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

iterated elliptic and hypergeometric integrals for feynman diagrams we calculate 3 loop master integrals for heavy quark correlators and the 3 loop quantum chromodynamics corrections to the parameter they obey non factorizing differential equations of second order with more than three singularities which cannot be factorized in mellin n space either the solution of the homogeneous equations is possible in terms of 2f1 gaus hypergeometric functions at rational argument in some cases integrals of this type can be mapped to complete elliptic integrals at rational argument this class of functions appears to be the next one arising in the calculation of more complicated feynman integrals following the harmonic polylogarithms generalized polylogarithms cyclotomic harmonic polylogarithms square root valued iterated integrals and combinations thereof which appear in simpler cases the inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals where the new innermost letter itself is not an iterative integral a new class of iterative integrals is introduced containing letters in which multiple definite integrals appear as factors for the elliptic case we also derive the solution in terms of integrals over modular functions and also modular forms using q product and series representations implied by jacobi s i functions and dedekind s function the corresponding representations can be traced back to polynomials out of lambert eisenstein series having representations also as elliptic polylogarithms a q factorial 1 k logarithms and polylogarithms of q and their q integrals due to the specific form of the physical variable x q for different processes different representations do usually appear numerical results are also presented

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

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

the andrews sellers family of partition congruences in 1994 james sellers conjectured an infinite family of ramanujan type congruences for 2 colored frobenius partitions introduced by george e andrews these congruences arise modulo powers of 5 in 2002 dennis eichhorn and sellers were able to settle the conjecture for powers up to 4 in this article we prove sellers conjecture for all powers of 5 in addition we discuss why the andrews sellers family is significantly different from classical congruences modulo powers of primes

totally reflexive modules and poincare series abstract we study cohen macaulay non gorenstein local rings r m k admitting certain totally reflexive modules more precisely we give a description of the poincare series of k by using the poincare series of a non zero totally reflexive module with minimal multiplicity our results generalize a result of yoshino to higher dimensional cohen macaulay local rings moreover from a quasi gorenstein ideal satisfying some conditions we construct a family of non isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators

arithmetic properties of the partition function let p n denote the number of partitions of the positive integer n p n is the number of representations of n as a non increasing sequence of positive integers by convention we agree that p 0 1 and that p n 0 if n 0 the study of the arithmetic properties of p n has a long and rich history see for example the works of andrews atkin dyson garvan kim stanton and swinnerton dyer an an g at1 at swd d g k s these works have their origins in the groundbreaking observations of ramanujan r1 r2 r3 r4 in ramanujan s own words r3

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

maximal families of gorenstein algebras the purpose of this paper is to study maximal irreducible families of gorenstein quotients of a polynomial ring r let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h we prove there is a close relationship between the irreducible components of gradalg h r whose general member is a gorenstein codimension c 1 quotient and the irreducible components of gradalg h r whose general member b is a codimension c cohen macaulay algebra of hilbert function h related to h if the castelnuovo mumford regularity of the gorenstein quotient is large compared to the castelnuovo mumford regularity of b this relationship actually determines a well defined injective mapping from such cohen macaulay components of gradalg h r to gorenstein components of gradalg h r in which generically smooth components correspond moreover the dimension of the gorenstein components is computed in terms of the dimension of the corresponding cohen macaulay component and a sum of two invariants of b using linkage by a complete intersection we show how to compute these invariants linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem

koszul homology and the structure of low codimension cohen macaulay ideals the relationship between the properties of the koszul homology modules of two ideals connected by linkage is studied if the ideal i is either i a cohen macaulay ideal of codimension 3 or ii a gorenstein ideal of codimension 4 the one dimensional koszul module carries considerable information on the struct ural nature of the linkage class of i in case i or on the conormal module of i in case ii emphasis is given to the verification of the properties by computation introduction the aim of this paper is to examine the structure of cohen macaulay ideals of low i e 3 and 4 codimension as it is expressed by the koszul homology modules more specifically given a perfect ideal i of a gorenstein nng r we are interested in how i its numerical data e g betti numbers and degrees of its generating syzygies in the case of a homogeneous ideal ii properties of the full linkage class of i and iii whether i is suitably generated e g generated by a d sequence are mirrored on the module and the algebra structures of the homology of the koszul complex build on a generating set of i the man result theorem 2 4 is the proof that for a perfect ideal i of codimension threes the condition that the 1 dimensional koszul homology module h1 i r be cohen macaulay is an invanant of the full linkage class of i the framework for this result is provided by several facts first the classical descnptions of cohen macaulay ideals of codimension two and of gorenstein ideals of heit ht three as belonging to the linkage classes of complete intersections a be ga w then a theorem of huneke hu1 asserting that the property hi i r is cohen macaulay for i ms is an invariant of the eren linkage class of i finally in case of ideals of polynomial rings data on the koszul homology has the advantage of being in principle known at the outset that is hypotheses on them can be venfied once the ideal has been specified by a set of generators in the last section we indicate how symbolic computation packages may be used to obtain this informa tion to contrast already in the case of codimension four gorenstein ideals the depths of the koszul homology modules of directly linked ideals may differ sharply furthermore in codimension three we show that this invanance does not extend to the next koszul module to further enhance the role of h1 i r as a carrier of

on the refined maximum principle for degenerate elliptic and parabolic problems abstract we extend the refined maximum principle in h berestycki l nirenberg s r s varadhan the principal eigenvalue and the maximum principle for second order elliptic operators in general domains comm pure appl math 47 1994 47 92 to degenerate elliptic and parabolic equations with unbounded coefficients then we discuss the well posedness of the corresponding dirichlet boundary value problems

strongly cohen macaulay schemes and residual intersections this paper studies the local properties of closed subschemes y in cohen macaulay schemes x such that locally the defining ideal of y in x has the property that its koszul homology is cohen macaulay whenever this occurs y is said to be strongly cohen macaulay in x this paper proves several facts about such embeddings chiefly with reference to the residual intersections of y in x the main result states that any residual intersection of y in x is again cohen macaulay introduction our purpose in this paper is to investigate a property of a closed subscheme y in a noetherian cohen macaulay scheme x which we call strong cohen macaulayness this is a local property which we first describe for a local ring let x spec r where r is a cohen macaulay local ring and let y spec r i

the koszul homology of an ideal recently many researchers have worked on problems connected with various graded algebras associated to an ideal i in a local ring r two algebras in particular have received the most attention the associated graded algebra of z gr r r z z z and the rees algebra of z defined to be r zt brodmann 2 and goto and shimoda 9 have studied the local cohomology of r zr for certain primary ideals z herzog has obtained new results in the case where z is the maximal ideal eisenbud and huneke 7 studied the cohenmacaulayness of these algebras while the concepts of d sequences 111 and hodge algebras s have been used to understand these graded algebras recently simis and vasconcelos w 211 related the cohen macaulayness torsion freeness and normality of these algebras to the koszul homology of i if z is an ideal we let zz z r denote the jth koszul homology of the ideal z with respect to somejixed system of generators for i we denote the symmetric algebra of a module a4 by sym m and denote thejth graded piece of this algebra by sym m in 20 and 21 simis and vasconcelos construct a complex a z or simply a with the following properties

on the dimension and integrality of symmetric algebras the symmetric algebra of an r module e sym e has a relatively simple description since it is given as a quotient of a polynomial ring over r modulo an ideal j generated by linear homogeneous polynomials our first result relates the grade of j to certain determinantal ideals associated to such presentation in particular at least when r is cohen macaulay the krull dimension of sym e can be easily estimated from such formulation the expression of the grade of j is also useful in setting up conditions for the integrality and or factoriality of sym e it turns out that the integrality of sym e for r regular is related to phenomena of cohen macaulayness of the symmetric algebra there are however considerable difficulties in translating the details of a projective resolution of e into properties of the ideal 3 this analysis is carried out for some modules finally we consider for an ideal i the symmetric algebra over the ring

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

maximal cohen macaulay modules and gorenstein algebras abstract let b be a graded cohen macaulay quotient of a gorenstein ring r it is known that sections of the dual of the canonical module kb can be used to construct gorenstein quotients of r the purpose of this paper is to place this method of construction into a broader context if m is a maximal cohen macaulay b module whose sheafified top exterior power is a twist of kb and if m satisfies certain additional homological conditions then regular sections of m can again be used to construct gorenstein quotients of r on cohen macaulay quotients the normal module the first koszul homology module and several other associated modules all have sheafified top exterior power equal to a twist of kb if additional restrictions are placed on the cohen macaulay quotients then these modules will satisfy the required additional homological conditions this places the canonical module within a broad family of easily manipulated maximal cohen macaulay modules whose sections can be used to construct gorenstein quotients of r

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

links of prime ideals and their rees algebras abstract in a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in gorenstein rings are equimultiple ideals with reduction number 1 this led to the construction of large families of cohen macaulay rees algebras the first goal of this paper is to extend this result to arbitrary cohen macaulay rings the means of the proof are changed since one cannot depend so heavily on linkage theory we then study the structure of the rees algebra of these links more specifically we describe their canonical module in sufficient detail to be able to characterize self linked prime ideals in the last section multiplicity estimates for classes of such ideals are established

congruence properties for a certain kind of partition functions in light of the modular equations of fifth and seventh order we derive some congruence properties for a certain kind of partition functions a n which satisfy n 0 a n qn q q k modm where k is a positive integer with 1 k 24 and m 2 3 in view of these properties we obtain many infinite families of congruences for c k n the number of generalized frobenius partitions of n with k colors and c k n the number of generalized frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors meanwhile we also apply the main theorems to some other kinds of partition functions