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

indecomposable cohen macaulay modules and their multiplicities the main aim of this paper is to find a large class of rings for which there are indecomposable maximal cohen macaulay modules of arbitrary high multiplicity or rank in the case of domains

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

stability of reducing subspaces abstract we characterize the stability of reducing subspaces of a rectangular matrix pencil of complex matrices b a except for the special case in which the pencil has no eigenvalues and only has one row and one column minimal indices and both are different from zero

on the lipschitz stability of a b invariant subspaces abstract let t be the set of triples a b s where s is an a b invariant subspace and let f be the matrix having minimum norm such that a bf s s then if t m m n is the map defined by a b s f and is continuous at a b s a simple necessary and sufficient condition is given for the lipschitz stability of s it is shown that this continuity condition is satisfied in an open and dense subset of t and that the set of triples a b s such that s is lipschitz stable is also open and dense in t

on the vanishing of local cohomology of the absolute integral closure in positive characteristic abstract the aim of this paper is to extend the main result of c huneke and g lyubeznik in adv math 210 2007 498 504 to the class of rings that are images of cohen macaulay local rings namely let r be a local noetherian domain of positive characteristic that is an image of a cohen macaulay local ring we prove that all local cohomology of r below the dimension maps to zero in a finite extension of the ring as a direct consequence we obtain that the absolute integral closure of r is a big cohen macaulay algebra since every excellent local ring is an image of a cohen macaulay local ring this result is a generalization of the main result of m hochster and huneke in ann of math 135 1992 45 79 with a simpler proof

stable subspaces of matrix pairs abstract let a b c n n c n m and let m be an a b invariant subspace in this paper the following results are presented i if dim m rank b n then m is stable ii if m im b 0 under some additional hypotheses necessary and sufficient conditions for the stability of m are given iii if m contains the controllability subspace of the pair a b sufficient conditions are given in terms of the brunovsky r numbers and the weyr characteristic of the eigenvalues of a quotient endomorphism associated to the noncontrollable part of the subspace m

stability of controlled invariant subspaces abstract let a b c n n c n m and m be an a b invariant subspace let c a b be the controllability subspace of the pair a b if m c a b m im b under additional hypotheses necessary and sufficient conditions for the stability of the subspace m are given

modules with finite cousin cohomologies have uniform local cohomological annihilators let a be a noetherian ring it is shown that any finite a module m of finite krull dimension with finite cousin complex cohomologies has a uniform local cohomological annihilator the converse is also true for a finite module m satisfying s2 which is over a local ring with cohen macaulay formal fibers

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

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

test ideals in local rings it is shown that certain aspects of the theory of tight closure are well behaved under localization let j be the parameter test ideal for r a complete local cohen macaulay ring of positive prime characteristic for any multiplicative system u c r it is shown that ju 1r is the parameter test ideal for u 1 r this is proved by proving more general localization results for the here introduced classes of f ideals of r and f submodules of the canonical module of r which are annihilators of r modules with an action of frobenius it also follows that the parameter test ideal cannot be contained in any parameter ideal of r tight closure has produced surprising new results for instance that the absolute integral closure of a complete local domain r of prime characteristic is a cohen macaulay algebra for r as well as tremendously simple proofs for otherwise difficult theorems such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is cohen macaulay the definition of tight closure is recalled in section 1 but we refer the reader to the papers of hochster and huneke listed in the bibliography for more about its applications while primarily a prime characteristic notion tight closure offers insight into arbitrary commutative rings containing q by fairly standard reduction to characteristic p techniques despite its successes the tight closure operation which in its principal setting is a closure operation performed on ideals in a commutative noetherian ring of prime characteristic remains poorly understood for example tight closure is not known to behave well under localization given an ideal i of a ring r letting i denote the tight closure of i it is not known whether or not

absolute integral closure in positive characteristic let r be a local noetherian domain of positive characteristic a theorem of hochster and huneke m hochster c huneke infinite integral extensions and big cohen macaulay algebras ann of math 135 1992 53 89 states that if r is excellent then the absolute integral closure of r is a big cohen macaulay algebra we prove that if r is the homomorphic image of a gorenstein local ring then all the local cohomology below the dimension of such a ring maps to zero in a finite extension of the ring as a result there follow an extension of the original result of hochster and huneke to the case in which r is a homomorphic image of a gorenstein local ring and a considerably simpler proof of this result in the cases where the assumptions overlap e g for complete noetherian local domains

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

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

policy iteration for customer average performance optimization of closed queueing systems we consider the optimization of queueing systems with service rates depending on system states the optimization criterion is the long run customer average performance which is an important performance metric different from the traditional time average performance we first establish with perturbation analysis a difference equation of the customer average performance in closed networks with exponentially distributed service times and state dependent service rates then we propose a policy iteration optimization algorithm based on this difference equation this algorithm can be implemented on line with a single sample path and does not require knowing the routing probabilities of queueing systems finally we give numerical experiments which demonstrate the efficiency of our algorithm this paper gives a new direction to efficiently optimize the customer centric performance in queueing systems

extension of weakly and strongly f regular rings by flat maps this is a preprint of an article published in the journal of algebra vol 241 2001 no 2 799 807

uniform annihilators of local cohomology of excellent rings abstract the main purpose of this paper is to give an affirmative answer to a conjecture of huneke c huneke uniform bounds in noetherian rings invent math 107 1992 203 223 conjecture 2 13

max min optimality of service rate control in closed queueing networks in this technical note we discuss the optimality properties of service rate control in closed jackson networks we prove that when the cost function is linear to a particular service rate the system performance is monotonic w r t with respect to that service rate and the optimal value of that service rate can be either maximum or minimum we call it max min optimality when the second order derivative of the cost function w r t a particular service rate is always positive negative which makes the cost function strictly convex concave the optimal value of such service rate for the performance maximization minimization problem can be either maximum or minimum to the best of our knowledge this is the most general result for the optimality of service rates in closed jackson networks and all the previous works only involve the first conclusion moreover our result is also valid for both the state dependent and load dependent service rates under both the time average and customer average performance criteria

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

non noetherian cohen macaulay rings in this paper we investigate a property for commutative rings with identity which is possessed by every coherent regular ring and is equivalent to cohen macaulay for noetherian rings we study the behavior of this property in the context of ring extensions of various types and rings of invariants

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

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

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

dimension and singularity theory for local rings of finite embedding dimension abstract in this paper we develop an algebraic theory for local rings of finite embedding dimension several extensions of krull dimension are proposed which are then used to generalize singularity notions from commutative algebra finally variants of the homological theorems are shown to hold in equal characteristic this theory is then applied to noetherian local rings in order to get i over a cohen macaulay local ring uniform bounds on the betti numbers of a cohen macaulay module in terms of dimension and multiplicity and similar bounds for the bass numbers of a finitely generated module ii a characterization for being respectively analytically unramified analytically irreducible unmixed quasi unmixed normal cohen macaulay pseudo rational or weakly f regular in terms of certain uniform arithmetic behavior iii in mixed characteristic the improved new intersection theorem when the residual characteristic or ramification index is large with respect to dimension and some other numerical invariants

representation theoretic properties of balanced big cohen macaulay modules let r mathfrak m k be a complete cohen macaulay local ring in this paper we assign a numerical invariant for any balanced big cohen macaulay module called underline mathsf h length among other results it is proved that for a given balanced big cohen macaulay r module m with an mathfrak m primary cohomological annihilator if there is a bound on the underline mathsf h length of all modules appearing in mathsf cm support of m then it is fully decomposable i e it is a direct sum of finitely generated modules while the first brauer thrall conjecture fails in general by a counterexample of dieterich dealing with multiplicities to measure the size of maximal cohen macaulay modules our formalism establishes the validity of the conjecture for complete cohen macaulay local rings in addition the pure semisimplicity of a subcategory of balanced big cohen macaulay modules is settled namely it is shown that r is of finite mathsf cm type if and only if r is an isolated singularity and the category of all fully decomposable balanced big cohen macaulay modules is closed under kernels of epimorphisms finally we examine the mentioned results in the context of cohen macaulay artin algebras admitting a dualizing bimodule omega as defined by auslander and reiten it will turn out that omega gorenstein projective modules with bounded mathsf cm support are fully decomposable in particular a cohen macaulay algebra lambda is of finite mathsf cm type if and only if every omega gorenstein projective module is of finite mathsf cm type which generalizes a result of chen for gorenstein algebras our main tool in the proof of results is gabriel roiter co measure an invariant assigned to modules of finite length and defined by gabriel and ringel this in fact provides an application of the gabriel roiter co measure in the category of maximal cohen macaulay modules

closure operations that induce big cohen macaulay modules and classification of singularities abstract geoffrey dietz introduced a set of axioms for a closure operation on a complete local domain r so that the existence of such a closure operation is equivalent to the existence of a big cohen macaulay module these closure operations are called dietz closures in complete rings of characteristic p 0 tight closure and plus closure satisfy the axioms we define module closures and discuss their properties for many of these properties there is a smallest closure operation satisfying the property in particular we discuss properties of big cohen macaulay module closures and prove that every dietz closure is contained in a big cohen macaulay module closure using this result we show that under mild conditions a ring r is regular if and only if all dietz closures on r are trivial finally we show that solid closure in equal characteristic 0 integral closure and regular closure are not dietz closures and that all dietz closures are contained in liftable integral closure

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

an algebraic proof of zak s inequality for the dimension of the gauss image a classical theorem in complex algebraic geometry states that for any smooth projective variety the gauss map is finite in particular a smooth variety and its gauss image have the same dimension with the obvious exception of a linear space furthermore even when the variety is not smooth zak proved a lower bound on the dimension of its gauss image in terms of the dimension of its singular locus our purpose in this note is to reinterpret gauss maps within a more general algebraic framework and thus recover zak s bound on the dimension of the gauss image as a special case of an interesting new bound on the analytic spread of a module of kahler differentials this connects that classical subject in complex geometry to recent research in commutative algebra concerning integral closures of modules in particular we give a new purely algebraic proof of zak s theorem we recall a precise version of zak s theorem let x be an irreducible projective variety of dimension d defined over an algebraically closed field k considered with a fixed embedding x p v for some finite dimensional k vector space v the gauss map is the rational map from x to the grassmannian of projective d planes in p v assigning to each smooth k point of x the projective tangent plane there

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

prime ideals in ultraproducts of commutative rings abstract we describe classes of prime ideals in ultraproducts of commutative rings we consider in particular prime ideals in ultraproducts of noetherian rings krull domains finite character rings and qr domains

decentralized control of service rates in a closed jackson network the authors consider a closed jackson network with m nodes the service rate at each node is controllable in a decentralized manner i e it will be controlled based on local information extracted from that node only for each node there is a holding cost and an operating cost it is assumed that both costs are time homogeneous and that the operating cost is a linear function of the service rate however both costs are arbitrary functions of the number of jobs at the node the objective is to minimize the time average expected total cost it is shown that there exists an optimal control characterized by a set of thresholds one for each node such that it is optimal for each node to serve at zero rates if the number of jobs there is below or at the threshold and serve at maximum allowed rates when the number of jobs exceeds the threshold this structure of the optimal control is identified by using the product form solution of the jackson network and the duality theory of linear programming

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

on the vanishing of tor of the absolute integral closure let r be an excellent local domain of positive characteristic with residue field k and let r be its absolute integral closure if tor r r k vanishes then r is weakly f regular if r has at most an isolated singularity or has dimension at most two then r is regular

ideals in a perfect closure linear growth of primary decompositions and tight closure this paper is concerned with tight closure in a commutative noetherian ring r of prime characteristic p and is motivated by an argument of k e smith and i swanson that shows that if the sequence of frobenius powers of a proper ideal a of r has linear growth of primary decompositions then tight closure of a commutes with localization at the powers of a single element it is shown in this paper that provided r has a weak test element linear growth of primary decompositions for other sequences of ideals of r that approximate in a certain sense the sequence of frobenius powers of a would not only be just as good in this context but in the presence of a certain additional finiteness property would actually imply that tight closure of a commutes with localization at an arbitrary multiplicatively closed subset of r work of m katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the frobenius powers of a has only finitely many maximal members this paper develops through a careful analysis of the ideal theory of the perfect closure of r strategies for showing that tight closure of a specified ideal a of r commutes with localization at an arbitrary multiplicatively closed subset of r and for showing that the union of the associated primes of the tight closures of the frobenius powers of a is actually a finite set several applications of the strategies are presented in most of them it was already known that tight closure commutes with localization but the resulting affirmative answers to katzman s question in the various situations considered are believed to be new

on the non existence of small cohen macaulay algebras abstract by finding a p adic obstruction we construct many examples of complete noetherian local normal f p algebras r such that no module finite extension r s is cohen macaulay these examples should be contrasted with a result of hochster huneke the directed union of all such extensions is always cohen macaulay

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

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

descent via koszul extensions abstract let r be a commutative noetherian local ring with completion r we apply differential graded dg algebra techniques to study descent of modules and complexes from r to r where r is either the henselization of r or a pointed etale neighborhood of r we extend a given r complex to a dg module over a koszul complex we describe this dg module equationally and apply artin approximation to descend it to r this descent result for koszul extensions has several applications when r is excellent we use it to descend the dualizing complex from r to a pointed etale neighborhood of r this yields a new version of p roberts theorem on uniform annihilation of homology modules of perfect complexes as another application we prove that the auslander condition on uniform vanishing of cohomology ascends to r when r is excellent henselian and cohen macaulay

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

f singularities via alterations we give characterizations of test ideals and f rational singularities via regular alterations formally the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions lastly we establish nadel type vanishing theorems up to finite maps for test ideals and further demonstrate how these vanishing theorems may be used to extend sections

integral perfectoid big cohen macaulay algebras via andre s theorem the main result of this article is to prove that any noetherian local domain of mixed characteristic maps to an integral perfectoid big cohen macaulay algebra the proof of this result is based on the construction of almost cohen macaulay algebras in mixed characteristic due to yves andre moreover we prove that the absolute integral closure of a complete noetherian local domain of mixed characteristic maps to an integral perfectoid big cohen macaulay algebra

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

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

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

linear equivalence of ideal topologies it is proved that whenever p is a prime ideal in a commutative noetherian ring such that the p adic and thep symbolic topologies are equivalent then the two topologies are equivalent linearly several explicit examples are calculated in particular for all prime ideals corresponding to non torsion points on nonsingular elliptic cubic curves

lipschitz stability of controlled invariant subspaces abstract let a b c n n c n m and m be an a b invariant subspace in this paper the following results are presented i if m im b 0 necessary and sufficient conditions for the lipschitz stability of m are given ii if m contains the controllability subspace of the pair a b sufficient conditions for the lipschitz stability of the subspace m are given