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

modules of finite projective dimension with negative intersection multiplicities let x be the image of x in r the main object of this paper is to study modules of finite length and finite projective dimension over r one consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension let denote length if s is a noetherian ring m n are finitely generated s modules l m is finite and d pdsm we may define following serre is d zs m n z m n 1 tors m n

the exact distribution of the stein rule estimator abstract this paper derives the exact density of the stein rule estimator in the setting of the general linear regression the derivation is facilitated by the author s development of new algebraic methods that involve an extension of the weyl calculus general formulae for the moments of the estimator are also provided

a finite element method for second order nonvariational elliptic problems we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form the method is of galerkin type using conforming finite elements and applied directly to the nonvariational nondivergence form of a second order linear elliptic problem the key tools are an appropriate concept of finite element hessian and a schur complement approach to solving the resulting linear algebra problem the method is illustrated with computational experiments on three linear and one quasi linear pde all in nonvariational form

on finite element methods for fully nonlinear elliptic equations of second order for the first time we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard c 1 finite element method fem we consider throughout the paper two cases in parallel for convex bounded polyhedral domains in bf r n or for c 2 bounded domains in bf r 2 we prove stability and convergence for the corresponding conforming or nonconforming c 1 fem respectively the results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere the classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator the consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated the stability has to be proved in an unusual way this is the hard core of the paper essential tools are linearization a compactness argument the interplay between the weak and strong form of the linearized operator and a new regularity result for solutions of finite element equations an essential basis for our proofs are davydov s results for c 1 fes on polyhedral domains in bf r n or of local degree 5 for c 2 domains in bf r 2 better convergence and extensions to bf r n for c 2 domains are to be expected from his forthcoming results on curved domains our proof for the second case in bf r n includes the first essentially as a special case the method applies to quasi linear elliptic problems not in divergence form as well a discrete newton method is shown to converge locally quadratically essentially independently of the actual grid size by the mesh independence principle

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

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

a finite element method for nonlinear elliptic problems we present a galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations a key tool is the discretization proposed in lakkis and pryer 2011 allowing us to work directly on the strong form of a linear pde an added benefit to making use of this discretization method is that a recovered finite element hessian is a byproduct of the solution process we build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear pdes benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the monge ampere equation and the pucci equation

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

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

discrete methods for fully nonlinear elliptic equations this paper exhibits and proves the stability of discrete approximations for the solution of the dirichlet problem for fully nonlinear uniformly elliptic partial differential equations in situations where classical solutions are not known to exist the resulting solutions are characterized in the viscosity sense of crandall and lions traps amer math soc 177 1983 pp 1 42 and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients

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

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

families of artinian and one dimensional algebras abstract the purpose of this paper is to study families of artinian or one dimensional quotients of a polynomial ring r with a special look to level algebras let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h let b a be any graded surjection of quotients of r with hilbert function h b 1 h 1 h j and h a respectively if dim a 0 respectively dim a depth a 1 and a is a truncation of b in the sense that h a 1 h 1 h j 1 0 0 respectively h a 1 h 1 h j 1 for some h j then we show there is a close relationship between gradalg h a r and gradalg h b r concerning e g smoothness and dimension at the points a and b respectively provided b is a complete intersection or provided the castelnuovo mumford regularity of a is at least 3 sometimes 2 larger than the regularity of b in the complete intersection case we generalize this relationship to non truncated artinian algebras a which are compressed or close to being compressed for more general artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of cohen macaulay type 2 this description and a linkage theorem for families allow us to prove a conjecture of iarrobino on the existence of at least two irreducible components of gradalg h r h 1 3 6 10 14 10 6 2 whose general elements are artinian level algebras of type 2

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

the regularity of tor and graded betti numbers let s k x1 xn let a b be finitely generated graded s modules and let m x1 xn s we give bounds for the regularity of the local cohomology of tork a b in terms of the graded betti numbers of a and b under the assumption that dim tor1 a b 1 we apply the results to syzygies grobner bases products and powers of ideals and to the relationship of the rees and symmetric algebras for example we show that any homogeneous linearly presented m primary ideal has some power equal to a power of m and if the first n 1 2 steps of the resolution of i are linear then i 2 is a power of m