On the Associated Primes of Matlis Duals of Local Cohomology Modules II

In continuation of [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zöschinger on coassociated primes of arbitrary modules with results from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] 4-6 Hellus , M. , Stückrad , J. ( 2008 ). On endomorphism rings of local cohomology modules . Proceedings of the American Mathematical Society 136 : 2333 – 2341 . Hellus , M. , Stückrad , J. ( 2008 ). Matlis duals of top local cohomology modules . Proceedings of the American Mathematical Society 136 : 489 – 498 . Hellus , M. , Stückrad , J. ( 2009 ). Artinianness of local cohomology . Journal of Commutative Algebra 1 : 269 – 274 . ], and obtain partial answers to questions which were left open in [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. These partial answers give further support for conjecture (*) from [1 Hellus , M. ( 2005 ). On the associated primes of Matlis duals of top local cohomology modules . Communications in Algebra 33 : 3997 – 4009 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zöschinger, we prove some non-finiteness results of local cohomology.     

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of , where is a -dimensional local ring and an ideal such that and .

     

Matlis Duals of Top Local Cohomology Modules and the Arithmetic Rank of an Ideal

Let I be an ideal of a local ring (R, 𝔪). Using local cohomology, we present new criteria (see 1.4, respectively 1.5) for the conditions ara (I) ≤ 1 respectively ara (I) ≤ 2, where ara (I) stands for the number of generators of I up to radical. Though this works equally well for the local and for the graded case, we show some subtle differences between the local and the graded situation in Section 2. Finally, in Section 3, we show that the Matlis dual of certain local cohomology modules, though not finite, is well behaved in some sense.     

On the Associated Primes of Matlis Duals of Top Local Cohomology Modules

After motivating the question, we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases, we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.     

On the finiteness properties of Matlis duals of local cohomology modules

Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D _R (−) := Hom _R (−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext _R ^j (R/a, D _R (H _a ^t (M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom _R (R/a, D _R (H _a ⁿ (M))) is also finitely generated. Specially, the set of prime ideals in Coass _R (H _a ⁿ (M)) which contains a is finite. Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D _R (H _a ^r (R)) where r is the least integer i such that H _a ^r (R) is not Artinian.     

Generalized concavity and duality with a square root term

Duality results for a class of nondifferentiable mathematical programming problems are given. These results allow for the weakening of the usual convexity conditions required for duality to hold. A pair of symmetric and self dual nondifferentiable programs under weaker convexity conditions are also given. A subgradient symmetric duality is proposed and its limitations discussed. Finally, a pair of nondifferentiable mathematical programs containing arbitrary norms is presented.     

Strongly Flat Modules over Matlis Domains

Strongly flat modules were introduced by Bazzoni–Salce [3 Bazzoni , S. , Salce , L. ( 2003 ). Almost perfect domains . Colloq. Math. 95 : 285 – 301 .[Crossref] , [Google Scholar]] and used to characterize almost perfect domains. Here we wish to study strongly flat modules, more generally, over Matlis domains; these are integral domains R such that the field of quotients Q has projective dimension 1. In Section 2, criteria are proved for strong flatness. We also prove that over arbitrary domains, strongly flat submodules of projective modules are projective (Theorem 3.2), in particular, strongly flat ideals are projective (Corollary 3.4) and use these results to show that the strongly flat dimension (which makes sense over Matlis domains) coincides with the projective dimension whenever it is > 1.     

The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

Wolfe duality and Mond-Weir duality via perturbations

Considering a general optimization problem, we attach to it by means of perturbation theory two dual problems having in the constraints a subdifferential inclusion relation. When the primal problem and the perturbation function are particularized different new dual problems are obtained. In the special case of a constrained optimization problem, the classical Wolfe and Mond-Weir duals, respectively, follow as particularizations of the general duals by using the Lagrange perturbation. Examples to show the differences between the new duals are given and a gate towards other generalized convexities is opened.     

Finiteness Properties of Duals of Local Cohomology Modules

We investigate Matlis duals of local cohomology modules and prove that, in general, their zeroth Bass number with respect to the zero ideal is not finite. We also prove that, somewhat surprisingly, if we apply local cohomology again (i.e., to the Matlis dual of the local cohomology module), we get (under certain hypotheses) either zero or E, an R-injective hull of the residue field of the local ring R.     

Constrained duality via unconstrained duality in generalized geometric programming

A specialization of unconstrained duality (involving problems without explicit constraints) to constrained duality (involving problems with explicit constraints) provides an efficient mechanism for extending to the latter many important theorems that were previously established for the former.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-73-2516.     

Multiobjective duality for convex ratios

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel-Rockafellar duality theory for a scalar optimization problem associated to the multiobjective primal, a dual problem is derived. This scalar dual problem is formulated in terms of conjugate functions and its structure gives an idea about how to construct a multiobjective dual problem in a natural way. Weak and strong duality assertions are presented.     

Generalized local duality,canonical modules,and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.     

Conjugate duality for vector-maximization problems

In this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n-dimensional orthant. Using this duality we obtain a zero-gap duality for a vector-maximization problem.     

Vector optimization and generalized Lagrangian duality

In this paper, foundations of a new approach for solving vector optimization problems are introduced. Generalized Lagrangian duality, related for the first time with vector optimization, provides new scalarization techniques and allows for the generation of efficient solutions for problems which are not required to satisfy any convexity assumptions.     

Generalized roof duality

The roof dual bound for quadratic unconstrained binary optimization is the basis for several methods for efficiently computing the solution to many hard combinatorial problems. It works by constructing the tightest possible lower-bounding submodular function, and instead of minimizing the original objective function, the relaxation is minimized. However, for higher-order problems the technique has been less successful. A standard technique is to first reduce the problem into a quadratic one by introducing auxiliary variables and then apply the quadratic roof dual bound, but this may lead to loose bounds.We generalize the roof duality technique to higher-order optimization problems. Similarly to the quadratic case, optimal relaxations are defined to be the ones that give the maximum lower bound. We show how submodular relaxations can efficiently be constructed in order to compute the generalized roof dual bound for general cubic and quartic pseudo-boolean functions. Further, we prove that important properties such as persistency still hold, which allows us to determine optimal values for some of the variables. From a practical point of view, we experimentally demonstrate that the technique outperforms the state of the art for a wide range of applications, both in terms of lower bounds and in the number of assigned variables.     

Full duality among graph algebras and flat graph algebras

We prove that among finite graph algebras and among finite flat graph algebras, dualizability, full dualizability, strong dualizability and entropicity are all equivalent. Any finite (flat) graph algebra which is not dualizable must be inherently non--dualizable for every infinite cardinal . A new, general method for proving strong duality is presented. Received August 30, 1999; accepted in final form September 22, 1999.     

Conjugate duality for multiobjective composed optimization problems

Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity conditions we prove the existence of strong duality. Finally, some particular cases of this problem are presented.