generating a residual intersection let i be an ideal of height g 0 in a commutative noetherian ring r and let 2i be a proper sub ideal of i generated by 3 g elements if the ideal j 21 i has height at least s then j is called an s residual intersection of i see artin and nagata 1 huneke 19 or huneke and ulrich 22 the case s g corresponds to the theory of linkage and is well understood at least from the homolo cal point of view the cornerstone of this theory is that if i is perfect then a free resolution of r j can be produced from a free resolution of r z and the koszul complex on a set of generators of cu very little is known about resolutions of s residual intersections for s g a few special cases have been resolved s 10 301 and the end of the resolution or in other words a set of generators for the canonical module of r j has been described 22 however in general even the generators of j remain unknown here we show that j can be approximated by a sum of links of i if s g 1 and we give conditions under which this approximation in fact gives a full set of generators of j since these generators arise from comparison maps from koszul complexes to known resolutions at least assuming that the resolution of r i is in some sense known it is feasible to count them compute their degrees in the graded case or even to give explicit formulas for them after stating our main results theorem 2 15 and 2 16 and corollary 2 18 precisely in section 2 we lay out the argument in sections 3 4 and 5 in section 6 we are concerned with the question of when a g t 1 residual intersection j is

tight double linkage of gorenstein algebras abstract a gorenstein ideal k in a local ring r is in the class if there is a sequence of linked ideals i 0 i 1 i 2 n k with i 0 a complete intersection and i 2 i gorenstein for all i if k is in then k is linked to a complete intersection by a sequence of tight double links this means there is a sequence i 0 j 0 j 1 j 2m k such that for each p j 2 p 1 is an almost complete intersection of the form b y w and its immediate neighbors are linked to it by the similar regular sequences b y and b w respectively in this sense j 2 p and j 2 p 2 are tightly linked gorenstein ideals let k be an algebraically closed field and a k x 1 x q k with k in the k algebra a is rigid if and only if it is linked to a regular complete intersection by a sequence of semi generic tight double links it follows that if a is rigid then it is regular in codimension six

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

algebra structures on resolutions of rings defined by grade four almost complete intersections abstract let r m k be a local ring in which 2 is a unit and let j be an indeal in r the minimal r free resolution of r j is called a dg algebra if it has the structure of an associative differential graded commutative algebra over r if j is a grade four almost complete intersection ideal in r then j is necessarily linked to a grade four gorenstein ideal i by a grade four complete intersection k it is well known that the minimal r free resolutions of r i and r k admit dg algebra structures in this paper we show that if w denotes the image of the map torr1 r k k torr1 r i k and w2 0 for example if k m i then the minimal r free resolution of s r j has a dg algebra structure in the case where the ring r is regular local and k m i we prove that the poincare series pns t of every finitely generated s module n is rational and the ring s satisfies the eisenbud conjecture

evolutions symbolic squares and fitting ideals given a reduced local algebra t over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections r t which induce isomorphisms r k t t k such maps called evolutions arise naturally in the study of hecke algebras as they implicitly do in the recent work of wiles taylor wiles and flach we show that the existence of non trivial evolutions of an algebra t can be characterized in terms of the symbolic square of an ideal defining t we give a characterization of the symbolic square in terms of fitting ideals using this and other techniques we show that certain classes of reduced algebras codimension 2 cohen macaulay codimension 3 gorenstein licci algebras in general and some others admit no nontrivial evolutions on the other hand we give examples showing that non trivial evolutions of reduced cohenmacaulay algebras of codimension 3 do exist in every positive characteristic

perfect linkage of cohen macaulay modules over cohen macaulay rings abstract in this paper we introduce and study the notion of linkage by perfect modules which we call perfect linkage for cohen macaulay modules over cohen macaulay local rings we explore perfect linkage in connection with syzygies maximal cohen macaulay approximations and yoshino isogawa linkage we recover a theorem of yoshino and isogawa and analyze the structure of double perfect linkage moreover we establish a criterion for two cohen macaulay modules of codimension one to be perfectly linked and apply it to the classical linkage theory for ideals we also construct various examples of linkage of modules and ideals

structure theory for a class of grade four gorenstein ideals asstracr an ideal i in a commutative noetherian ring r is a gorenstein ideal of grade g if pdr r i grade i g and the canonical module extr r i r is cyclic serre showed that if g 2 then i is a complete intersection and buchsbaum and eisenbud proved a structure theorem for the case g 3 we present generic resolutions for a class of gorenstein ideals of grade 4 and we illustrate the structure of the resolution with various specializations among these examples there are gorenstein ideals of grade 4 in k x y z v that are n generated for any odd integer n 7 we construct other examples from almost complete intersections of grade 3 and their canonical modules in the generic case the ideals are shown to be normal primes finally we conclude by giving an explicit associative algebra structure for the resolutions it is this algebra structure that we use to classify the different gorenstein ideals of grade 4 and which may be the key to a complete structure theorem

linkage and sums of ideals it is shown under mild conditions that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection for a sum of geometrically linked cohen macaulay ideals we compute the depths of the conormal module arld the first koszul homology as applications we construct general examples of ideals which are strongly cohenmacaulay strongly nonobstructed but not in the linkage class of a complete intersection and gorenstein ideals which are strongly nonobstructed but not syzygetic

heights of ideals of minors we prove new height inequalities for determinantal ideals in a regular local ring or more generally in a local ring of given embedding codimension our theorems extend and sharpen results of faltings and bruns

deformation and linkage of gorenstein algebras general double linkage of gorenstein algebras is defined rigidity genericity and regularity up to codimension six all pass across general double linkage rigid strongly unobstructed codimension four gorenstein algebras which lie in different herzog classes are produced introduction we begin by reviewing the classical structure theorems let p k xi x 41 a p i and f be the minimal resolution of a by free p modules f 1 if a has codimension 1 and i is unmixed then f is given by 0 p p for some fin p 2 hilbert burch 7 schaps 28 if a has codimension 2 and is cohen macaulay then f is given by x x 0 p n i ppnp where xis an n x n 1 matrix xi l 1l i x and a i x is the determinant formed by omitting the ith row from x 3 buchsbaum and eisenbud 4 if a has codimension 3 and its gorenstein then f is given by xt x x o p pn pn p where xis an n x n alternating matrix xi l i lpfi x and pfi x denotes the pfaffian of the alternating submatrix of x formed by omitting row and column i in these structure theorems as the codimension of a increases so does the number of conditions on a it is reasonable to suspect that a structure theorem for codimension four gorenstein algebras a will require that an additional hypothesis be imposed on a we observe that in each case a there is a structure theorem b all deformations are unobstructed see herzog 13 and c each algebra a is in the linkage class of a complete intersection see peskine and szpiro 27 and watanabe 30 historically these three facts were found independently with hindsight we now see that b follows from c and a follows in turn from b in his thesis received by the editors june 10 1982 and in revised form march 29 1983 some of the contents of this paper were presented to the special session on commutative algebra at the american mathematical society meeting in cincinnati on january 13 1982 1980 mathematics subject classification primary 13c05 13d25 13h10 secondary 14b07 14d115 14m05 16ao3 both authors were supported in part by the national science foundation 1984 american mathematical society 0002 9947 84 1 00 25 per page

classification of the tor algebras of codimension four almost complete intersections let r m k be a local ring in which 2 is a unit assume that every element of k has a square root in k we classify the algebras tor r r j k as j varies over all grade four almost complete intersection ideals in r the analogous classification has already been found when j varies over all grade four gorenstein ideals 21 and when j varies over all ideals of grade at most three 5 30 the present paper makes use of the classification in 21 of the tor algebras of codimension four gorenstein rings as well as the usually nonminimal dg algebra resolution of a codimension four almost complete intersection which is produced in 25 and 26

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

conjugacy growth of commutators abstract for the free group f r on r 1 generators respectively the free product g 1 g 2 of two nontrivial finite groups g 1 and g 2 we obtain the asymptotic for the number of conjugacy classes of commutators in f r respectively g 1 g 2 with a given word length in a fixed set of free generators respectively the set of generators given by the nontrivial elements of g 1 and g 2 our result is proven by using the classification of commutators in free groups and in free products by wicks and builds on the works of rivin and sharp who asymptotically counted the conjugacy classes of commutator subgroup elements in f r with a given word length

a riemann hilbert boundary value problem in a triangle abstract in this article a riemann hilbert boundary value problem on an isosceles orthogonal triangle is considered using explicit schwarz poisson type formulae for the triangle schwarz type and pompeiu type operators are obtained boundary behaviors of these operators are discussed in detail finally we investigate the riemann hilbert boundary value problem for both homogeneous and inhomogeneous cauchy riemann equations an explicit solvability of the problem is obtained

rees algebras of modules we study rees algebras of modules within a fairly general framework we introduce an approach through the notion of bourbaki ideals that allows the use of deformation theory one can talk about the essentially unique generic bourbaki ideal i e of a module e which in many situations allows one to reduce the nature of the rees algebra of e to that of its bourbaki ideal i e properties such as cohen macaulayness normality and being of linear type are viewed from this perspective the known numerical invariants such as the analytic spread the reduction number and the analytic deviation of an ideal and its associated algebras are considered in the case of modules corresponding notions of complete intersection almost complete intersection and equimultiple modules are examined in some detail special consideration is given to certain modules which are fairly ubiquitous because interesting vector bundles appear in this way for these modules one is able to estimate the reduction number and other invariants in terms of the buchsbaum rim multiplicity

on the cohen macaulayness of the conormal module of an ideal abstract in the present paper we investigate a question stemming from a long standing conjecture of vasconcelos given a generically complete intersection perfect ideal i in a regular local ring r when is it true that the cohen macaulayness of i i 2 or r i 2 implies that r i is gorenstein this property is known to hold for licci ideals and essentially squarefree monomial ideals we show that a positive answer actually holds for every monomial ideal we then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result as they lie in the first class of ideals not covered by it the homogeneous examples have been found thanks to the help of j c migliore as a by product we exhibit several classes of cohen macaulay ideals whose square is not cohen macaulay our methods work both in the homogeneous and in the local settings

liaison classes of modules abstract we propose a concept of module liaison that extends gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative gorenstein ring analyzing the resulting equivalence classes we show that several results known for gorenstein liaison are still true in the more general case of module liaison in particular we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules as a consequence we show that the intermediate cohomology modules and properties like being perfect cohen macaulay buchsbaum or surjective buchsbaum are preserved in even module liaison classes furthermore we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one

on the structure of free resolutions of length 3 the sources of structure theory of finite free resolutions go back to hilbert in his famous paper on invariant theory he proved that every ideal of projective dimension 1 over a polynomial ring is determinantal this theorem was then extended by burch to arbitrary commutative rings there are various motivations for extending this results one of them is a variety of applications of the hilbert burch theorem in deformation theory and algebraic geometry cf for example s another reason comes from serre s conjecture one would like to understand in terms of resolutions why the lifting property of grothendieck does not hold in 1974 buchsbaum and eisenbud in bel introduced so called structure theorems various factorisations which occur for acyclic complexes of arbitrary length then hochster in h generalized the hilbert burch theorem to arbitrary free complexes of length 2 his results were then improved in br pw still the structure of complexes of length bigger than 2 is not understood the first steps in this direction are the results of b and the construction of a generic ring for complexes of length 3 of type 1 rz 1 1 given in pw in this paper i construct a ring which i believe is the generic ring for complexes of length 3 the construction involves new structure theorems and a new idea the notion of defects of structure theorems the paper is organized as follows section 1 is introductory i give there the background from representation theory and the cohomology groups of a lie algebra section 2 contains the proof of the new structure theorems we need the construction of the generic ring r and of the defect lie algebra l in section 3 i show that genericity of r follows from vanising of the homology of a certain family of complexes over u l the enveloping algebra of l in section 4 i apply the results of section 2 to study multi

betti numbers of modules of exponent two over regular local rings abstract let r m k be a regular local ring of dimension n and let m be a finite length module over r this paper gives an affirmative answer to horrocks questions when m 2 m 0 that is in this case the rank of the i th syzygy of m is at least formula and the i th betti number of m is at least formula

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

symbolic squares in regular local rings in this paper we prove for several classes of ideals in regular local rings of equicharacteristic 0 that the symbolic square of the ideal contains no minimal generator of the ideal our techniques come from residual intersections integral closures of ideals and differentials

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

linkage theory for algebras with pure resolutions abstract let s be a graded cohen macaulay quotient r i of a polynomial ring r k x 1 x n over an infinite field k if s is in the linkage class of a complete intersection then its koszul homology and twisted conormal module are cohen macaulay by passing to zero and one dimensional specializations of s one can convert these properties into numerical test conditions for a graded algebra to be in the linkage class of a complete intersection we prove that if s is generically a complete intersection and the minimal free solution of s is pure and almost linear then s is not in the linkage class of a complete intersection apart from a few obvious exceptions for algebras of small codimension the results are stronger if s has codimension three and has a pure resolution then s is in the linkage class of a complete intersection if and only if s is gorenstein if s is a codimension four gorenstein algebra with a pure resolution then s is in the linkage class of a complete intersection if and only if s is a complete intersection

on the projective dimension and the unmixed part of three cubics let r be a polynomial ring over a field in an unspecified number of variables we prove that if j r is an ideal generated by three cubic forms and the unmixed part of j contains a quadric then the projective dimension of r j is at most 4 to this end we show that if k r is a three generated ideal of height two and l r an ideal linked to the unmixed part of k then the projective dimension of r k is bounded above by the projective dimension of r l plus one

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

a deformation theory approach to betti numbers of finite length modules let r k x xd be a polynomial ring in d variables with d at least 2 over a field k and let a4 be a finite length r module buchsbaum and eisenbud b e conjecture that the ith betti number pi should be at least f in their work on linkage huneke and ulrich h u prove this if m r i where i is a licci ideal by observing on pages 317 318 that after deforming and localizing they can assume that z is a complete intersection this paper extends this remark to ideals generated by monomials the technique works because deformation preserves the betti numbers exactly while localization can only make them go down in this paper we consider a sequence of localizations and deformations until we arrive at a module whose betti numbers we can compute in c one of us showed that if r kcx i xd and z was an ideal generated by monomials in the xi such that r z co and z is not an r sequence then the pi satisfy stronger inequality that pi 2 f ii in this paper we give a deformation theoretic proof of this by showing theorem l that if z is a finite colength ideal generated by monomials and is not a complete intersection then after a sequence of deformations and localizations we arrive at a new monomial ideal generated by d 1 elements consisting of powers of each of the xi and one other monomial

singularities of generic linkage of algebraic varieties let y be a generic link of a subvariety x of a nonsingular variety a we give a description of the grauert riemenschneider canonical sheaf of y in terms of the multiplier ideal sheaves associ ated to x and use it to study the singularities of y as the first application we give a criterion when y has rational singularities and show that log canonical threshold increases and log canonical pairs are preserved in generic linkage as another application we give a quick and simple liaison method to generalize the results of de fernex ein and chardin ulrich on the castelnuovo mumford regularity bound for a projective variety

tight closure and linkage classes in gorenstein rings we study the relationship between the tight closure of an ideal and the sum of all ideals in its linkage class

heavy hydroelastic travelling waves this is a mathematical study of steady two dimensional waves of prescribed period which propagate without change of form on the surface of an infinitely deep expanse of fluid that is moving under gravity and bounded above by heavy frictionless thin elastic sheet the flow which is supposed irrotational is at rest at infinite depth and its velocity is stationary relative to a frame moving with the wave in that frame the elastic sheet coincides with the zero streamline and its material points move according to the equations of hyperelasticity from the mechanics of the surface material the pressure on a point of the sheet in the moving frame is shown to be a function of its height its slope and derivatives with respect to arc length of these quantities independently according to classical fluid mechanics the pressure in the fluid at the same point is determined by its height and the fluid velocity tangent to the zero streamline therefore this is a free boundary problem to find a non self intersecting curve in the plane which is the zero contour of a harmonic function the stream function and at which the normal derivative of the same harmonic function is a prescribed function of the shape of the curve with the wavelength fixed the parameters are the density of the undeformed sheet the wave velocity c the mean velocity c of the surface sheet relative to the wave and the gravitational acceleration g the case of a weightless sheet in which 0 and c does not appear is a special case under quite general hypotheses on the elastic response of the sheet the existence of rapidly propagating or very slow depending on parameter values steady waves is established by finding a critical point of the lagrangian this is a saddle point problem in the calculus of variations

links of symbolic powers of prime ideals in this paper we prove the following let r frak m be a cohen macaulay local ring of dimension d 2 suppose that either r is not regular or if r is regular assume that d 3 let t 2 be a positive integer if alpha 1 dots alpha d is a regular sequence contained in frak m t then alpha 1 dots alpha d frak m t subseteq frak m t this result gives an affirmative answer to a conjecture raised by polini and ulrich

steady periodic hydroelastic waves this is a study of steady periodic travelling waves on the surface of an infinitely deep irrotational ocean when the top streamline is in contact with a light frictionless membrane that strongly resists stretching and bending and the pressure in the air above is constant it is shown that this is a free boundary problem for the domain of a harmonic function the stream function which is zero on the boundary and at which its normal derivative is determined by the boundary geometry with the wavelength fixed at 2 we find travelling waves with arbitrarily large speeds for a significant class of membranes our approach is based on zakharov s hamiltonian theory of water waves to which elastic effects at the surface have been added however we avoid the hamiltonian machinery by first defining a lagrangian in terms of kinetic and potential energies using physical variables a conformal transformation then yields an equivalent lagrangian in which the unknown function is the wave height once critical points of that lagrangian have been shown to correspond to the physical problem the existence of hydroelastic waves for a class of membranes is established by maximization hardy spaces on the unit disc and the hilbert transform on the unit circle play a role in the analysis

numerical invariants of liaison classes 302 c huneke intersection what properties are then enjoyed by v for instance buchweitz was able to show that all such v are strongly nonobstructed see 5 for details the two questions above have been completely answered if x spec r with r a regular local ring and if codim v 2 in this case

steady periodic water waves under nonlinear elastic membranes this is a study of two dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending and the pressure in the air above is constant it is not supposed that the waves have small amplitude the problem of existence of such waves is addressed using methods from the calculus of variations the analysis involves the hilbert transform and a riemann hilbert formulation

the minimal free resolutions of the huneke ulrich deviation two gorenstein ideals structure theorems exist for gorenstein ideals of small grade a different parameter one can use for organizing ideals is deviation or complete intersection defect the deviation of the ideal i is the minimal number of generators of i minus the grade of i a deviation zero ideal is a complete intersection kunz s proved that there are no deviation one gorenstein ideals presently people are trying to find all deviation two gorenstein ideals see for example 3 a hypersurface section of a grade g 1 deviation two gorenstein ideal is a grade g deviation two gorenstein ideal we consider this to be a trivial example very few non trivial examples of deviation two gorenstein ideals are known the buchsbaum eisenbud structure theorem 2 describes all grade three gorenstein ideals in particular every deviation two grade three gorenstein ideal is generated by the maximal order pfaflians of a 5 x 5 alternating matrix no non trivial deviation two gorenstein ideals of even grade are known herzog and miller 3 have taken a step toward proving that all grade four deviation two gorenstein ideals are trivial let i be a grade four deviation two gorenstein ideal if i is a generic complete intersection and i i is cohenmacaulay then i is a hypersurface section the only known non trivial deviation 2 gorenstein ideals of grade at least four are the ideals of huneke and ulrich 4 which are the ideals studied in this paper for each integer s let y be a 1 x s matrix of indeterminates x be an s x s alternating matrix of indeterminates g be the product y x and ct be the sequence gp g let a be the ideal of

sums of linked ideals it is shown that the sum of two geometrically linked ideals in the linkage class of a complete intersection is again an ideal in the linkage class of a complete intersection conversely every gorenstein ideal of height at least two in the linkage class of a complete intersection can be obtained as a generalized localization of a sum of two geometrically linked ideals in the linkage class of a complete intersection we also investigate sums of doubly linked gorenstein ideals as an application we construct a perfect prime ideal which is strongly nonobstructed but not strongly cohen macaulay and a perfect prime ideal which is not strongly nonobstructed but whose entire linkage class is strongly cohen macaulay

the growth of free products abstract we improve estimates for the exponential growth rates of free products with or without amalgamation hnn extensions one relator groups and groups of positive deficiency some examples of groups for which the growth rates are realized are given