homological properties of the perfect and absolute integral closures of noetherian domains for a noetherian local domain r let r be the absolute integral closure of r and let r be the perfect closure of r when r has prime characteristic in this paper we investigate the projective dimension of residue rings of certain ideals of r and r in particular we show that any prime ideal of r has a bounded free resolution of countably generated free r modules also we show that the analogue of this result is true for the maximal ideals of r when r has residue prime characteristic we compute global dimensions of r and r in some cases some applications of these results are given

fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

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

vectorized multigrid poisson solver for the cdc cyber 205 the full multigrid fmg method is applied to the two dimensional poisson equation with dirichlet boundary conditions this has been chosen as a relatively simple test case for examining the efficiency of fully vectorizing of the multigrid method data structure and programming considerations and techniques are discussed accompanied by performance details

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

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

etale cohomological dimension and the topology of algebraic varieties the purpose of this paper is twofold to develop a theory of etale cohomological dimension in the context of schemes of finite type over a separably closed field that would be analogous to the well known theory of quasicoherent cohomological dimension and to apply our theory to prove new results about the topology of algebraic varieties of small codimension in n space the cohomological dimension of a scheme x relative to a closed subscheme y denoted by cd x y is the largest integer r such that the local cohomology group h x f 0 for some abelian torsion sheaf f on xet where f consists only of torsion prime to all the residual characteristics of x the gist of our theory is a technique for proving various upper bounds on cd x y especially in the case where y has small codimension in x local bounds are the most important as well as the easiest to state accordingly for the purposes of this introduction let x spec a where a is a strictly henselian local ring of a finite type scheme over a separably closed field just what kind of bounds on cd x y one should expect is indicated by the theory of quasicoherent cohomological dimension which has been developed by several authors see for example fa grl hal ha2 hasp huly 01 and pessz the quasicoherent cohomological dimension of x relative to y denoted by qccd x y is the largest integer r such that there exists a quasicoherent sheaf f on xzar with h x f 54 0 in the above stated local case the main results of the theory of quasicoherent cohomological dimension are the following where n dim x i qccd x y n see gri 1 12 ii qccd x y t if y is set theoretically defined by t equations see gri 2 3

on a cluster category of infinite dynkin type and the relation to triangulations of the infinity gon let k be a field and let d be a k linear algebraic triangulated category with split idempotents let be the suspension functor of d and let s be a 2 spherical object of d that is the morphism space d s i s is k for i 0 and i 2 and vanishes otherwise assume that s classically generates d that is each object of d can be built from s using de suspensions direct sums direct summands and distinguished triangles it was proved in 15 thm 2 1 that d is uniquely determined by these properties as we will explain d is a good candidate for a cluster category of dynkin type a for instance we show that there is a bijection between the cluster tilting subcategories of d and certain triangulations of the gon we use this to give an example of a subcategory a which is weak cluster tilting that is satisfies a 1a a but fails to be functorially finite perpendicular

quasilength latent regular sequences and content of local cohomology abstract the quasilength of a finitely generated module that is killed by a power of a finitely generated ideal i is introduced it is the length of a shortest filtration of the module with factors that are cyclic modules killed by i this notion is then used to define a notion of content for the dth local cohomology module of a ring or module with support in an ideal generated by d elements in the case of a ring which is central this content is a real number between 0 and 1 it is not known whether it is independent of the choice of the d generators nor whether it can change if the generators are replaced by powers in positive prime characteristic it is shown that the content is always 0 or 1 this is an open question in equal characteristic 0 and in mixed characteristic it is conjectured that if the elements form a system of parameters in a local ring of dimension d then the content is 1 this is proved if the ring contains a field in mixed characteristic it is an open question in dimension 3 or more and implies the direct summand conjecture the relationship between the notions of quasilength and content and the property that a sequence of elements can become a regular sequence on a ring or module after base change is explored

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

on the support of local cohomology we examine the question of whether the support of an arbitrary local coho mology module of a finitely generated module over a noetherian ring with support in a given ideal must be closed in the zariski topology several results giving an armative answer to this question are given in particular we show that the support is closed when the given ideal has cohomological dimension at most two or the ambient ring is local of dimension at most four

on the direct summand conjecture and its derived variant andre recently gave a beautiful proof of hochster s direct summand conjecture in commutative algebra using perfectoid spaces his two main results are a generalization of the almost purity theorem the perfectoid abhyankar lemma and a construction of certain faithfully flat extensions of perfectoid algebras where discriminants acquire all p power roots in this paper we explain a quicker proof of hochster s conjecture that circumvents the perfectoid abhyankar lemma instead we prove and use a quantitative form of scholze s hebbarkeitssatz the riemann extension theorem for perfectoid spaces the same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de jong

on superheight conditions for the affineness of open subsets in this paper we consider the open complement u of a hypersurface y v a in an affine scheme x we study the relations between the affineness of u the intersection of y with closed subschemes the property that every closed surface in u is affine the property that every analytic closed surface is stein and the superheight of a defining ideal a

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

a comparison of vectorized methods for solving the two dimensional diffusion equation multigrid versus polynomial preconditioned conjugate gradient this paper presents the result of a study to develop efficient vectorized multigrid mg methods for solving the two dimensional diffusion equation on a rectangular domain with dirichlet neumann boundary conditions for particular application to groundwater flow problems three different algorithms have been implemented on the cdc cyber 205 the best one being selected on the basis of the degree of continuity and isotropy of the coefficients of the particular problem at hand to demonstrate the superiority of our algorithms we compare performances for several sample problems with those resulting from application of the polynomial preconditioned conjugate gradient ppcg method except for small problems mg solution rates are found to be substantially greater than those obtained using ppcg and the ratio of the rates increases dramatically with problem size

cohomological dimension lyubeznik numbers and connectedness in mixed characteristic abstract we establish a second vanishing theorem for local cohomology modules over regular rings of unramified mixed characteristic which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology applying this and new results on the mixed characteristic lyubeznik numbers we further study connectedness properties of the spectra of a certain class of rings

some restrictions on the maps in minimal resolutions abstract we show that there are certain restrictions on the entries of the maps in the minimal free resolutions of finitely generated modules of infinite projective dimension over noetherian local rings these restrictions provide a way to slightly improve herzog s characterization of modules of finite projective and injective dimensions in characteristic p 0 we also discuss other homological situations related to these restrictions including previously known results on the existence of free summands in syzygy modules

analytic cohomology groups in top degrees of zariski open sets in mathbb p n we show that if a is a closed analytic subset of mathbb p n of pure codimension q then h i mathbb p n setminus a mathcal f are finite dimensional for every coherent algebraic sheaf mathcal f and every i geq n left frac n 1 q right if n 1 geq 2q rm we show that h n 2 mathbb p n setminus a mathcal f 0

a connectedness result in positive characteristic let r m be a complete local ring of dimension at least two which contains a separably closed coefficient field of positive characteristic using a vanishing theorem of peskine szpiro lyubeznik proved that the local cohomology module h 1 m r is frobenius torsion if and only if the punctured spectrum of r is connected in the zariski topology we give a simple proof of this theorem and more generally a formula for the number of connected components in terms of the frobenius action on h 1 m r

dagger closure in regular rings containing a field abstract we prove that dagger closure is trivial in regular domains containing a field and that graded dagger closure is trivial in polynomial rings over a field we also prove that heitmann s full rank one closure coincides with tight closure in positive characteristic under some mild finiteness conditions furthermore we prove that dagger closure is always contained in solid closure and that the forcing algebra for an element contained in dagger closure is parasolid

maximal rigid subcategories and cluster tilting subcategories in 2 cy categories abstract let c be a connected hom finite 2 cy triangulated category we prove that if t is a functorially finite maximal rigid subcategory of c without loops in its quiver then t is cluster tilting in particular this gives a positive answer to a conjecture proposed in 5

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

local cohomology with support in ideals of maximal minors abstract suppose that k is a field of characteristic zero x is an r s matrix of indeterminates where r s and r k x is the polynomial ring over k in the entries of x we study the local cohomology modules h i i r where i is the ideal of r generated by the maximal minors of x we identify the indices i for which these modules vanish compute h i i r at the highest nonvanishing index i r s r 1 and characterize all nonzero ones as submodules of certain indecomposable injective modules these results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings

reduced bass numbers auslander s δ invariant and certain homological conjectures in the category of finitely generated modules over a local cohen macaulay ring with dualizing module one can following auslander buchweitz consider maximal cohen macaulay approximations and hulls of finite injective dimension the former give rise to y and for a gorenstein ring invariants the latter to vd invariants which are in some sense dual this last type of invariant has in the guise of what we call a reduced bass number been encountered in the study of the canonical element conjecture cec by hochster in this paper we establish the basic properties of these invariants some known some not we go on to explore the connection with cec the latter is through linkage seen to imply a fact about delta invariants this in turn shows that cec guarantees the existence of certain maximal cohen macaulay modules over a gorenstein ring

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

how to rescue solid closure abstract we define a closure operation for ideals in a commutative ring which has all the good properties of solid closure at least in the case of equal characteristic but such that also every ideal in a regular ring is closed this gives in particular a kind of tight closure theory in characteristic zero without referring to positive characteristic

on the notion of cohen macaulayness for non noetherian rings abstract there exist many characterizations of noetherian cohen macaulay rings in the literature these characterizations do not remain equivalent if we drop the noetherian assumption the aim of this paper is to provide some comparisons between some of these characterizations in non noetherian case toward solving a conjecture posed by glaz we give a generalization of the hochster eagon result on cohen macaulayness of invariant rings in the context of non noetherian rings

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

lower bounds on projective levels of complexes abstract for an associative ring r the projective level of a complex f is the smallest number of mapping cones needed to build f from projective r modules we establish lower bounds for the projective level of f in terms of the vanishing of homology of f we then use these bounds to derive a new version of the new intersection theorem for level when r is a commutative noetherian local ring

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

extended plus closure and colon capturing abstract three results concerning the colon capturing property for extended plus closures in excellent mixed characteristic rings are demonstrated it is shown that the extended plus closure has the colon capturing property for arbitrary sets of three parameters it is also seen that establishing the colon capturing property for the extended plus closure is sufficient to guarantee the existence of balanced big cohen macaulay algebras also for three dimensional complete domains the second local cohomology of r is annihilated by arbitrarily small powers of every non unit

koszul and local cohomology and a question of dutta for a mathfrak m a local ring we study the natural map from the koszul cohomology module h dim a mathfrak m a to the local cohomology module h dim a mathfrak m a we prove that the injectivity of this map characterizes the cohen macaulay property of the ring a we also answer a question of dutta by constructing normal rings a for which this map is zero

depth for complexes and intersection theorems this paper introduces a new notion of depth for complexes it agrees with the classical definition for modules and coincides with earlier extensions to complexes whenever those are defined techniques are developed leading to a quick proof of an extension of the improved new intersection theorem this uses hochster s big cohen macaulay modules and also a generalization of the depth formula for tensor product of modules properties of depth for complexes are established extending the usual properties of depth for modules

the monomial conjecture and order ideals ii abstract let i be an ideal of height d in a regular local ring r m k r m of dimension n and let denote the canonical module of r i in this paper we first prove the equivalence of the following the non vanishing of the edge homomorhpism d ext r n d k ext r n k r the validity of the order ideal conjecture for regular local rings and the validity of the monomial conjecture for all local rings next we prove several special cases of the order ideal conjecture monomial conjecture

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

fontaine rings and local cohomology abstract we use a construction due to fontaine to construct rings with almost vanishing local cohomology from rings of mixed characteristic and discuss the question of using this method to construct almost cohen macaulay algebras over the original ring we also show that the existence of almost cohen macaulay algebras implies the monomial conjecture and give an example to show how this procedure can be carried out in a nontrivial case

axiomatic closure operations phantom extensions and solidity abstract in this article we generalize a previously defined set of axioms for a closure operation that induces balanced big cohen macaulay modules while the original axioms were only defined in terms of finitely generated modules these new ones will apply to all modules over a local domain the new axioms will lead to a notion of phantom extensions for general modules and we will prove that all modules that are phantom extensions can be modified into balanced big cohen macaulay modules and are also solid modules as a corollary if r has characteristic p 0 and is f finite then all solid algebras are phantom extensions if r also has a big test element e g if r is complete then solid algebras can be modified into balanced big cohen macaulay modules hochster and huneke have previously demonstrated that there exist solid algebras that cannot be modified into balanced big cohen macaulay algebras we also point out that tight closure over local domains in characteristic p generally satisfies the new axioms and that the existence of a big cohen macaulay module induces a closure operation satisfying the new axioms

on some local cohomology modules we give an explicit description of a certain high order local cohomology module with support in an ideal in terms of some combinatorial information about the minimal primes of the ideal as a consequence we give a combinatorial necessary and sufficient condition for an upper bound on the cohomological dimension of the ring with support in the ideal

a short note on standard parallel multigrid algorithms for 3d problems we present some results concerning parallel multigrid mg algorithms for 3d problems for certain practically important anisotropic cases plane relaxation is needed for smoothing if standard mg coarsening etc is used parallel versions of these sophisticated mg algorithms are described and their time complexity and multiprocessor efficiency is studied

on the arithmetical rank of certain segre embeddings we study the number of set theoretically defining equations of segre products of projective spaces times certain projective hypersurfaces extending results by singh and walther meanwhile we prove some results about the cohomological dimension of certain schemes in particular we solve a conjecture of lyubeznik about an inequality involving the cohomological dimension and the etale cohomological dimension of a scheme in the characteristiczero case and under a smoothness assumption furthermore we show that a relationship between depth and cohomological dimension discovered by peskine and szpiro in positive characteristic also holds true in characteristic zero up to dimension three

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway

a vectorized multigrid solver for the three dimensional poisson equation a vectorized version of the multigrid mg method has been implemented on the cdc cyber 205 for solving the three dimensional poisson equation on a rectangular domain with dirichlet boundary conditions this paper describes the techniques employed in vectorizing the mg processes of relaxation residual transfer and interpolation it discusses the data structures and programming concepts employed and presents resulting performance details it concludes with some suggestions for future applications efforts