A direct limit representation for Abelian groups with an application to tensor sequences

Abstract

Each group may be regarded as a direct limit of its finitely generated subgroups. We develop in w 1 a slightly different direct limit representation valid for all abelian groups. The advantage of this alternate representation lies in the fact that some properties of abetian groups are preserved under direct limits but are not inherited by arbitrary finitely generated subgroups. We will be concerned with the following such property which an abelian group G may have: for a particular subgroup A of a particular group B, the canonical homomorphism G| ~GQB is a monomorphism. In w we apply the direct limit representation of w 1 to discuss the exactness of tensor sequences. Since w l and w allow a quick description of those classes of abelian torsion groups which are closed under direct sums and direct limits, we insert this description as w In w we discuss some details of the splitting question for tensor sequences. We use Z(n) to denote the group of integers modulo n and Z(p ~) to denote the p-primary component of the group of rationals modulo I. Otherwise our general reference for terms and notations is [1]. 1. A direct limit representation. Let S, (~ E A) be the family of those subgroups of G which are isomorphic to the direct sum of a finitely generated free abelian group and a finite number of indecomposable primary groups each isomorphic to a direct summand of G. This family is directed by set inclusion. It is a consequence of our first theorem that G is isomorphic to the direct limit of this directed system of groups S~ (~EA). THEOREM 1. Each finitely generated subgroup A of each abelian group G is contained in a subgroup S of G which is the direct sum of a finitely generated free abelian group and a finite number of indecomposable primary groups where each of the indecomposable primary groups is isomorphic to a summand of G. PROOF. Case I: Let G be a p-primary group. Then A is finite and we may use finite induction on the order of A. The theorem is trivial for A = 0. Suppose g is an element of order p in A. (i) If g is in the maximal divisible subgroup D of G, let H (~-Z(p=)) be a divisible hull of {g} in D. Then A§ = AI~3HcPOH=G for some summands A1 of A and P of G for which PDA~. (ii) Ifg is of finite height n in G, let hEG be such that g=p"h and let H={h}. Then A+H = Aa| cP| = G for some summands A 1 of A and P of G for which P~A~. (iii) If g is of infinite height but is not an element of D then the basic subgroup of G must be unbounded. These conditions imply the existence of an integer n and an element