the approximate tubular neighborhood theorem skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations under dimension and compactness assumptions in general the stratified approximate fibrations cannot be replaced by bundles and the teardrops cannot be replaced by mapping cylinder neighborhoods thus this is the best possible topological tubular neighborhood theorem in the stratified setting

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

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

ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

eigenvalue estimates for the dirac operator on quaternionic kahler manifolds we consider the dirac operator on compact quaternionic kahler manifolds and prove a lower bound for the spectrum this estimate is sharp since it is the first eigenvalue of the dirac operator on the quaternionic projective space

embeddings of open manifolds abstiract let top m be the simplicial group of homeo morphisms of m the following theorems are proved theorem a let m 1 be a topological manifold of dim 5 with a finite number of tame ends ei 1 r be manifold approximate fibrations then there exists a map f top p m f2i top p wi such that the homotopy fiber of f is equivalent to topcs m the simplicial group of homeomorphisms of m which have compact support theorem b let m be a compact topological manifold of dim 5 with connected boundary m and denote the interior of m by int m let f top m top int m be the restriction map and let g be the homotopy fiber of f over id nt v then ri g is isomorphic to 7ri c m for i 0 where c m is the concordance space of om theorem c let q w r be a manifold approximate fibration with dim w 5 then there exist maps ca iri top1p w 1 ri top w and 3 iri top w iri top p w for i 1 such that 3 o cx id where w is a compact and connected manifold and w is the infinite cyclic cover of w

simply connected 4 manifolds with a given boundary let m be a closed oriented connected 3 manifold for each bilinear symmetric pairing zn l our goal is to calculate the set vl m of all oriented homeomorphism types of compact 1 connected oriented 4manifolds with boundary m and intersection pairing isomorphic to zn l for each pair zn l which presents h m we construct a double coset space bl m and a function ct vl m bl m the set bt m is the quotient of the group of all link pairing preserving isomorphisms of the torsion subgroup of h1 m by two naturally occuring subgroups when zn l is an even pairing we construct another double coset space bl m a function 6l vl m b fl m and a projection p2 bl m bl m such that p2 c l ct our main result states that when zn l is even the function c l is injective as is the function c x a vl m bl m x z 2 when z l is odd here a is a kirby siebenmann obstruction to smoothing it follows that the sets vl m are finite and of an order bounded above by a constant depending only on h1 m we also show that when hi m q 0 and z l is even ct p2 c 6l is injective it seems likely that via the functions cl x a and 6l the sets bl m x z 2 and bl m calculate vl m when zn l is respectively odd and even we verify this in several cases most notably when h1 m is free abelian the results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two 1connected 4 manifolds extending a given homeomorphism of their boundaries the theory developed is then applied to show that there is an m 0 depending only on h1 m such that for any self homeomorphism f of m fm extends to a self homeomorphism of any 1 connected compact 4 manifold with boundary m introduction in m freedman s fundamental paper fr he classified closed simply connected oriented 4 manifolds up to orientation preserving homeomorphism he showed that modulo the kirby siebenmann invariant these manifolds are in bijective correspondence through their intersection pairings with the set of unimodular bilinear symmetric pairings over z see theorem 1 5 of fr and corollary 2 2 3 of q in this paper we begin a classification of simply connected 4 manifolds with connected boundary our results with regard to this problem are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between 1 connected 4 manifolds extending a given homeomorphism of their boundaries received by the editors september 19 1985 1980 mathematics subject classification 1985 revision primary 57n15 research supported by an nserc canada post doctoral fellowship 1986 american mathematical society 0002 9947 86 1 00 25 per page

stratified path spaces and fibrations the main objects of study are the homotopically stratified metric spaces introduced by quinn closed unions of strata are shown to be stratified forward tame stratified fibrations between spaces with stratifications are introduced paths which lie in a single stratum except possibly at their initial points form a space with a natural stratification and the evaluation map from that space of paths is shown to be a stratified fibration applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers

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

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 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

a conformal lower bound for the smallest eigenvalue of the dirac operator and killing spinors on a riemannian spin manifold we give a lower bound for the square of the eigenvalues of the dirac operator by the smallest eigenvalue of the conformal laplacian the yamabe operator we prove in the limiting case that the eigenspinor field is a killing spinor i e parallel with respect to a natural connection in particular if the scalar curvature is positive the eigenspinor field is annihilated by harmonic forms and the metric is einstein

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

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

kahlerian twistor spinors on a kahler spin manifold kahlerian twistor spinors are a natural analogue of twistor spinors on riemannian spin manifolds they are defined as sections in the kernel of a first order differential operator adapted to the kahler structure called kahlerian twistor penrose operator we study kahlerian twistor spinors and give a complete description of compact kahler manifolds of constant scalar curvature admitting such spinors as in the riemannian case the existence of kahlerian twistor spinors is related to the lower bound of the spectrum of the dirac operator

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

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

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

eingenvalues of the dirac operator on compact kahler manifolds kahlerian twistor operators are introduced to get lower bounds for the eigenvalues of the dirac operator on compact spin kahler manifolds in odd complex dimensions manifolds with the smallest eigenvalues are characterized by an over determined system of differential equations similar to the riemannian case in these dimensions we show the existence of a unique natural kahlerian twistor operator it is also proved that on a kahler manifold with nonzero scalar curvature the space of riemannian twistor spinors is trivial

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

spin c geometry of kahler manifolds and the hodge laplacian on minimal lagrangian submanifolds from the existence of parallel spinor fields on calabi yau hyper kahler or complex flat manifolds we deduce the existence of harmonic differential forms of different degrees on their minimal lagrangian submanifolds in particular when the submanifolds are compact we obtain sharp estimates on their betti numbers which generalize those obtained by smoczyk in 49 when the ambient manifold is kahler einstein with positive scalar curvature and especially if it is a complex contact manifold or the complex projective space we prove the existence of kahlerian killing spinor fields for some particular spin c structures using these fields we construct eigenforms for the hodge laplacian on certain minimal lagrangian submanifolds and give some estimates for their spectra these results also generalize some theorems by smoczyk in 50 finally applications on the morse index of minimal lagrangian submanifolds are obtained

bounded homotopy equivalences of hilbert cube manifolds let m and f be hubert cube manifolds with f compact the purpose of this paper is to study homotopy equivalences m r x f which have bounded control in the redirection roughly these homotopy equivalences form a semi sim plicial complex 4 rm x f the controlled whitehead space using results about approximate fibrations rm x f is related to the semi simplicial complex of bounded concordances on r x f then the homotopy groups of rm x f are computed in terms of the lower algebraic at theoretic functors k i 1 introduction let f be a compact hilbert cube manifold we are interested in homotopy equivalences m rm x f which are controlled in the redirection where m is also a hilbert cube manifold to say is controlled in the redirection means that pf m rm is an approximate fibration where p rm x f rm is projection the collection of all such homotopy equivalences which are additionally given to be retractions onto the collared submanifold rm x f of m form the vertices of a semi simplicial complex ta rm x f see 2 for the precise definition the main result of this paper is the computation of the homotopy groups of wa rm x f see corollary 1 below in order to do this we relate wa rm x f to the semi simplicial complex h rm x f of bounded concordances on rm x f an simplex of eb rm x f is a homeomorphism

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

the first eigenvalue of the dirac operator on quaternionic kahler manifolds in 15 we proved a lower bound for the spectrum of the dirac operator on quaternionic kahler manifolds in the present article we show that the only manifolds in the limit case i e the only manifolds where the lower bound is attained as an eigenvalue are the quaternionic projective spaces we use the equivalent formulation in terms of the quaternionic killing equation introduced in 16 and show that a nontrivial solution defines a parallel spinor on the associated hyperkahler manifold

rigidity theorems for right angled reflection groups let t be a right angled reflection group let m and m be coxeter manifolds then any t map f m fm is t homotopic to a homeo morphism introduction this paper contains the results given in the author s ph d the sis davis in 9 showed that in every dimension 4 there exists a cocompact reflection group on a contractible manifold not homeomorphic to an euclidean space also he gave the first example of a closed aspherical manifold not covered by euclidean space davis construction provides an infinite coxeter group acting locally smoothly effectively and properly discontinuously on a contractible manifold with compact quotient

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

la premiere valeur propre de l operateur de dirac sur les varietes kahleriennes compactes k d kirchberg ki1 gave a lower bound for the first eigenvalue of the dirac operator on a spin compact kahler manifoldm of odd complex dimension with positive scalar curvature we prove that manifolds of real dimension 8l 6 satisfying the limiting case are twistor space cf sa of quaternionic kahler manifold with positive scalar curvature and that the only manifold of real dimension 8l 2 satisfying the limiting case is the complex projective spacecp4l 1

compact six dimensional kahler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the dirac operator si m 6 est une variete de spin de kahler fermee de courbure scalaire positive r pour laquelle r 0 3 1 2 r 0 min r est une valeur propre de l operateur de dirac alors m 6 est analytiquement isometrique soit a l espace projectif p 3 c soit a la variete drapeau f c 3 par rapport a leurs structures de kahler naturelles

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

approximately fibering a manifold over an aspherical one the paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers i e is homotopic to manifold approximate fibration we define obstructions in algebraic k theory their vanishing is necessary and under certain conditions sufficient basic ingredients are quinn s thin h cobordism theorem and end theorem and knowledge about the farrell jones conjectures in algebraic k and l theory and the maf rigidity conjecture by hughes taylor williams

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4