Krull-Schmidt Theorem Research Articles

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration \({\cal X}\) is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of \({\cal X}\) comes from a certain Cartesian decomposition. It is proved that if the coherent configuration \({\cal X}\) is thick, then there is a unique maximal Cartesian decomposition of \({\cal X}\), i.e., there is exactly one internal tensor decomposition of \({\cal X}\) into indecomposable components. In particular, this implies an analog of the Krull-Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.

UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS

In this article, we present the classical Krull-Schmidt Theorem for groups, its statement for modules due to Azumaya, and much more modern variations on the theme, like the so-called weak Krull-Schmidt Theorem, which holds for some particular classes of modules. Also, direct product of modules is considered. We present some properties of the category of G-groups, a category in which Remak's results about the Krull-Schmidt Theorem for groups can be better understood. In the last section, direct-sum decompositions and factorisations in other algebraic structures are considered.

Several Generalizations of the Wedderburn-Artin Theorem with Applications

We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) $R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})$ where k,n 1,…,n k ∈ ℕ and each D i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m 1 ⊕… ⊕ R m k where each R m i is either a simple R-module or a virtually simple direct summand of R.

Rationality Problem for Algebraic Tori

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq {\rm GL}(n,\mathbb{Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.

The Krull-Schmidt Theorem Holds for Finite Direct Products of Biuniform Groups

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups $\mathcal {N}(G)$ has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.

A Krull–Schmidt theorem for infinite products of modules

We prove a unique decomposition theorem for direct products of finitely generated modules over certain classes of rings, which is analogous to the classical Krull–Schmidt–Remak–Azumaya theorem for direct-sum decompositions of modules.

Automorphisms of real Lie algebras of dimension five or less

The Lie algebra version of the Krull–Schmidt theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. For finite-dimensional Lie algebras, there is a well-known algorithm for finding such components, so the theorem considerably simplifies the problem of classifying the automorphism groups. We illustrate this by classifying the automorphisms of all indecomposable real Lie algebras of dimension five or less. Our results are presented very concisely, in tabular form.

Direct-sum decompositions of modules with semilocal endomorphism rings

According to the classical Krull–Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J(E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of uniserial modules are described via their monogeny classes and their epigeny classes up to two permutations of the factors.

Some structure theory of table algebras and applications to association schemes

In this paper we prove some basic structure theorems for arbitrary table algebras. In particular, we prove a Krull–Schmidt theorem for arbitrary table algebras, and give applications to association schemes.

KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

The Krull-Schmidt Theorem in the Case Two

We study what happens if, in the Krull-Schmidt Theorem, instead of considering modules whose endomorphism rings have one maximal ideal, we consider modules whose endomorphism rings have two maximal ideals. If a ring has exactly two maximal right ideals, then the two maximal right ideals are necessarily two-sided. We call such a ring of type 2. The behavior of direct sums of finitely many modules whose endomorphism rings have type 2 is completely described by a graph whose connected components are either complete graphs or complete bipartite graphs. The vertices of the graphs are ideals in a suitable full subcategory of Mod-R. The edges are isomorphism classes of modules. The complete bipartite graphs give rise to a behavior described by a Weak Krull-Schmidt Theorem. Such a behavior had been previously studied for the classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, kernels of morphisms between indecomposable injective modules, and couniformly presented modules. All these modules have endomorphism rings that are either local or of type 2. Here we present a general theory that includes all these cases.

$${\mathsf{N}}$$ -Complexes as Functors, Amplitude Cohomology and Fusion Rules

We consider \({\mathsf{N}}\) -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated \({\mathsf{N}}\) -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive \({\mathsf{N}}\) -complexes is proved to be isomorphic to an Ext functor of an indecomposable \({\mathsf{N}}\) -complex inside the abelian functor category. Finally we show that for the monoidal structure of \({\mathsf{N}}\) -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for \({\mathsf{N}}\) -complexes can be determined.

Representations of additive categories and direct-sum decompositions of objects

The aim of this paper is to embed additive categories in which direct-sum decompositions into indecomposables are not unique but have a regular geometric behavior into categories in which the Krull-Schmidt Theorem holds, that is, to give a representation of additive categories into categories with unique direct-sum decompositions into indecomposables. Cf. Theorems 4.8, 6.1, 6.2, 7.2, and 8.2.

Geometric regularity of direct-sum decompositions in some classes of modules

In this paper, we show that modules with semilocal endomorphism rings appear in abundance in applications, that their direct-sum decompositions are described by the so-called Krull monoids, and that this implies a geometric regularity of the direct-sum decompositions of these modules. Their direct-sum decompositions into indecomposables are not necessarily unique in the sense of the Krull-Schmidt theorem. The application of the theory of Krull monoids to the study of direct-sum decompositions of modules has been developed during the last five years. After a quick survey of the results obtained in this direction, we concentrate in particular on the abundance of examples. At present, these examples are scattered in the literature, and we try to collect them in a systematic way.

Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory \({\mathcal M}(G,R)\) of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that \({\mathcal M}(G,R)\) is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in \({\mathcal M}(G,R).\) We prove that the Krull--Schmidt theorem holds in many cases.

A Krull–Schmidt theorem for one-dimensional rings of finite Cohen–Macaulay type

This paper determines when the Krull–Schmidt property holds for all finitely generated modules and for maximal Cohen–Macaulay modules over one-dimensional local rings with finite Cohen–Macaulay type. We classify all maximal Cohen–Macaulay modules over these rings, beginning with the complete rings where the Krull–Schmidt property is known to hold. We are then able to determine when the Krull–Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen–Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.

A Version of the Weak Krull–Schmidt Theorem for Infinite Direct Sums of Uniserial Modules

We show a version of the weak Krull–Schmidt theorem concerning infinite families of uniserial modules.

Failure of Krull-Schmidt for invertible lattices over a discrete valuation ring

Let p p be a prime greater than 3 3 , and let N N be the semi-direct product of a group H H of order p p by a cyclic C C group of order p − 1 p-1 , which acts faithfully on H H . Let R R be the localization of Z Z at p p . We show that the Krull-Schmidt Theorem fails for the category of invertible R N RN -lattices.

On uniserial modules that are not quasi-small

We prove ℵ 0 th root property for uniserial modules and we show that the Krull–Schmidt theorem holds for direct sums of uniserial modules that are not quasi-small.

Quasi-decompositions for Self-Small Abelian Groups#

We will prove a Krull-Schmidt type theorem for the quasi decompositions of a self-small abelian group of finite torsion free rank, which extends the classical result proved for finite rank torsion free abelian groups.