalized cohen macaulayness then we apply this result to prove a cohomological characterization for sequentially cohen macaulay and sequentially generalized cohen macaulay modules

notes on local cohomology and duality abstract we provide a formula see theorem 1 5 for the matlis dual of the injective hull of r p where p is a one dimensional prime ideal in a local complete gorenstein domain r m this is related to results of enochs and xu see 4 and 5 we prove a certain dual version of the hartshorne lichtenbaum vanishing see theorem 2 2 we prove a generalization of local duality to cohomologically complete intersection ideals i in the sense that for i m we get back the classical local duality theorem we determine the exact class of modules to which a characterization of cohomologically complete intersection from 7 generalizes naturally see theorem 4 4

toward an efficient algorithm for deciding the vanishing of local cohomology modules in prime characteristic let r k x 1 dots x n be a ring of polynomials over a field k of characteristic p 0 there is an algorithm due to lyubeznik for deciding the vanishing of local cohomology modules h i i r where i subset r is an ideal this algorithm has not been implemented because its complexity grows very rapidly with the growth of p which makes it impractical in this paper we produce a modification of this algorithm that consumes a modest amount of memory

finite dimensional invariant subspace property and amenability for a class of banach algebras anthony to ming lau and yong zhangabstract motivated by a result of ky fan in 1965 we establish a charac terization of a left amenable f algebra which includes the group algebra andthe fourier algebra of a locally compact group and quantum group algebras or more generally the predual algebra of a hopf von neumann algebra interms of a nite dimensional invariant subspace property this is done by rstrevealing a xed point property for the semigroup of norm one positive lin ear functionals in the algebra our result answers an open question posted intokyo in 1993 by the rst author see 25 problem 5 we also show that theleft amenability of an ideal in an f algebra may determine the left amenabilityof the algebra

on the structure of the endomorphism ring of a certain local cohomology module let r m denote an n dimensional gorenstein ring for an ideal i r of height c we are interested in the endomorphism ring b homr hic r hic r it turns out that b is a commutative ring in the case of r m a regular local ring containing a field b is a cohen macaulay ring its properties are related to the highest lyubeznik number l dimkextrd k hic r d dimr i in particular r b if and only if l 1 moreover we show that the natural homomorphism extrd k hic r k is non zero

the pinwheel tilings of the plane prototiles in euclidean n space en for n 2 the prototiles are usually required to be rather nice topologically at least homeomorphs of the closed unit ball one then makes arbitrarily many congruent copies called tiles of these prototiles and considers all ways called tilings that such tiles may provide a simultaneous covering and packing of en a tiling is thus an unordered collection of tiles for which the union is all of en but such that the interiors of each pair of tiles do not intersect wang s original problem was to determine if it was possible to have a finite set of prototiles with associated tilings of space for which all the tilings were nonperiodic a tiling is nonperiodic if it is not invariant under any single simultaneous translation of its tiles this was settled in the affirmative in the thesis of wang s student robert berger 1

approximately cohen macaulay rings 1 introduction let a be a noetherian local ring with maximal ideal m and d dim a the purpose of this note is to study when there is an element a of m such that a a 4 is a cohen macaulay ring of dimension d 1 for every integer n 0 recall that a is gorenstein if and only if there is an element a of m such that a a a is a gorenstein ring of dimension d 1 for every integer n 0 cf e g 131 clearly this is not true in the cohen macaulay case however since such rings are so to speak approximately cohen macaulay and may be pretty special it seems interesting to clarify when a given ring a has this property let assh a p e ass a dim a p d and put u o n i p

fixed point properties for semigroup of nonexpansive mappings on frechet spaces abstract in this paper we shall establish a fixed point property on frechet spaces for left reversible semitopological semigroups generalizing some classical results

time continuity in cohesive finite element modeling we introduce the notion of time continuity for the analysis of cohesive zone interface finite element models we focus on initially rigid models in which an interface is inactive until the traction across it reaches a critical level we argue that methods in this class are time discontinuous unless special provision is made for the opposite time discontinuity leads to pitfalls in numerical implementations oscillatory behavior non convergence in time and dependence on nonphysical regularization parameters these problems arise at least partly from the attempt to extend uniaxial traction displacement relationships to multiaxial loading we also argue that any formulation of a time continuous functional traction displacement cohesive model entails encoding the value of the traction components at incipient softening into the model we exhibit an example of such a model most of our numerical experiments concern explicit dynamics

algebraic mesh quality metrics quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics the theory based on the jacobian and related matrices provides a means of constructing classifying and evaluating mesh quality metrics the jacobian matrix is factored into geometrically meaningful parts a nodally invariant jacobian matrix can be defined for simplicial elements using a weight matrix derived from the jacobian matrix of an ideal reference element scale and orientation invariant algebraic mesh quality metrics are defined the singular value decomposition is used to study relationships between metrics equivalence of the element condition number and mean ratio metrics is proved the condition number is shown to measure the distance of an element to the set of degenerate elements algebraic measures for skew length ratio shape volume and orientation are defined abstractly with specific examples given two combined metrics shape volume and shape volume orientation are algebraically defined and examples of such metrics are given algebraic mesh quality metrics are extended to nonsimplicial elements a series of numerical tests verifies the theoretical properties of the metrics defined

an algorithm for two dimensional mesh generation based on the pinwheel tiling we propose a new two dimensional meshing algorithm called pinw able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges this technique is based on an extension of pinwheel tilings proposed by radin and conway see c radin ann of math 2 139 1994 pp 661 702 we prove that the algorithm produces triangles of bounded aspect ratio this kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation path is an outcome of a simulation process

semigroup of nonexpansive mappings on a hilbert space soit s un semi groupe semi topologique soit h un espace de hilbert et s t s s s une representation continue de s comme applications non dilatation sur un sous ensemble convexe clos c de h dans c soit x c on etudie la convergence de t s x s s vers un point fixe commun z de s dans c

spatial convergence of crack nucleation using a cohesive finite element model on a pinwheel based mesh we consider the use of initially rigid cohesive interface models in a two dimensional dynamic finite element solution of a fracture process our focus is on convergence of finite element solutions to a solution of the undiscretized medium as the mesh spacing x and therefore time step t tends to zero we propose the use of pinwheel meshes which possess the isoperimetric property that for any curve c in the computational domain there is an approximation to c using mesh edges that tends to c including a correct representation of its length as the grid size tends to zero we suggest that the isoperimetric property is a necessary condition for any possible spatial convergence proof in the general case that the crack path is not known in advance conversely we establish that if the pinwheel mesh is used the discrete interface first activated in the finite element model will converge to the initial crack in the undiscretized medium finally we carry out a mesh refinement experiment to check convergence of both nucleation and propagation our results indicate that the crack path computed in the pinwheel mesh is more stable as the mesh is refined compared to other types of meshes copyright 2006 john wiley sons ltd

perverse coherent sheaves and the geometry of special pieces in the unipotent variety let x be a scheme of finite type over a noetherian base scheme s admitting a dualizing complex and let ux be an open set whose complement has codimension at least 2 we extend the deligne bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension or intersection cohomology functor from perverse sheaves on u to perverse sheaves on x may be defined for a much broader class of perversities than has previously been known we also introduce a derived category version of the coherent intermediate extension functor under suitable hypotheses we introduce a construction called s2 extension in terms of perverse coherent sheaves of algebras on x that takes a finite morphism to u and extends it in a canonical way to a finite morphism to x in particular this construction gives a canonical s2 ification of appropriate x the construction also has applications to the macaulayfication problem and it is particularly well behaved when x is gorenstein our main goal however is to address a conjecture of lusztig on the geometry of special pieces certain subvarieties of the unipotent variety of a reductive algebraic group the conjecture asserts in part that each special piece is the quotient of some variety previously unknown for the exceptional groups and in positive characteristic by the action of a certain finite group we use s2 extension to give a uniform construction of the desired variety

bockstein homomorphisms in local cohomology let r be a polynomial ring in finitely many variables over the integers fix an ideal a of r we prove that for all but finitely prime integers p the bockstein homomorphisms on local cohomology h a r pr h k 1 a r pr are zero this vanishing of bockstein homomorphisms is predicted by lyubeznik s conjecture which states that when r is a regular ring the modules h a r have finitely many associated prime ideals we further show that when r is replaced by a hypersurface ring the vanishing of bockstein homomorphisms as well as the analogue of lyubeznik s conjecture need not hold bockstein homomorphisms have their origins in algebraic topology we establish a connection between algebraic and topological bockstein homomorphisms through stanley reisner theory

invariant submeans and semigroups of nonexpansive mappings on banach spaces with normal structure abstract in this paper we study the relation between invariant submean and normal structure in a banach space this is used to give an improvement and different proof of a fixed point theorem of lim also of belluce and kirk for commutative semigroups for left reversible semigroup of nonexpansive mappings on weakly compact convex subsets of a banach space with normal structure

fourier stieltjes algebra of a topological group abstract this paper is an invitation to the study of the fourier stieltjes algebra b g the linear span of the continuous positive definite complex valued functions on a topological group g we study b g when g has a host algebra or a group c algebra the analogue of the group c algebra of a locally compact group our main challenge is that a topological group g cannot have a positive regular borel measure which is left translation invariant unless g is locally compact

2m weak amenability of group algebras a common fixed point property for semigroups is applied to show that the group algebra l1 g of a locally compact group g is 2m weakly amenable for each integer m 1

constructing arbitrary torsion elements for a local cohomology module local cohomology modules are in general not finitely generated over the ambient ring towards a better understanding of the structure of these modules huneke 1992 hu asked if they have finitely many associated prime ideals the answer is negative several authors have constructed a ring whose local cohomology module has infinitely many associated primes in certain situations one can prove that a local cohomology module has infinitely many associated primes by showing that z pz embeds into the local cohomology module for each prime integer p we show that in fact each finitely generated abelian group embeds into a local cohomology module