Cotorsion Groups Research Articles

Homology of the pronilpotent completion and cotorsion groups

For a non-cyclic free group F, the second homology of its pronilpotent completion H2(F^)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${H_2}(\\widehat F)$$\\end{document} is not a cotorsion group.

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

For separable p-groups, the questions concerning the full transitivity and description of the lattice of fully invariant subgroups of the cotorsion hull are discussed. We consider the case in which the cotorsion hull is not fully transitive, and construct a function, different from the indicator, that makes it possible to study the lattice of fully invariant subgroups of a cotorsion group.

The abelianization of inverse limits of groups

The abelianization is a functor from groups to abelian groups, which is left adjoint to the inclusion functor. Being a left adjoint, the abelianization functor commutes with all small colimits. In this paper we investigate the relation between the abelianization of a limit of groups and the limit of their abelianizations. We show that if T is a countable directed poset and G: T → Grp is a diagram of groups that satisfies the Mittag-Leffler condition, then the natural map $$Ab(\mathop {lim}\limits_{t \in T} {G_t}) \to \mathop {lim}\limits_{t \in T} Ab({G_t})$$ is surjective, and its kernel is a cotorsion group. In the special case of a countable product of groups, we show that the Ulm length of the kernel does not exceed ℵ1.

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

In this paper, we consider the lattice of fully invariant subgroups of the cotorsion hull of a separable p-group, which is assumed to be an arbitrary direct sum of torsion-complete p-groups. It is shown that this lattice is isomorphic to the lattice of filters of a lower semilattice made up of infinite matrices.

Generalized Cotorsion Locally Compact Abelian Groups

This paper is concerned with the generalization of the concept of cotorsion abelian group. A locally compact abelian group L is called generalized cotorsion if L contains a compact open subgroup K such that the character group of K and the group L/K are cotorsion groups. Some properties and homological characteristics of generalized cotorsion groups are obtained. The classification problem of generalized cotorsion groups is discussed.

The lattice of fully invariant subgroups of a reduced cotorsion group

We consider some problems of the theory of abelian groups; the term “group” always means an additively written abelian group.

Splitting kernels into small summands

Let λ be a regular cardinal. An epimorphism between abelian groups is λ-pure if it is projective with respect to abelian groups of size less than λ. We show that cotorsion groups A have λ-pure projective dimension greater than 1 for all uncountable λ ≤ |A/tA|, where tA denotes the torsion subgroup of A. For λ > |A/tA|, cotorsion groups A are λ-pure projective.

Fuchs’ problem 34 for mixed Abelian groups

This paper investigates the extent to which an Abelian group A is determined by the homomorphism groups Hom(A. G). A class C of Abelian groups is a Fuchs 34 class if A and C in C are isomorphic if and only if Hom(A,G)≅ Hom(C,G) for all G ∈ C. Two p-groups A and C satisfy Hom(A, G) ≅ Hoot(C, G) for all groups G if and only if they have the same n th -Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class G introduced by Glaz and Wickless. While G is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.

Large indecomposable roots of Ext

Let D be any p-cotorsion free (abelian) group. We construct a proper class of indecomposable torsion free groups G such that Ext( D, G)=0. This contrasts with the classical case of D= Q where the only indecomposable torsion free cotorsion groups are Q , and the groups of all p-adic integers for each prime p.

Cotilting and a Hierarchy of Almost Cotorsion Groups

We describe the structure of all cotilting and partial cotilting groups, and of all cotilting torsion-free classes of groups. Under V=L, we also describe all tilting and partial tilting groups. As a by-product, we introduce new classes of groups—called almost cotorsion groups. Unlike the well-known cotorsion groups, the almost cotorsion ones have a rather complex structure: we prove that any cotorsion free ring can be realized as an endomorphism ring of an almost cotorsion group.

Group-Valued Charges: Common Extensions and the Infinite Chinese Remainder Property

We exhibit two consistent, integer-valued charges (finitely additive measures) which do not have a common, integer-valued extension. More generally, after introducing the notion of an infinitary Chinese remainder property for Abelian groups, we show that if a group has the common extension property, then the group must have the infinite Chinese remainder property. The class of groups with the common extension property is characterised as coincident with the class of cotorsion groups.

Abelian groups with small cotorsion images

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

Group-valued charges: common extensions and the infinite Chinese remainder property

We exhibit two consistent, integer-valued charges (finitely additive measures) which do not have a common, integer-valued extension. More generally, after introducing the notion of an infinitary Chinese remainder property for Abelian groups, we show that if a group has the common extension property, then the group must have the infinite Chinese remainder property. The class of groups with the common extension property is characterised as coincident with the class of cotorsion groups.

BIVARIANT LONG EXACT SEQUENCES I

Abstract Given a pair of short exact sequences in an abelian category A, extending results due to Pressman [7]. We construct nine bivariant long exact sequences that involve the groups extn(δ,β), extn(δ,α) and extn(γ,α) that interlock with the univariant hom-ext sequences determined by (i). Two applications of these sequences in the category of abelian groups are given. Firstly we describe an isomorphism that yields a “constructive” way to compute the adjusted cotorsion part of a reduced cotorsion group and secondly, given a torsion-free, reduced group G with no free summands, we indicate how to determine the p-ranks of the divisible group Ext(G,Z).

On Almost totally injective p-groups and cotorsion groups

On Almost totally injective p-groups and cotorsion groups

Projective classes of cotorsion groups

Projective classes of cotorsion groups

Ext(Q,Z) is the additive group of real numbers

Standard homological methods and a theorem of Harrison on cotorsion groups are used to prove the result mentioned.In this note Z denotes an infinite cyclic group, Q the additive group of rational numbers, Zp ∞ a p–quasicyclie group, and Ip the group of p–adic integers.Pascual Llorente proves in [3] that Ext(Q,z) is an uncountable group, and gives explicitly a countably infinite subset. Very little extra effort produces the result embodied in the title, as follows.

Infinite Abelian Groups and Homological Methods

This paper has two sections concerned with the characterization of the algebraic structures of certain types of infinite abelian groups such as compact groups, or tensor products of torsion groups. It also has a section which gives an approach to the problems of mixed abelian groups (those which are neither torsion nor torsion-free) by establishing a duality between torsion groups and those abelian groups which are as mixed as (that is, they are as far away as possible from being just a direct sum of a torsion group and a torsion-free group). There is a final section containing some miscellaneous results on torsion-free abelian groups. For the reader unfamiliar with homological algebra, we give in Section 1 a brief outline of some results and methods of that theory. Almost all our proofs will depend heavily on homological methods. Section 2 is devoted to the duality of which we have spoken. A reduced group is called co-torsion if it is always a direct summand whenever it appears as a subgroup with a torsion-free quotient group. In the same way that Pontrjagin duality reduces the problems of compact groups to those of discrete groups, the problems of co-torsion groups are reduced to those of torsion groups by a one-to-one duality between the groups of these two classes. A method is then given by which all the mixed groups with a given torsion group can be constructed from the co-torsion group dual to that torsion group. The main result of Section 3 is that an abelian group can be a compact topological group if and only if it is isomorphic to a direct product (unrestricted direct sum) of copies of finite cyclic groups, p-adic integers, the reals, and the groups Z(p-) where, for each prime p, the number of copies of Z(p-) does not exceed the number of copies of the reals. It is also proved that if G is an abelian group with the multiples n! G considered as neighborhoods of the identity, then essentially G is a direct summand of a direct product of finite cyclic groups if and only if it is complete in the metric defined by these neighborhoods, while G is a direct sum of cyclic groups if and only if it is as far from complete as possible in this metric. At the very end of Section 3 it is shown that a torsion-free group is a direct summand of a direct product of finite 366