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Completely decomposable summands of almost completely decomposable groups
Abstract
Almost completely decomposable groups are torsion-free finite extensions of completely decomposable groups of finite rank. We answer completely and in a constructive fashion the question when an almost completely decomposable group has non-zero completely decomposable direct summands. A new invariant, the rank-width difference of X at τ, given by $$rw{d_\tau }\left( X \right) = rk\frac{{X\left( \tau \right)}} {{{X^\# }\left( \tau \right)}} - width\frac{{{X^\# }\left[ \tau \right]}} {{X\left( \tau \right) + X\left( \tau \right)}}$$ is the exact rank of a maximal τ-homogeneous (completely decomposable) direct summand of X. We show that the rank-width difference can be effectively computed for an almost completely decomposable group given in “standard description” X = A + f ℕN
-1 αl provided that A is regulating in X. We also establish an algorithmic criterion for deciding whether an almost completely decomposable group given in standard description is completely decomposable.
... For the most part, we adopt the notation of [7]. The definitions and properties of almost completely decomposable, rigid and block-rigid cyclic regulator quotient groups can be found in [10]. The properties we need are described in Section 2. ...
... To our knowledge all published examples of groups with "pathological" decompositions are almost completely decomposable groups. The essential simplification occurs by passing from isomorphism to Lady's near-isomorphism ( [9], [10,Chapter 9]). A theorem by David Arnold ([2, Corollary 12.9], [10, Theorem 10.2.5]) says that passing from isomorphism to the coarser equivalence relation of Lady does not mean a loss of generality for our line of questioning: If G and G are nearly isomorphic groups, then P(G) = P(G ). ...
... We say that G is nearly isomorphic to H, denoted G ∼ =nr H, if there is a group K such that G ⊕ K ∼ = H ⊕ K. The properties of near isomorphism are described in [2,Section 12] and in [10,Chapter 10]. In our context, the most important such property is [2, Corollary 12.9], [10, Theorem 10.2.5]: for any groups G and H, if G ∼ =nr H and G = G 1 ⊕ · · · ⊕ G t , then H = H 1 ⊕ · · · ⊕ H t with H i ∼ =nr G i , in particular P(G) = P(H). ...
An indecomposable decomposition of a torsion-free abelian group $G$ of rank $n$ is a decomposition $G=A_1\oplus\cdots\oplus A_t$ where $A_i$ is indecomposable of rank $r_i$ so that $\sum_i r_i=n$ is a partition of $n$. The group $G$ may have decompositions that result in different partitions of $n$. We address the problem of characterising those sets of partitions of $n$ which can arise from indecomposable decompositions of a torsion-free abelian group.
... r ′ ℓ respectively? The second problem of Fuchs was solved by Blagoveshchenskaya, [20,Theorem 13.1.19] for a restricted class C of groups: let P and Q be partitions of n. ...
... A more general question was settled by Lee Lady for almost completely decomposable groups (defined below) [17,Corollary 7], [20,Theorem 9.2.7]. A group G is clipped if it has no direct summands of rank 1. Lady's "Main Decomposition Theorem" says that every almost completely decomposable group G has a decomposition G = G cd ⊕ G cl where G cd is completely decomposable, G cl is clipped, G cd is unique up to isomorphism, and G cl is unique up to near-isomorphism. ...
... Near isomorphism is a weakening of isomorphism due to Lady [18]. There are several equivalent definitions, see for example [20,Chapter 9], the most useful one for us being that a group A is nearly isomorphic to ...
Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.
... A torsion-free abelian group G of finite rank is completely decomposable if G is isomorphic to a finite direct sum of subgroups of Q, the additive group of rational numbers, and almost completely decomposable if G has a completely decomposable subgroup C with G/C a finite group. Almost completely decomposable groups are a notoriously complicated class of torsion-free abelian groups of finite rank, [9,1,10], the source of many pathological examples, [8], and have been generalized to infinite rank, [12]. ...
... Given a finite partially ordered set S of types and an integer m 1, an S-group with p m -regulator quotient is an almost completely decomposable group G with critical typeset T cr (G) ⊆ S and p m G ⊆ R(G), the regulator subgroup of G, e.g. see [14,15,11,13,6,10]. Let w(S) denote the width of S, the length of a maximal antichain contained in S. Define Sgroups with p m -regulator quotients to have unbounded representation type if there are indecomposable S-groups with p m -regulator quotients of arbitrarily large finite rank. ...
... Define cdrep(S, Z p m ) to be the collection of objects U = (U 0 , U s , U * : s ∈ S) such that for each s ∈ S, there is a finitely generated free Z p m -module V s with U 0 = s∈S V s , U s = s t∈S V t , U t ⊆ U s whenever s t (note the reversal of the order), and U * a submodule of U 0 with U s ∩ U * = 0 for each s ∈ S. Notice that U * is finitely generated but need not be a free Z p m -module. An object U of cdrep(S, Z p m ) is called an anti-representation in [10]. ...
Almost completely decomposable groups with a regulating regulator and a p-primary regulator quotient are studied. It is shown that there are indecomposable such groups of arbitrarily large rank provided that the critical typeset contains some basic configuration and the exponent of the regulator quotient is sufficiently large.
... These embeddings generalize more complicated embeddings given in [7] and are used to construct examples of indecomposable modules in AH of arbitrarily large finite rank (Example 4) and to determine the representation type of AH(S, j ) in case S is an antichain (Corollary 3). Just as for almost completely decomposable abelian groups and regulator subgroups [16], direct sum decompositions and classifications of special classes of modules M in AH can be found by exploiting the pi-decomposable modules D M and p j D M and their characterizations in terms of ranks of quotients of radicals. Also, many of the results in this paper can be dualized by considering the co-purely indecomposable modules [1]. ...
... ✷ The τ -socle of M ∈ T F for τ = [X] ∈ PI is defined to be M(τ ) = Hom(X, M)X, the image of the evaluation homomorphism Hom(X, M) ⊗ R X → M. As a consequence of Lemma 2(d), if M is reduced, then Hom(X, M) = 0 (equivalently M(τ ) = 0) if and only if X is isomorphic to a pure submodule of M. Let M # (τ ) be the pure submodule of M generated by {M(σ ): τ < σ ∈ PI}. By comparison with completely decomposable abelian groups [16], one might expect that pi-decomposable modules would also be characterized by the ranks of the M(τ )/M # (τ )'s. However, it need not even be the case that M # (τ ) ⊆ M(τ ). ...
... These are the finite rank modules M for which there is a pi-decomposable module C with M ⊆ C and C/M bounded torsion and finitely generated (since M has finite rank). It turns out that these modules are perfect analogues of the almost completely decomposable abelian groups (acd groups) and we shall state some of their salient properties without proofs since they are easy modifications of the proofs of the analogous results for acd groups [4,16,17]. Correspondences between subcategories of AH and categories of R-representations of finite partially ordered sets are defined and used in constructing indecomposable modules of arbitrary finite rank in AH. Since R is a discrete valuation ring, a finitely generated bounded R-module T is a direct sum of torsion cyclic modules, say Proof. ...
Finite direct sums of finite rank purely indecomposable modules (called pi-decomposable modules) are characterized by the ranks of quotients of radicals. Torsion-free homomorphic images of pi-decomposable modules form an interesting class admitting a nice homological characterisation. Torsion-free modules quasi-isomorphic to a pi-decomposable module have properties reminiscent of almost completely decomposable abelian groups, including characterizations of representation type via correspondences with representations of finite partially ordered sets. This leads to the construction of indecomposable modules.
... The regulating subgroups can be defined as the completely decomposable subgroups of least index in an almost completely decomposable group. Details on the subsequent developments can be found in the survey article [12] or the monograph [13]. ...
... By a rational group we mean a subgroup of the additive group Q that contains Z. It is well known ( [2,13,Chapter 11] or Lemma 4.1) that any almost completely decomposable group can be specified by a list of rational groups [σ 1 , . . . , σ r ] and a pair of matrices (N, M) where N is a non-singular integral k × k-matrix, the structure matrix, and M is a k × r integral matrix, the coordinate matrix. ...
... We take it for granted that the reader is familiar with the usual type subgroups G(τ ), G * (τ ), and G (τ ) = G * (τ ) G * . If A is a completely decomposable group, then A = ρ∈T cr (A) A ρ is assumed to be a decomposition of A into ρ-homogeneous components A ρ = 0, so that T cr (A) is the critical typeset of A. For background on torsion-free abelian groups and almost completely decomposable groups, see [1,12,13]. Maps are written on the right. The set inclusion symbol ⊂ allows equality. ...
Based on a constructive proof of a theorem of Dittmann, an algorithm is developed for finding a regulating subgroup of an almost completely decomposable group given in standard description. The algorithm is implemented in the Maple 8 computer algebra system and involves properties and manipulations of matrices with integer entries.
... The regulator is a fully invariant subgroup. Burkhardt [5] showed that for any regulating subgroup A = ρ∈T cr (X) A ρ of the almost completely decomposable group X, the regulator is obtained in the form R(X) = ρ∈T cr (X) β X ρ A ρ where the "Burkhardt invariants" β X τ = exp(X(τ )/R(X(τ ))) depend only on X and not on the particular regulating subgroup A. See [10] for proofs and other background material. ...
... We begin with a short proof of a useful lemma on abelian groups that will enable us to give a new proof of a theorem of Lady on regulating indices, and to prove its analogue for co-regulating indices. [10,Exercise 16.8.8].) (1), with A changed to B + C, and the splitting follows as before. 2 ...
... We will need a result of Lady for which we provide a new proof. [9] or [10,Theorem 4.2.13.2] There is a "dual" proof showing that the co-regulating index divides any finite index of an almost completely decomposable group in a completely decomposable group. ...
We establish a duality for near-isomorphism categories of almost completely decomposable groups through a combination of Butler duality and Warfield duality. We are then able to dualize results in the literature to obtain classifications and structure theories for new classes of almost completely decomposable groups. For example, known results on p-local crq-groups, those groups X that contain a completely decomposable subgroup B with X/B a cyclic p-group, can be dualized to results on p-local co-crq groups, those groups Y that embed in a completely decomposable group B so that the quotient B/Y is a cyclic p-group. A variety of other applications is given.
... The group G is indecomposable if and only if G is nearly isomorphic to an indecomposable group, [1]. Moreover, an almost completely decomposable G with G/ R(G) p-primary is, up to near-isomorphism, uniquely a direct sum of indecomposable groups, [13], [16,Corollary 10.4.6]. Consequently, a classification of all indecomposable (S, p k )-groups up to near isomorphism results in a classification of all (S, p k )-groups up to near isomorphism. ...
... The following terminology is used in this paper. Details, equivalent formulations, and confirmation of assertions can be found in [1] or [16]. Let G be an almost completely decomposable group. ...
The class of almost completely decomposable groups with a critical typeset of type (1,4) and a homocyclic regulator quotient of exponent p(3) is shown to be of bounded representation type. There are precisely four near-isomorphism classes of indecomposables, all of rank 6.
... If G is a strongly indecomposable bracket group then it is known that End(G) ⊂ Q is a pid. See [2,13,15]. ...
... We may then assume that End(G) ⊂ End(G), [13]. Then G is a balanced semi-primary rtffr group. ...
LetE(G)=End(G)/N(End(G)). Our goal in this paper is to study direct sum decompositions of certain reduced torsion-free finite rank (rtffr) abelian groups by introducing an ideal τ of E(G) called a conductor of G. This ideal induces a natural ring decomposition E(G)=E(G)(τ)×E(G)τ and a natural direct sum decomposition G=G(τ)⊕Gτ for an rtffr group G.Let {G1,…,Gt} be a set of strongly indecomposable rtffr groups such that Gi≇˙Gj for each i≠j∈{1,…,t}, and such that E(Gi) is a Dedekind domain for each i∈{1,…,t}. Let n1,…,nt>0 be integers and letG¯=G1n1⊕⋯⊕Gtnt. We say that G has semi-primary index inG¯ if for each i=1,…,t there is a primary ideal Pi⊂End(Gi) such thatP1G1n1⊕⋯⊕PtGtnt⊂G⊂G¯. The group G is balanced inG¯ if G⊂G¯ and if E(G)⊂E(G¯). We say that G is a balanced semi-primary group if there is a balanced embedding G⊂G¯ such that G has semi-primary index in G¯. TheoremIf G is a balanced semi-primary rtffr group then G has a locally unique decomposition and the local refinement property.
... 4. If B is a regulating subgroup of X τ , X * τ , or X τ , then there exists a regulating subgroup A of X such that A τ = B, A * τ = B, or A τ = B respectively. 5. T cr X τ = σ ∈ T cr X σ ≥ τ and T cr X * τ = T cr X τ = σ ∈ T cr X σ > τ . 6. Let S be any subset of T cr X , let X S = σ∈S X σ and A S = σ∈S A σ . ...
... As in the finite rank case (see [6,Proposition 3.4]) we have the following result. ...
... When the regulating index is a prime power and the group is clipped, this property is equivalent to indecomposability ([3],[7]). Blagoveshchenskaya–Mader [1] studied decompositions of crq–groups whose critical typesets are antichains. Such groups have a single regulating subgroup and hence the quotients modulo regulating subgroups are all cyclic. ...
... Such groups have a single regulating subgroup and hence the quotients modulo regulating subgroups are all cyclic. Examples in [1] show that such groups need neither be indecomposable nor clipped. We will have to deal with different regulating subgroups A, B, . . . in this section, and several characteristics χ A n , χ B n , . . . ...
... As an application we show that an almost completely decomposable group whose typeset is a garland is a direct sum of rankone and rank-two groups (Theorem 3.14). Details on the theory of almost completely decomposable groups can be found in the monograph [5]. Rather than citing the original sources we will quote [5], which contains an extensive bibliography. ...
... Details on the theory of almost completely decomposable groups can be found in the monograph [5]. Rather than citing the original sources we will quote [5], which contains an extensive bibliography. ...
A generalized and canonical definition of "sharp type" is given and a decomposition theorem is proved for arbitrary almost completely decomposable groups. As an application we show that an almost completely decomposable group whose critical typeset is a garland is a direct sum of rank-one and rank-two groups.
... Let us recall the Hermite normal form of an integer matrix with nonzero determinant (see e.g. [Mad00]). ...
We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out that the structure of these modules depends in a crucial way on the properties of the function spaces. Moreover, for a class of subalgebras related to compact manifolds, we provide a construction procedure for the corresponding fuzzy spaces, i.e. sequences of finite dimensional modules of increasing dimension as the deformation parameter tends to zero, as well as infinite dimensional modules related to fuzzy non-compact spaces.
... Theorem 4. 12. Let G be a fully factorable compact group and G D H K. Then the factor H is fully factorable. ...
A protorus is a compact connected abelian group of finite dimension. We use a result on finite-rank torsion-free abelian groups and Pontryagin duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus free complementary factor. We also classify direct products of protori of dimension 1 by means of canonical “type” subgroups. In addition, we produce the duals of some fundamental theorems of discrete abelian groups.
... M.C.R. Butler [3] defined a finite rank Butler group as a pure subgroup or equivalently as an epimorphic image of an abelian completely decomposable group of finite rank. In the theory of Abelian groups the diversity of finite rank Butler groups has played an important role and has attracted much attention (see [1] or [9]). However, an attempt by Bican and Salce [2] to extend Butler's notion to infinite rank torsion-free abelian groups led to competing definitions. ...
Finite rank Butler groups are pure subgroups of completely decomposable groups of finite rank and were defined by M.C.R. Butler. Extending this concept to infinite rank groups, Bican and Salce gave various possible descriptions: A B2-group G is a union of an ascending chain of pure subgroups Gα such that for every α we have Gα+1 = G α+Hα for some finite rank Butler group Ha. A B1-group is a torsion-free group G satisfying Bext 1/ℤ(G,T) = 0 for all torsion groups T. While the class of B2-groups is contained in the class of B1-groups, it is in general undecidable in ZFC if the two classes coincide. In this paper we study the endomorphism rings of B 1-groups which are not B2-groups working in a model of ZFC that satisfies 2N0 = N4.
... Except of the motivation presented above, one of our main motivations for the study is the fact that the category fspr(I, F m ) is playing an important role in the representation theory of finitedimensional algebras (see [31]), in the study of lattices over orders (see [29], [31,Chapter 13], [34,35,[38][39][40][41]47,48]) and in the investigation of categories of abelian groups (see [1][2][3]6,7,17,21]). Some application of our results presented here can be found in a recent papers [4,5]. ...
We determine the representation type (wild, tame, polynomial growth) of the category fspr(I,Fm)fspr(I,Fm) of filtered subprojective FmFm-representations of a finite poset I in terms of m and I, where Fm=K[t]/(tm)Fm=K[t]/(tm), m⩾1m⩾1, and K is an algebraically closed field. Criteria for tameness, wildness and tameness of non-polynomial growth of fspr(I,Fm)fspr(I,Fm) are given in Theorems 1.1 and 1.2. As an application, a solution of Birkhoff's type problem [G. Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. 38 (1934) 385–401] for the category repft(I,Fm)repft(I,Fm) of filtered I-chains of FmFm-modules is given in Section 5, by determining the representation type repft(I,Fm)repft(I,Fm).
... Without loss of generality, we can write X = A + * Z N −1 Mv such that V = {v 1 ; : : : ; v r } is a p-basis of A, and N = diag(p d1 ; : : : ; p d k ) with 1 ≤ d 1 ≤ · · · ≤ d k . Since X is clipped it follows from [15,Lemma 3:7] that rkA i = k. We now assume that the basis elements v i are so listed that the ÿrst k form a p-basis of ...
Almost completely decomposable groups can be described in terms of integral matrices and in terms of anti-representations in finite modules over proper quotient rings of the ring of integers. The anti-representations are described by the so-called representating matrices. Representing matrices and their interrelationship with the integral matrices describing a group are studied in general. It is shown that the integral and representing matrices may be assumed to have a special form. Two applications demonstrate the usefulness of the results.
This review paper is a continuation of two previous review papers devoted to properties of modules over discrete valuation domains. The first part of this work was published in the volumes 121 (Part I, Chaps. 1–4, Secs. 1–22) and 122 (Part II, Chaps. 5–8, Secs. 23–39) of the series Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory (English translation: Journal of Mathematical Sciences (New York), vol. 145, No. 4, pp. 4997–5117 (2007), and vol. 151, No. 5, pp. 3225–3371 (2008)). In this review paper, we preserve the numeration of chapters and sections of parts I and II. The first part consists of Chapter 9 “Appendix,” Secs. 40–42.
In Sec. 40, we consider p-adic torsion-free modules with isomorphic automorphism groups. Section 41 is devoted to torsion-free modules over a complete discrete valuation domain with isomorphic radicals of their endomorphism rings.
The volume of the paper does not allow us to provide the proofs of all the results that appeared after publication of the previous parts and directly related to the issues under consideration in it. In the final Sec. 42, we describe some of these new results.
The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group {G/(\phi-1_{G})(G)} , and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G . In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called {R_{\infty}} -property. We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the {R_{\infty}} -property.
For a prime p and a poset of types, p-reduced almost completely decomposable groups with critical typeset and a p-power as regulating index are called -groups. The number of near-isomorphism types of indecomposable -groups depends on the exponent of the regulator quotient. It is shown that indecomposable -groups with a regulator quotient of exponent have rank ⩽4, and if the types and the prime p are fixed, then there are precisely four near-isomorphism types of indecomposable groups. It is unknown for which exponent of the regulator quotient exist infinitely many near-isomorphism types of indecomposable -groups.
The goal of this paper is to present a comprehensive survey of recent results addressing the question for which Abelian groups G the func-tors Hom(G, −), Hom(−, G), Ext(G, −) or Ext(−, G) preserve or invert direct products or sums.
We study a class of Butler groups of infinite rank, called Hawaiian groups.They are defined as subgroups of a rational vector space and contain parameters that provide for flexibility but are concrete enough to allow for the computation of certain crucial subgroups and quotient groups, to exhibit endomorphisms and describe the endomorphism
rings. Most Hawaiian groups are finitely Butler; under stronger assumptions they are not finitely filtered and hence not $B_2$-groups.
Uniform groups are extensions of rigid completely decomposable groups by a finite direct sum of cyclic primary groups all of the same order. The direct decompositions of uniform groups are completely determined by an algorithm that is realised by a MAPLE procedure.
In an earlier paper the authors introduced a K0-like construction that produces, for each torsion-free abelian group A of finite rank, a finitely generated abelian group G(A). In this note, we show that for any finite abelian group S, there is an almost completely decomposable (acd) group A such that G(A) has torsion subgroup isomorphic to S. In addition, if S is a finitely generated abelian group satisfying a certain condition on the torsion-free rank, then there is an almost completely decomposable group A such that G(A)≃S. In the usual K0 construction for acd groups, one always obtains a trivial torsion subgroup. Thus, G(A) appears to be a more versatile tool than K0 for the study of acd's.
The torsion-free abelian groups of finite rank are the additive subgroups of finite-dimensional vector spaces over the field ℚ of rational numbers. In the following “group” will mean torsion-free abelian group of finite rank unless specifically modified.
M. C. R. Butler, in 1965, proved that if G is a finite rank torsion free abelian group then the following statements are equivalent: (i) G is a pure subgroup of a finite rank completely decomposable group C; (ii) G is a homomorphic image of a finite rank completely decomposable group D; (iii) typeset(G) is finite and for each type τ, G(τ) = Gτ ⨁ < G*(τ) >*, for some τ-homogenous completely decomposable group Gτ, and < G*(τ) >*/G*(τ) is finite.
A finite rank torsion free abelian group G is almost completely decomposable if there exists a completely decomposable subgroup C with finite index in G. The minimum of [ G: C] over all completely decomposable subgroups C of G is denoted by i(G). An almost completely decomposable group G has, up to isomorphism, only finitely many summands. If i(G) is a prime power, then the rank 1 summands in any decomposition of G as a direct sum of indecomposable groups are uniquely determined. If G and H are almost completely decomposable groups, then the following statements are equivalent: (i) $G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group L. (ii) i(G) = i(H) and H contains a subgroup G' isomorphic to G such that [ H: G'] is finite and prime to i(G). (iii) $G \oplus L \approx H \oplus L$ where L is isomorphic to a completely decomposable subgroup with finite index in G.