Topological Modules Research Articles

Orbites coadjointes et variétés caractéristiques

The purpose of the present work is to describe a dequantization procedure for topological modules over a deformed algebra. We define the characteristic variety of a topological module as the common zeroes of the annihilator of the representation obtained by setting the deformation parameter to zero. On the other hand, the Poisson characteristic variety is defined as the common zeroes of the ideal obtained by considering the annihilator of the deformed representation, and only then setting the deformation parameter to zero. Using Gabber’s theorem, we show the involutivity of the characteristic variety. The Poisson characteristic variety is indeed a Poisson subvariety of the underlying Poisson manifold. We compute explicitly the characteristic variety in several examples in the Poisson-linear case, including the dual of any exponential solvable Lie algebra. In the nilpotent case, we show that any coadjoint orbit appears as the Poisson characteristic variety of a well chosen topological module. Résumé: Nous présentons dans ce travail un procédé de déquantification pour des modules topologiques sur une algèbre déformée. Nous définissons la variété caractéristique d’un module topologique comme l’ensemble des zéros communs de l’annulateur de la représentation obtenue en annulant le paramètre de déformation. Nous définissons par ailleurs la variété de Poisson caractéristique comme l’ensemble des zéros communs de l’idéal obtenu par quotient en annulant le paramètre de déformation dans l’annulateur de la représentation déformée. Nous montrons à l’aide du théorème de Gabber l’involutivité de la variété caractéristique. La variété de Poisson caractéristique est une sous-variété de Poisson de la variété de Poisson sous-jacente. Nous explicitons la variété caractéristique dans plusieurs exemples, incluant le dual des algèbres de Lie résolubles exponentielles. Dans le cas nilpotent nous montrons que toute orbite coadjointe est la variété de Poisson caractéristique d’un module topologique bien choisi.

Universal covers of topological modules and a monodromy principle

Let $R$ be a simply connected topological ring and $M$ be a topological left $R$-module in which the underling topology is path connected and has a universal cover. In this paper, we prove that a simply connected cover of $M$ admits the structure of a topological left $R$-module, and prove a Monodromy Principle, that a local morphism on $M$ of topological left $R$-modules extends to a morphism of topological left $R$-modules.

Topological Morita contexts

In synthesis, this paper presents a generalization of the theory of Morita contexts from the case of abstract modules over abstract rings to that of complete l. t. (linearly topological) modules over complete l. t. rings. To begin with, given three complete right l. t. rings ( R, ρ), ( S, σ) and ( T, τ), and two complete l. t. bimodules ( R A S , α) and ( S B T , β) satisfying suitable hypotheses, we introduce the “topological tensor product” ( A, α)⊗ u S ( B, β). Next, we define a topological Morita context to be a family made up of two complete l. t. rings ( R, ρ) and ( S, σ), two bimodules ( S A R , α) and ( R B S , β) of the above kind, and two continuous bilinear maps μ :(B,β)⊗ u S(A,α)→(R,ρ) and ν :(A,α)⊗ u R(B,β)→(S,σ) ; the context is called dense if both μ and ν have dense image. We then prove that such a dense Morita context yields an equivalence of categories between CLT -( R, ρ) and CLT -( S, σ), in such a way to induce an equivalence between Mod-( R, ρ) and Mod-( S, σ). Finally, we give a “topological” parallel of the notion of progenerator, and we show that such a “topological progenerator” gives rise to a dense context, and hence to an equivalence of the above-mentioned kind. Conversely, we show that every such equivalence arises in this way.

Pontryagin Duality in the Theory of Topological Vector Spaces and in Topological Algebra

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements:

A modeling methodology for finite element mesh generation of multi-region models with parametric surfaces

This paper presents a description of the reorganization of a geometric modeler, MG, designed to support new capabilities of a topological module (CGC) that allows the detection of closed-off solid regions described by surface patches in non-manifold geometric models defined by NURBS. These patches are interactively created by the user by means of the modeler's graphics interface, and may result from parametric–surface intersection in which existing surface meshes are used as a support for a discrete definition of intersection curves. The geometry of realistic engineering objects is intrinsically complex, usually composed by several materials and regions. Therefore, automatic and adaptive meshing algorithms have become quite useful to increase the reliability of the procedures of a FEM numerical analysis. The present approach is concerned with two aspects of 3D FEM simulation: geometric modeling, with automatic multi-region detection, and support to automatic finite-element mesh generation.

Polynomials and multilinear mappings in topological modules

Criteria for the equicontinuity of sets of multilinear mappings between topological modules are studied, as well as topological modules of continuous multilinear mappings. As a consequence, criteria for the equicontinuity of sets of homogeneous polynomials between topological modules are also studied, as well as topological modules of continuous homogeneous polynomials.

A Characterization for an L(μ, K)-Topological Module to Admit Enough Canonical Module Homomorphisms

Let K be the scalar field of all real or complex numbers, let (Ω,A,μ) be a σ-finite measure space, and let L(μ,K) be the algebra of the μ-equivalence classes of all K-valued μ-measurable functions defined on (Ω,A,μ). L(μ,K) is a topological algebra over K when endowed with the topology of convergence locally in measure; topological modules over this topological algebra L(μ,K) (briefly, L(μ,K)-topological modules) are an extensive class of topological modules, which arise naturally in the course of the study of the theory of probabilistic normed spaces. The purpose of this paper is to show that an arbitrary regular L(μ,K)-topological module admits enough canonical module homomorphisms if and only if all of its quasi-free submodules of finite rank are complemented in the sense of topological modules.

Topological Hochschild homology of algebras in characteristic p

We compute the topological Hochschild homology modules of finitely generated commutative algebras over finite fields provided their singularities are sufficiently mild.

Classifying modules over K-theory spectra

The category of modules over an S-algebra ( A ∞ or E ∞ ring spectrum) has many of the good properties of the category of spectra. When the homotopy groups of the S-algebra in question form a sufficiently nice ring, it is possible to see the deviation of the category of modules over an S-algebra from the corresponding algebraic module category. In particular, many algebraic modules are realized as homotopy groups of topological modules over S-algebras. Examples studied include real and complex K-theory, both connective and periodic. Further, Bousfield localization by a smashing spectrum is shown to yield a category of modules over the localized sphere. For periodic K-theory, these methods yield an algebraic criterion to determine when a local spectrum is a module over the K-theory S-algebra, real or complex.

On modules of continuous linear mappings

Modules of continuous linear mappmgs with values n topological modules of continuous mappings are studied.

Fractional Power Series and Pairings on Drinfeld Modules

Let C C be an algebraically closed field containing F q \mathbb {F}_q which is complete with respect to an absolute value | | |\;| . We prove that under suitable constraints on the coefficients, the series f ( z ) = ∑ n ∈ Z a n z q n f(z) = \sum _{n \in \mathbb {Z}} a_n z^{q^n} converges to a surjective, open, continuous F q \mathbb {F}_q -linear homomorphism C → C C \rightarrow C whose kernel is locally compact. We characterize the locally compact sub- F q \mathbb {F}_q -vector spaces G G of C C which occur as kernels of such series, and describe the extent to which G G determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of f f of a given valuation, given the valuations of the coefficients. The “adjoint” series f ∗ ( z ) = ∑ n ∈ Z a n 1 / q n z 1 / q n f^\ast (z) = \sum _{n \in \mathbb {Z}} a_n^{1/q^n} z^{1/q^n} converges everywhere if and only if f f does, and in this case there is a natural bilinear pairing \[ ker ⁡ f × ker ⁡ f ∗ → F q \ker f \times \ker f^\ast \rightarrow \mathbb {F}_q \] which exhibits ker ⁡ f ∗ \ker f^\ast as the Pontryagin dual of ker ⁡ f \ker f . Many of these results extend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module.

Pontryagin duality in the theory of topological modules

Pontryagin duality in the theory of topological modules

THE IMPACT OF CLOSURE OPERATORS ON THE STRUCTURE OF A CONCRETE CATEGORY

Abstract We study the question of whether morphisms of a concrete category X can be determined by means of closure operators of X in the sense of [DT]. In particular, we show that closure operators cannot detect uniformly continuous maps in case X is the category of uniform spaces. We have analogous result for proximity spaces and topological modules.

OPERATORS IN A TOPOLOGICAL MODULE OF ENTIRE FUNCTIONS COMMUTING WITH DIFFERENTIATION ON A SUBMODULE

The general form is found of continuous linear operators in a certain topological module of entire functions of several complex variables, commuting with differentiation on a finitely generated submodule. Bibliography: 5 titles.

VECTOR-VALUED DUALITY FOR MODULES OVER BANACH ALGEBRAS

Pairs of topological modules , over algebras , are considered that are in duality, with values in an -bimodule . An important example: if an arbitrary -module is regarded as an -bimodule, where , then for any -module the pair , is in a natural -duality. Conditions on the -bimodule are found under which the bipolar theorem and certain other results in convex analysis carry over to -valued duality. In several cases this enables one to describe the structure of the closed submodules and (in terms of graphs) the closed homomorphisms. Among the applications are results on commutation systems, unbounded derivations, left Hilbert algebras, spaces with an indefinite metric, and multipliers of -algebras.

A note on the continuity of multilinear mappings in topological modules

In the present note, we obtain a criterion for the equicontinuity of families of multilinear mappings between topological modules. We also give an example which shows that the hypothesis imposed on the neighborhoods of zero is essential for the validity of our theorem.

On the existence and the stability of the solutions of certain nonlinear equations in topological modules of mappings

On the existence and the stability of the solutions of certain nonlinear equations in topological modules of mappings

On barrelled topological modules

Some of the main theorems concerning ultrabarrelled topological vector spaces are extended to the context of topological modules.

Formal moduli of modules over local $k$-algebras

We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local $k$-algebra $R$, not necessarily commutative, where $k$ is a field. The class of topological modules we consider include all those of finite rank over $k$ and some of infinite rank as well, namely those with a Schauder basis in the sense of $\S 1$. This generalizes the results of [Sh], where the result was obtained in a different way in case the ring $R$ is the completion of the local ring of a plane curve singularity and the module is ${k^n}$. Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called "growth functions" to handle explicit epsilonics involving the convergence of formal power series in noncommuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of [Sch] which were used in [Sh] and is more conceptual.

Differentiation in modules over topological ∗-algebras

In recent years, there has been a systematic development of the theory of A-bundles, that is, vector bundles where the field of coefficients is replaced by a topological algebra A, usually an involutive one. The above extension is justified by its applications in pure mathematics (differential topology [15]) as well as in relativistic quantum mechanics, via, e.g., the Serre-Swan theorem [22]. Bundles with coefficients in a commutative unital Banach algebra, or yet a C*-algebra, were considered by T. K. Kandelaki [6, 71, K. Fujii [S], and A. S. MiSEenko and Yu. P. Solov’ev [ 15, 161. Furthermore, A. Mallios studied A-bundles, with A being a locally m-convex (*-)algebra with unit [l&12]. Note also that certain results are still valid for more general topological algebras [ 13, Sect. 53, or even topological rings [21]. The results already obtained are mainly within the framework of topological A-bundles. The basic difficulty in the development of a theory of differential A-bundles is the lack of a suitable differential calculus in the libres, the latter being topological modules over A. One might, of course, use the existing methods of differentiation in the underlying topological vector spaces (cf. for instance [18]). However, besides the intrinsic weaknesses of these methods (regarding the higher order chain rule, continuity of differentiable maps, differentiability of the composition map and evaluation map, which are necessary for differential geometric considerations on bundles), the additional algebraic operations (scalar multiplication by elements of A, involution) should be taken into account, because they are essentially involved in the structure of A-bundles (cf. [ 11, 183). For the same reason (incompatibility with involution), the 255 0022-247X/92 55.00