Cotorsion group

Abstract: 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. 0. Introduction We are concerned with finitely additive measures ("charges") taking values in a group G. All groups will be assumed Abelian, and we shall employ the usual additive notation for these groups, writing, for example, nx—y for xny~ and indicating the neutral element of G with 0. Let X be a nonempty set, and let sf be a field of subsets of X. A function p: sf —> G is a ( G-valued) charge if p(0) = 0 and p(Ax U A2) = p(Ax) + p(A2) whenever Ax and A2 are disjoint sets in sf . Now suppose that sf and 3§ are fields of subsets of X and that p: sf —> G and v : ⧠—> G are (7-valued charges. We say that p and v are consistent if p(C) = v(C) whenever Cej/flJ1. For a given G, we are interested in whether any two consistent charges p and v have a common extension, i.e. whether there is a charge p such that p(A) — p(A) if A £ sf and p(B) = v(B) if B £ f%. The charge p is to be defined on sf V 3§, the field generated by sf u J? . Say that a group G has the common extension property if every pair of consistent G-valued charges has a common extension. It is known that the group R of real numbers has the extension property. In [1], it was shown that every algebraically compact group has this property. In [2], the authors exhibited an example (inspired by G. Bergman) of a group G without the extension Received by the editors May 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28B10; Secondary 20K99.

Abstract: 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.

Abstract: Let $H_n$ be the $n$-th group homology functor (with integer coeffcients) and let $\{G_i\} _ {i \in \mathbb{N}}$ be any tower of groups such that all maps $G_{i+1} \to G_i$ are surjective. In this work we study kernel and cokernel of the following natural map: $$H_n(\varprojlim G_i) \to \varprojlim H_n(G_i)$$ For $n=1$ Barnea and Shelah [BS] proved that this map is surjective and its kernel is a cotorsion group for any such tower $\{G_i\} _ {i \in \mathbb{N}}$. We show that for $n=2$ the kernel can be non-cotorsion group even in the case when all $G_i$ are abelian and after it we study these kernels and cokernels for towers of abelian groups in more detail.