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

degreep extensions of an unramified regular local ring of mixed characteristicp abstract we investigate the case of degreep extensions and certain degreep2 extensions of an unramified regular local ringr of mixed characteristicp and show thatr is a direct summand as anr module of those extensions some new examples of normal non cohen macaulay rings are given and a question related to the direct summand conjecture is discussed

collective phase description of coupled oscillators with general network structure we develop a collective phase description for a population of nonidentical limit cycle oscillators with any network structure undergoing fully phase locked collective oscillations the whole network dynamics can be described by a single collective phase variable we derive a general formula for the collective phase sensitivity which quantifies the phase response of the whole network to weak external perturbations applied to the constituent oscillators moreover we consider weakly interacting multiple networks and develop an effective phase coupling description for them several examples are given to illustrate our theory

tight closure and projective bundles abstract we formulate problems of tight closure theory in terms of projective bundles and subbundles this provides a geometric interpretation of such problems and allows us to apply intersection theory to them this yields new results concerning the tight closure of a primary ideal in a two dimensional graded domain

a new case of integrability of the general riccati equation and its application to relaxation problems a new case of integrability by quadratures of the general riccati equation is discussed some possibilities of its use in molecular physics are demonstrated by the example of vibrational translational or rotational translational relaxation of a small amount of diatomic molecules in a structureless gas

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

open sets with stein hypersurface sections in stein spaces let d c cn n 3 be an open set such that for any linear hyperplane h c cn the intersection h n d is stein according to a theorem of lelong 13 it then follows that d is itself stein using the solution of the levi problem by docquier grauert 7 this result can easily be generalized to open subsets in stein manifolds therefore it is natural to raise the following question see also 4

isostables isochrons and koopman spectrum for the action angle representation of stable fixed point dynamics abstract for asymptotically periodic systems a powerful phase reduction of the dynamics is obtained by computing the so called isochrons i e the sets of points that converge toward the same trajectory on the limit cycle motivated by the analysis of excitable systems a similar reduction has been attempted for non periodic systems admitting a stable fixed point in this case the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories instead the sets of interest that we call isostables are defined in the literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point however it turns out that this definition of the isostables holds only for systems with slow fast dynamics also efficient methods for computing the isostables are missing the present paper provides a general framework for the definition and the computation of the isostables of stable fixed points which is based on the spectral properties of the so called koopman operator more precisely the isostables are defined as the level sets of a particular eigenfunction of the koopman operator through this approach the isostables are unique and well defined objects related to the asymptotic properties of the system also the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action angle coordinates for the dynamics in addition an efficient algorithm for computing the isostables is obtained which relies on the evaluation of laplace averages along the trajectories the method is illustrated with the excitable fitzhugh nagumo model and with the lorenz model finally we discuss how these methods based on the koopman operator framework relate to the global linearization of the system and to the derivation of special lyapunov functions

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

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 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

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

phase resetting of collective rhythm in ensembles of oscillators phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations we consider globally coupled ensembles of sakaguchi kuramoto oscillators and use the ott antonsen theory of ensemble evolution to derive the analytical phase resetting equations we show the final phase reset value to be composed of two parts an immediate phase reset directly caused by the perturbation and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium analytical semianalytical and numerical approximations of the final phase resetting curve are constructed we support our findings with extensive numerical evidence involving identical and nonidentical oscillators the validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit

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

pure subrings of regular local rings endomorphism rings and frobenius morphisms abstract the aim of this paper is threefold first to prove that the endomorphism ring associated to a pure subring of a regular local ring is a noncommutative crepant resolution if it is maximal cohen macaulay second to see that in that situation a different but morita equivalent noncommutative crepant resolution can be constructed by using frobenius morphisms finally to study the relation between frobenius morphisms of noncommutative rings and the finiteness of global dimension as a byproduct we will obtain a result on wild quotient singularities if the smooth cover of a wild quotient singularity is unramified in codimension one then the singularity is not strongly f regular

purity of branch and critical locus abstract to a dominant morphism x s y s of n therian integral s schemes one has the inclusion c x y b x y of the critical locus in the branch locus of x y starting from the notion of locally complete intersection morphisms we give conditions on the modules of relative differentials x y x s and y s that imply bounds on the codimensions of c x y and b x y these bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by zariski nagata auslander and faltings and grothendieck and van der waerden s purity for birational morphisms

homology over local homomorphisms the notions of betti numbers and of bass numbers of a finite module n over a local ring r are extended to modules that are only assumed to be finite over s for some local homomorphism r s various techniques are developed to study the new invariants and to establish their basic properties in some cases they are computed in closed form applications go in several directions one is to identify new classes of finite r modules whose classical betti numbers or bass numbers have extremal growth another is to transfer ring theoretical properties between r and s in situations where s may have infinite flat dimension over r a third is to obtain criteria for a ring equipped with a contracting endomorphism such as the frobenius endomorphism to be regular or complete intersection these results represent broad generalizations of kunz s characterization of regularity in prime characteristic

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

condition for alternans and stability of the 1 1 response pattern in a memory model of paced cardiac dynamics we analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration thereby containing some degree of memory the model displays rate dependent restitution so that the dynamic and s1 s2 restitution curves are different a manifestation of memory in the model we derive a criterion for the stability of the 1 1 response pattern displayed by this model it is found that the stability criterion depends on the slope of both the dynamic and s1 s2 restitution curves and that the pattern can be stable even when the individual slopes are greater or less than one we discuss the relation between the stability criterion and the slope of the constant bcl restitution curve the criterion can also be used to determine the bifurcation from the 1 1 response pattern to alternans we demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model we illustrate our results by considering a specific map recently derived from a three current membrane model and find that the stability of the 1 1 pattern is accurately described by our criterion in addition a numerical experiment is performed using the three current model to illustrate the application of the pacing protocol and the evaluation of the criterion

cohen macaulay properties of ring homomorphisms abstract numerical invariants which measure the cohen macaulay character of homomorphisms r s of noetherian rings are introduced and studied comprehensive results are obtained for homomorphisms which are locally of finite flat dimension they provide a point of view from which a variety of phenomena receive a unified treatment the conceptual clarification and technical versatility of this approach leads among other things to a determination of those homomorphisms which preserve the cohen macaulay character of the rings to the discovery of new classes of homomorphisms with remarkable stability properties and to solutions of some problems on flat homomorphisms raised by grothendieck

computation of the phase response curve a direct numerical approach neurons are often modeled by dynamical systems parameterized systems of differential equations a typical behavioral pattern of neurons is periodic spiking this corresponds to the presence of stable limit cycles in the dynamical systems model the phase resetting and phase response curves prcs describe the reaction of the spiking neuron to an input pulse at each point of the cycle we develop a new method for computing these curves as a by product of the solution of the boundary value problem for the stable limit cycle the method is mathematically equivalent to the adjoint method but our implementation is computationally much faster and more robust than any existing method in fact it can compute prcs even where the limit cycle can hardly be found by time integration for example because it is close to another stable limit cycle in addition we obtain the discretized phase response curve in a form that is ideally suited for most applications we present several examples and provide the implementation in a freely available matlab code

quasi lie families schemes invariants and their applications to abel equations abstract this work analyses types of group actions on families of t dependent vector fields of a particular class the hereby called quasi lie families we devise methods to obtain the defined here quasi lie invariants namely a kind of functions constant along the orbits of the above mentioned actions our techniques lead to a deep geometrical understanding of quasi lie schemes and quasi lie systems giving rise to several new results our achievements are illustrated by studying abel and riccati equations we retrieve the liouville invariant and study other new quasi lie invariants of abel equations several abel equations with a superposition rule are described and we characterise abel equations via quasi lie schemes

determining individual phase response curves from aggregate population data phase reduction is an invaluable technique for investigating the dynamics of nonlinear limit cycle oscillators central to the implementation of phase reduction is the ability to calculate phase response curves prcs which describe an oscillator s response to an external perturbation current experimental techniques for inferring prcs require data from individual oscillators which can be impractical to obtain when the oscillator is part of a much larger population here we present a simple methodology to calculate prcs of individual oscillators using an aggregate signal from a large homogeneous population this methodology is shown to be accurate in the presence of interoscillator coupling and noise and can also provide a good estimate of an average prc of a heterogeneous population we also find that standard experimental techniques for prc measurement can produce misleading results when applied to aggregate population data

regularity over homomorphisms and a frobenius characterization of koszul algebras abstract let r be a standard graded algebra over an f finite field of characteristic p 0 let r r be the frobenius endomorphism for each finitely generated graded r module m let m be the abelian group m with the r module structure induced by the frobenius endomorphism the r module m has a natural grading given by deg x j if x m j p i for some 0 i p 1 in this paper we prove that r is koszul if and only if there exists a non zero finitely generated graded r module m such that reg r m this result supplies another instance for the ability of the frobenius in detecting homological properties as exemplified by kunz s famous regularity criterion the main technical tool is the notion of castelnuovo mumford regularity over certain homomorphisms between n graded algebras the latter notion is a common generalization of the relative and absolute castelnuovo mumford regularity of modules

extending phase reduction to excitable media theory and applications phase reduction methods have been tremendously useful for understanding the dynamics of nonlinear oscillators but have been difficult to extend to systems with a stable fixed point such as an excitable system using the notion of isostables which measure the time it takes for a given initial condition in phase space to approach a stable fixed point we present a general method for isostable reduction for excitable systems we also devise an adjoint method for calculating infinitesimal isostable response curves which are analogous to infinitesimal phase response curves for oscillatory systems through isostable reduction we are able to implement sophisticated control strategies in a high dimensional model of cardiac activity for the termination of alternans a precursor to cardiac fibrillation

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

new invariants of noetherian local rings abstract we investigate properties of certain invariants of noetherian local rings including their behavior under flat local homomorphisms we show that these invariants are bounded by the multiplicity for cohen macaulay local rings with infinite residue fields and they all agree with the multiplicity when such rings are hypersurfaces we also show that these invariants are all equal to 2 for a non regular cohen macaulay local ring a if and only if a has a minimal multiplicity provided its residue field is infinite

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

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

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 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

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

grobner deformations connectedness and cohomological dimension abstract in this paper we will compare the connectivity dimension c p i of an ideal i in a polynomial ring p with that of any initial ideal of i generalizing a theorem of kalkbrener and sturmfels m kalkbrener b sturmfels initial complex of prime ideals adv math 116 1995 365 376 we prove that c p lt i min c p i dim p i 1 for each monomial order as a corollary we have that every initial complex of a cohen macaulay ideal is strongly connected our approach is based on the study of the cohomological dimension of an ideal a in a noetherian ring r and its relation with the connectivity dimension of r a in particular we prove a generalized version of a theorem of grothendieck a grothendieck cohomologie locale des faisceaux coherents et theoremes de lefschetz locaux et globaux sga 2 in seminaire de geometrie algebrique du bois marie 1962 as consequence of these results we obtain some necessary conditions for an open subscheme of a projective scheme to be affine

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

special symmetries to standard riccati equations and applications by using symmetries associated to riccati equation in standard form sre we obtain a family which can be integrated by quadratures as a consequence we get a new integrability condition for the generalized riccati equation gre we illustrate the result with some examples and we give some applications in the solitons theory

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

a note on the monomial conjecture several cases of the monomial conjecture are proved an equivalent form of the direct summand conjecture is discussed let a m k be a noetherian local ring of dimension n m its maximal ideal and k a rn the monomial conjecture henceforth mc of hochster asserts that given any system of parameters henceforth s o p xi xn of a xix2 xn xv xn bt o hochster proved the conjecture in the equicharacteristic case hi h2 he also established the fact that in the mixed and the positive characteristic cases the direct summand conjecture henceforth dsc and hence mc is equivalent to the canonical element conjecture h12 henceforth cec thus mc occupies a central position in the study of several homological conjectures in i1l hochster pointed out that given any s o p xl xn of a xt xt satisfies mc for t sufficiently large next goto g proved mc for buchsbaum rings and koh proved dsc for degree p extensions k several special cases of cec were proved in di when deptha dima 1 in d2 the following result was established if ji annh j a and j jij2 jr where r dima deptha 0 then xl xn satisfies mc if j xl xn this in turn implies that given any s o p xl xn in a complete local normal domain a xi x2 x3 xt satisfies mc for t 0 we will have several more applications of this result in section 2 we recall that there is no loss of generality in assuming a to be a complete local normal domain in our most recent work we established the validity of cec over i a xa when a is a complete local normal domain and x e mj1 d3 ii any almost complete intersection domain a or any almost complete intersection ring a over which p the mixed characteristic is a non zero divisor d4 iii rings of the form r q where r is a complete gorenstein ring such that the complete local normal domain a is a homomorphic image of r dimr dima and q is the canonical module of a d4 and iv complete local normal domains a for which q is s3 thus mc holds in all the above cases the main results of this paper are arranged in the following way in section 1 first we reduce the study of mc to almost complete intersection rings recall that due to the results mentioned earlier the problem boils down to almost complete intersection rings of depth 0 next we prove the following received by the editors july 20 1996 1991 mathematics subject classification primary 13d02 secondary 13h1 0 this work was partially supported by an nsa grant and an nsf grant 1998 american mathematical society