A duality principle for lattices and categories of modules

Let R be a ring with 1, R^op the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories $\text{A}$ such that there exists an exact embedding functor $\text{A}$→R-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor $\text{A}$→R-Mod if and only if there exists an exact embedding functor $\text{A}$^op→R^op-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then R^op-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; $\text{L}$(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L¹ is representable by R^op-modules. That is, $\text{L}\text{(}\text{R}{}^{\mathrm{op}}\text{)={L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}}$.If $\text{R}$ is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, $\text{L}\text{(}\text{R}\text{)}$ is self-dual: $\text{L}\text{(}\text{R}\text{)= {L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}.}$     

A duality theorem for generalized local cohomology

We prove a duality theorem for graded algebras over a field that implies several known duality results: graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

     

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors ${\gamma }_W$ with supports in arbitrary subsets W of Spec R. If W is a specialization-closed subset, then ${\gamma }_W$ coincides with the right derived functor ${\mathrm{R}\mathrm{\Gamma }}_W$ of the section functor ${\mathrm{\Gamma }}_W$ with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for ${\gamma }_W$ with W being an arbitrary subset.     

A duality theorem for linear congruences

An analogous duality theorem to that for Linear Programming is presented for systems of linear congruences. It is pointed out that such a system of linear congruences is a relaxation of an Integer Programming model (for which the duality theorem does not hold). Algorithms are presented for both the resulting primal and dual problems. These algorithms serve to give a constructive proof of the duality theorem.     

A duality theorem for graph embeddings

A generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. If K, L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs K?,L? in orientable surfaces S?, such that S? are branched covers of S and the restrictions of the branched coverings to K?,L? are wqc's of K, L. the theory is applied to obtain genus embeddings of composition graphs G[nK₁] from embeddings of “quotient” graphs G.     

A subgradient duality theorem

A duality theorem of P. Wolfe for nonlinear differentiable programming is extended to the nondifferentiable case by replacing gradients by subgradients. The dual pair is further simplified in the case that nondifferentiability enters only in the objective functions and then only through a positively homogeneous convex function. A number of previously studied problems appear as special cases.     

A duality theorem for complex quadratic programming

A duality theory for complex quadratic programming over polyhedral cones is developed, following Dorn, by using linear duality theory.This research was partly supported by the National Science Foundation, Project No. GP-7550, and by the US Army Research Office, Durham, North Carolina, Contract No. DA-31-124-ARO-D-322. The authors are indebted to the referee for his helpful suggestions.     

A theorem of ordered duality

We associate, to any ordered configuration of n points or ordered arrangement of n lines in the plane, a periodic sequence of permutations of [1, n] in a way which reflects the order and convexity properties of the configuration or arrangement, and prove that a sequence of permutations of [1, n] is associated to some configuration of points if and only if it is associated to some arrangement of lines. We show that this theorem generalizes the standard duality principle for the projective plane, and we use it to derive duals of several well-known theorems about arrangements, including a version of Helly's theorem for convex polygons.     

A general duality theorem for marginal problems

Summary Given probability spaces (X _i ,A _i ,P _i ),i=1, 2 letM(P ₁,P ₂) denote the set of all probabilities on the product space with marginalsP ₁ andP ₂ and leth be a measurable function on (X ₁×X ₂,A ₁ A ₂). In order to determine supfh dP where the supremum is taken overP inM(P ₁,P ₂), a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.     

A Cohen-type theorem for Artinian modules

We show that a finitely embedded module M over a commutative ring R is Artinian if the factor module M/(0 :_M P) is finitely embedded for every prime ideal P of R. Received: 10 June 2005     

A duality theorem for continuous-time linear programming problems

Summary In dealing with dynamic economic policy models one encounters optimization problems whose objective function is an integral of a linear function of a finite number of continuous variables and whose constraints are linear integral inequalities. A set of intertemporal efficiency conditions (equilibrium conditions) yielding the optimal policy are given. By approximating the continuous problem by a set of discrete problems and appealing to a well known convergence theorem in functional analysis a continuous analog of the duality theorem is proved.

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Zusammenfassung Bei der Beschäftigung mit dynamischen Modellen der ökonomischen Politik stößt man auf Optimierungsprobleme, deren Zielfunktion ein Integral einer linearen Funktion von einer endlichen Anzahl stetiger Variablen ist und deren Beschränkungen lineare Integral-Ungleichungen sind. Eine Menge intertemporaler Effizienz-Bedingungen (Gleichgewichtsbedingungen), die zur optimalen Politik führen, sind gegeben. Durch Approximation des kontinuierlichen Problems mittels einer Menge von diskreten Problemen und Berufung auf einen wohlbekannten Konvergenzsatz aus der Funktionalanalysis wird ein stetiges Analogon des Dualitätstheorems bewiesen.

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The author is indebted to Mr.Arnold Faden for helpful suggestions and to ProfessorKarl A. Fox andGerhard Tintner for encouragement during the preparation of the paper. This research has been partially supported by a grant from the Ford Foundation to the School of Business Administration administered by the Center for Research in Management Science, University of California, Berkeley.

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Vorgel. v.:G. Tintner.     

A Dirichlet unit theorem for Drinfeld modules

We show that the module of integral points on a Drinfeld module satisfies an analogue of Dirichlet’s unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated sub-module of the module of integral points. We use the results to give a precise formulation of a conjectural analogue of the class number formula.