Elementary Duality Research Articles

A monoidal analogue of the 2-category anti-equivalence between ABEX and DEF

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to the 2-category of fp-hom-closed definable additive categories satisfying an exactness criterion. For a fixed finitely accessible category C with products and a monoidal structure satisfying the appropriate assumptions, we provide bijections between the fp-hom-closed definable subcategories of C , the Serre tensor-ideals of C fp - mod and the closed subsets of a Ziegler-type topology. For a skeletally small preadditive category A with an additive, symmetric, rigid monoidal structure we show that elementary duality induces a bijection between the fp-hom-closed definable subcategories of Mod - A and the definable tensor-ideals of A - Mod .

Dualities from dualities: the sequential deconfinement technique

It is an interesting question whether a given infra-red duality between quantum field theories can be explained in terms of other more elementary dualities. For example recently it has been shown that mirror dualities can be obtained by iterative applications of Seiberg-like dualities. In this paper we continue this line of investigation focusing on theories with tensor matter. In such cases one can apply the idea of deconfinement, which consists of trading the tensor matter for extra gauge nodes by means of a suitable elementary duality. This gives an auxiliary dual frame which can then be manipulated with further dualizations, in an iterative procedure eventually yielding an interesting dual description of the original theory. The sequential deconfinement technique has avatars in different areas of mathematical physics, such as the study of hypergeometric and elliptic hypergeometric integral identities or of 2d free field correlators. We discuss various examples in the context 4d mathcal{N} = 1 supersymmetric theories, which are related to elliptic hypergeometric integrals. These include a new self-duality involving a quiver theory which exhibits a non-trivial global symmetry enhancement to E6.

Flat cotorsion modules over Noether algebras

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.

The defect recollement, the MacPherson-Vilonen construction, and pp formulas

For any abelian category A, Auslander constructed a localisation w:fp(Aop,Ab)→A called the defect, which is the left adjoint to the Yoneda embedding Y:A→fp(Aop,Ab). If A has enough projectives, then this localisation is part of a recollement called the defect recollement. We show that this recollement is an instance of the MacPherson-Vilonen construction if and only if A is hereditary. We also discuss several subcategories of fp(Aop,Ab) which arise as canonical features of the defect recollement, and characterise them by properties of their projective presentations and their orthogonality with other subcategories. We apply some parts of the defect recollement to the model theory of modules. Let R be a ring and let ϕ/ψ be a pp-pair. When R is an artin algebra, we show that there is a smallest pp formula ρ such that ψ⩽ρ⩽ϕ which agrees with ϕ on injectives, and that there is a largest pp formula μ such that ψ⩽μ⩽ϕ and ψR=μR. When R is left coherent, we show that there is a largest pp formula σ such that ψ⩽σ⩽ϕ which agrees with ψ on injectives, and that the pp-pair ψ/ϕ is isomorphic to a pp formula if and only if ψ=σ, and that there is a smallest pp formula ν such that ψ⩽ν⩽ϕ and ϕR=νR. We also show that, for any pp-pair ϕ/ψ, w(ϕ/ψ)≅(Dψ)R/(Dϕ)R, where D is the elementary duality of pp formulas. We also give expressions for w(ϕ/ψ) in terms of free realisation of ϕ and ψ.

K-decompositions and 3d gauge theories

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, ℂ)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space ℒ K (M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of ℒ K (M) as a Lagrangian subvariety in the symplectic moduli space $$ {\mathcal{X}}_K^{\mathrm{un}}\left(\partial M\right) $$ of framed flat connections on the boundary — and more so, as a “K2-Lagrangian,” meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of ℒ K (M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, ℂ)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of ℒ K (M) is K2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that ℒ K (M) is K2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement. Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d $$ \mathcal{N}=2 $$ superconformal field theories T K [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories T K [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N f = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T K [M] grow cubically in K.

Almost Dual Pairs and Definable Classes of Modules

In [7] Holm considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his results by placing them in the context of elementary duality on definable subcategories. In doing so we also prove that dual modules have enough indecomposable direct summands.

Linearization of isometric embeddings between Banach spaces: An elementary approach

Mazur-Ulam’s classical theorem states that any isometric onto map between normed spaces is linear. This result has been generalized by T. Figiel [F] who showed that ifΦ is an isometric embedding from a Banach space X to a Banach space Y such that Φ(0) = 0 and vect [φ(X )] = Y , there exists a linear quotient map Q such that ‖Q‖ = 1 and Q ◦Φ= I dX . The third chapter of this short story is [GK] where it is shown that if a quotient map Q from a Banach space Y onto a separable Banach space X has a Lipschitz lifting, then it actually has a continuous linear lifting. Combining this statement with Figiel’s theorem provides another result from [GK] : if a separable Banach space X isometrically embeds into a Banach space Y , there exists an isometric linear embedding from X into Y . Nigel Kalton had an outstanding ability to set theorems and proofs in their proper frame. His articles are therefore fountains of ideas, irrigating each of the many fields to which he contributed : a non exhaustive survey on these contributions is [G]. The paper [GK] is no exception to this rule, and among other things it prepared the ground for far-reaching extensions, where e.g. the Lipschitz assumption is weakened to the Holder condition, leading to very different conclusions ( [K1], [K2], [K3] ). However, some readers could find daunting some arguments from [GK], whichwould probably turn down undergraduate students. Since the extension from [GK] of Mazur-Ulam’s theorem which is recalled above has a statement which is understandable to any student in mathematics, it seems appropriate to provide an elementary proof. This is the purpose of the present short note, where only basic functional analysis (elementary duality theory) and calculus (culminating at Fubini’s theorem for continuous functions on Rn) are used. Moreover this note is fully self-contained : even the generic smoothness of convex fuctions on Rn is shown, and a detailed proof of Figiel’s theorem is provided (following [BL], although Lemma 2 is not stated there). Themain result of the note is Theorem 5, whose proof follows the strategy of the proof of ( [GK], Corollary 3.2) but in such an elementary way that diagrams, free spaces and infinitedimensional integration or differentiation arguments are avoided. Thus our approach makes it clear that the core of the proof consists of finite-dimensional considerations. We can therefore teach at an undergraduate level Mazur-Ulam’s theorem and its natural extensions from [F] and [GK].

Semidefinite Representation of Convex Hulls of Rational Varieties

Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension.

Tensor product and theories of modules

Modules over a ring R, when tensored with an (R, S)-bimodule, are converted to S-modules. Here I consider, from the standpoint of the model theory of modules, the effect of this operation. The primary motivation arises from questions concerning representation type of algebras and interpretability of modules, where such tensor functors play a key role, but this paper is devoted to more general considerations. For instance, the elementary duality of [2] and [1] is generalised here. It is also shown that, although tensor product does not preserve elementary equivalence, one can define the tensor product of two complete theories of R-modules. The results in Section 1 grew out of a number of discussions with T. Kucera.This is our generic situation. We have an (R, S)-bimodule B and we are considering the functor –⊗RBS from Mod – R (the category of right R-modules) to Mod – S which is given on objects by MR ↦ (M ⊗RB)s and has the obvious action on morphisms. There is a somewhat more general situation: namely we may consider the effect of tensoring (T, R)-bimodules over R with B to obtain (T, S)-bimodules. For the intended applications this case is not needed. Moreover, although some results extend to this more general case, there are some which definitely do not (see the example after 2.1). Therefore we confine ourselves to the case first described. We will also consider B itself as a variable and so ask “What is the effect of tensoring R-modules with (R, S)-bimodules?”.

An Axiomatic Description of a Duality for Modules

Take for instance a k-algebra 4 over a field k. Then the usual k-dual DM=Homk(M, k) of a finite dimensional 4-module M satisfies all three properties. However for infinite dimensional 4-modules the condition (D1) does not hold, and for most rings there is even on the level of the finite length modules no duality functor between right and left modules available. The aim of this note is to describe without any restriction on the ring 4 a bijection between certain classes of right and left 4-modules which satisfies (D1) (D3) and which is uniquely determined by these properties. This bijection covers the k-duality mentioned above but usually there will be non-finitely generated modules on which D is defined. Our approach gives a new interpretation of a construction which is known as elementary duality amongst model theorists, and which is due to Herzog [6], see also [9]. We formulate now the main result. To this end denote by Mod(4) the category of (right) 4-modules and by mod(4) the full subcategory of all finitely presented 4-modules. We shall identify the category of left 4-modules with Mod(4). Recall that a 4-module M is pure-injective if every pure monomorphism starting in M splits. A map .: M N is a pure monomorphism if . 4 X : M 4 X N 4 X is a monomorphism for every X in mod(4). We call an indecomposable pure-injective 4-module article no. AI971660

Extensions of elementary duality

Elementary duality of pp-formulas for modules is extended to certain infinitary formulas and theories. Examples are given indicating the limits of such an extension. Some applications to absolutely pure and flat modules are made. Also considered are the extension of elementary duality to abelian structures and the relationship of elementary duality with a duality defined on categories of finitely presented functors from finitely presented modules to abelian groups.

Elementary Duality of Modules

Elementary Duality of Modules

Remarks on elementary duality

Elementary duality between left and right modules over a ring, especially its interpretation in terms of the relevant functor categories, is discussed, as is the relationship between these categories of functors and sorts in theories of modules. A topology on the set of indecomposable pure-injective modules over a ring is introduced. This topology is dual to the Ziegler topology and may be seen as a generalisation of the Zariski topology.

Elementary duality of modules

Let R R be a ring. A formula φ ( x ) \varphi ({\mathbf {x}}) in the language of left R R -modules is called a positive primitive formula (ppf) if it is of the form ∃ y ( A B ) ( x y ) = 0 \exists {\mathbf {y}}\left ({AB} \right )\left (\begin {array}{*{20}{c}}x\\y\\\end {array} \right ) = 0 where A A and B B are matrices of appropriate size with entries in R R . We apply Prest’s notion of D φ ( x ) D\varphi ({\mathbf {x}}) , the ppf in the language of right R R -modules dual to φ \varphi , to show that the model theory of left R R -modules as developed by Ziegler [Z] is in some sense dual to the model theory of right R R -modules. We prove that the topologies on the left and right Ziegler spectra are "isomorphic" (Proposition 4.4). When the lattice of ppfs is well behaved, there is a homeomorphism D D between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left R R -module U U the dual pure-injective indecomposable right R R -module D U DU . Theorem 6.6 asserts that given a complete theory T T of left R R -modules, there is a dual complete theory D T DT of right R R -modules with corresponding Baur-Garavaglia-Monk invariants. In the end, we give some conditions on a pure-injective indecomposable R U _RU which ensure that its dual D U DU may be represented as a hom set of the form Hom S ( R U S , E S ) {\operatorname {Hom}_S}{(_R}{U_S},{E_S}) where S S is some ring making R U S _R{U_S} into a bimodule and E S {E_S} is injective.