Direct Sum Decomposition Research Articles

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Abstract A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

The Common Basis Complex and the Partial Decomposition Poset

Abstract For a finite-dimensional vector space $V$, the common basis complex of $V$ is the simplicial complex whose vertices are the proper non-zero subspaces of $V$, and $\sigma $ is a simplex if and only if there exists a basis $B$ of $V$ that contains a basis of $S$ for all $S\in \sigma $. This complex was introduced by Rognes in 1992 in connection with stable buildings. In this article, we prove that the common basis complex is homotopy equivalent to the proper part of the poset of partial direct sum decompositions of $V$. Moreover, we establish this result in a more general combinatorial context, including the case of free groups, matroids, vector spaces with non-degenerate sesquilinear forms, and free modules over commutative Hermite rings, such as local rings or Dedekind domains.

Study on switching from position to stiffness control for musculoskeletal-inspired robot arm based on direct sum decomposition

Study on switching from position to stiffness control for musculoskeletal-inspired robot arm based on direct sum decomposition

External Direct Sum Invariant Subspace and Decomposition of Coupled Differential-Difference Equations

This article discusses the invariant subspaces that are restricted to be external direct sums. Some existence conditions are presented that facilitate finding such invariant subspaces. This problem is related to the decomposition of coupled differential-difference equations, leading to the possibility of lowering the dimensions of coupled differential-difference equations. As has been well documented, lowering the dimension of coupled differential-difference equations can drastically reduce the computational time needed in stability analysis when a complete quadratic Lyapunov-Krasovskii functional is used. Most known ad hoc methods of reducing the order are special cases of this formulation.

Comodule theories in Grothendieck categories and relative Hopf objects

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category S. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category SHA of relative (A,H)-Hopf modules in S, where H is a Hopf algebra and A is a right H-comodule algebra. We study the cohomological theory in SHA by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in SHA. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in SHA.

Isometries, direct sum decompositions, analytic type function spaces, and radial operators

The paper connects rather general statements about the direct sum decomposition of a Hilbert space generated by isometries with the study of analytic type subspaces in weighted L_2-spaces over the unit disk {mathbb {D}} or the complex plane {mathbb {C}}, and with characterization of the so-called radial operators acting on these spaces. We characterize the properties of the so-called (m, n)-analytic functions, those that are annihilated by frac{partial ^m }{partial z^m}frac{partial ^n }{partial {{overline{z}}}^n}, which as particular cases include analytic, anti-analytic, harmonic, and various related to them functions. We characterize also the so-called radial operators, with a particular attention to Toeplitz ones, that act on different subspaces of our weighted L_2-spaces.

Measurement of Angle Between Subspaces in Direct Sum Decomposition

The problems of Eigen structure assignment has been studied. As an immediate part of that, the problem addressed is the development of measures of “skewness” between subspaces which are in the direct sum decomposition of the state space and a concept of angle between a set of subspaces.

Conjugate variables approach tomixedq-Araki–Woods algebras: Factoriality and non-injectivity

We establish factoriality and non-injectivity in full generality for the mixed q-Araki–Woods von Neumann algebra associated to a separable real Hilbert space HR with dimHR≥2, a strongly continuous one parameter group of orthogonal transformations on HR, a direct sum decomposition HR=⊕iHR(i), and a real symmetric matrix (qij) with supi,j|qij| < 1. This is achieved by first proving the existence of conjugate variables for a finite number of generators of the algebras (when dimHR<∞), following the lines of Miyagawa–Speicher and Kumar–Skalski–Wasilewski. The conjugate variables belong to the factors in question and satisfy certain Lipschitz conditions.

Adjacency matrices over a finite prime field and their direct sum decompositions

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field Fp. We apply symmetric (row and column) elementary transformations to the adjacency matrix over Fp in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over Fp for any odd prime p. Our key tool is quadratic residues of Fp.

Reflexive homology

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

An elementary result on infinite and finite direct sums of modules

Let R be a ring, and consider an R-module M given with two (generally infinite) direct sum decompositions, A⊕(⨁i∈ICi)=M=B⊕(⨁j∈JDj), such that the submodules A and B, and the Dj, are all finitely generated. We show that there then exist finite subsets I0⊆I, J0⊆J, and a direct summand Y⊆⨁i∈I0Ci, such that A⊕Y=B⊕(⨁j∈J0Dj).We then note some ways that this result can and cannot be generalized, and pose some related questions.

Centers of multilinear forms and applications

The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.

Minimal encoders of convolutional codes with general cyclic structures

Roos’ direct sum decomposition of a cyclic convolutional code based upon a suitable construction of a minimal encoder is extended to a broader class of convolutional codes endowed with additional algebraic structure. The two (equivalent) usual concepts of a convolutional code are taken into account, as a vector space and as a module.

Ideals, bands and direct sum decompositions in mixed lattice vector spaces

A mixed lattice vector space is a partially ordered vector space with two partial orderings and certain lattice-type properties. In this paper we first give some fundamental results in mixed lattice groups, and then we investigate the structure theory of mixed lattice vector spaces, which can be viewed as a generalization of the theory of Riesz spaces. More specifically, we study the properties of ideals and bands in mixed lattice spaces, and the related idea of representing a mixed lattice space as a direct sum of disjoint bands. Under certain conditions, these decompositions can also be given in terms of order projections.

Two criteria for locally noetherian Grothendieck categories

We give two criteria for a Grothendieck category to be locally noetherian. The first of these is in terms of direct sum decompositions into uniform components. The second is related to the existence of finitely injective objects that cannot be expressed as a direct sum of injectives.

Krull-Remak-Schmidt decompositions in Hom-finite additive categories

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A Hom-finite category is an additive category A for which there is a commutative unital ring k, such that each Hom-set in A is a finite length k-module. The aim of this note is to provide a proof that a Hom-finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.

Tridiagonal pairs, alternating elements, and distance-regular graphs

The positive part Uq+ of Uq(slˆ2) has a presentation with two generators W0, W1 and two relations called the q-Serre relations. The algebra Uq+ contains some elements, said to be alternating. There are four kinds of alternating elements, denoted {W−k}k∈N, {Wk+1}k∈N, {Gk+1}k∈N, {G˜k+1}k∈N. The alternating elements of each kind mutually commute. A tridiagonal pair is an ordered pair of diagonalizable linear maps A,A⁎ on a nonzero, finite-dimensional vector space V, that each act in a (block) tridiagonal fashion on the eigenspaces of the other one. Let A, A⁎ denote a tridiagonal pair on V. Associated with this pair are six well-known direct sum decompositions of V; these are the eigenspace decompositions of A and A⁎, along with four decompositions of V that are often called split. In our main results, we assume that A, A⁎ has q-Serre type. Under this assumption A, A⁎ satisfy the q-Serre relations, and V becomes an irreducible Uq+-module on which W0=A and W1=A⁎. We describe how the alternating elements of Uq+ act on the above six decompositions of V. We show that for each decomposition, every alternating element acts in either a (block) diagonal, (block) upper bidiagonal, (block) lower bidiagonal, or (block) tridiagonal fashion. We investigate two special cases in detail. In the first case the eigenspaces of A and A⁎ all have dimension one. In the second case A and A⁎ are obtained by adjusting the adjacency matrix and a dual adjacency matrix of a distance-regular graph that has classical parameters and is formally self-dual.

Generalized lattices over one-dimensional noetherian domains

We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Drawing inspiration from the representation theory of orders in separable algebras, we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.

Quaternionic exponentially dichotomous operators through S-spectral splitting and applications to Cauchy problem

In this paper, based on the slice regular quaternionic functions and spectral theory on the S-spectrum for quaternionic operators, firstly, the exponential growth bound of quaternionic operator semigroups is studied and the generalized Hille Lemma is obtained through introducing Bochner and Pettis integrals in quaternionic Banach space. Secondly, we study the quaternionic bisemigroups and the direct sum decomposition of quaternionic Banach space, then we introduce the notion of the quaternionic exponentially dichotomous operator and obtain its integral representation formula. It is crucial to note that the quaternionic operators we consider do not necessarily commute. Meanwhile, through establishing the S-spectral splitting theorem, the characterizations of quaternionic exponentially dichotomous operators are demonstrated. Moreover, the slice regular quaternionic bisemigroups generated by the quaternionic bisectorial operator are studied. Thirdly, by introducing the slice quaternionic Banach algebra and multiplicative quaternionic linear functionals, the perturbation invariance of exponential dichotomy for the quaternionic operators is explored. Under the noncommutative setting, our obtained results are essentially different from their analogues in the complex setting. As applications, we consider the Cauchy problems of quaternionic non-homogeneous evolution equations involving the quaternionic semigroup generators and exponentially dichotomous quaternionic operators. Additionally, the slice exponential dichotomy of quaternionic operators is characterized via the corresponding Cauchy problem.

Eigenspace Decomposition of Mixed Hodge Structures on Alexander Modules

Abstract In previous work jointly with Geske, Maxim, and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of (one variable) Alexander modules, which generalizes the MHS on the cohomology of the Milnor fiber for weighted homogeneous polynomials. The cohomology of a Milnor fiber carries a monodromy action, whose semisimple part is an isomorphism of MHS. The natural question of whether this result still holds for Alexander modules was then posed. In this paper, we give a positive answer to that question, which implies that the direct sum decomposition of the torsion part of Alexander modules into generalized eigenspaces is in fact a decomposition of MHS. We also show that the MHS on the generalized eigenspace of eigenvalue $1$ can be constructed without passing to a suitable finite cover (as is the case for the MHS on the torsion part of the Alexander modules), and compute it under some purity assumptions on the base space. Further, we show a formula relating the Alexander module’s Hodge numbers to those of finite covers of the base space, under some assumptions.Dedicated to the memory of Georgia Benkart