Classification theory of abelian groups. I. Balanced projectives

Abstract

ABSTRACT. We introduce in this paper a class of Abelian groups which includes the torsion totally projective groups and those torsion-free groups which are direct sums of groups of rank one. Characterizations of the groups in this class are given, and a complete classification theorem, in terms of additive numerical invariants, is proved.By a height, we mean a formal product h = Tl p"^p\ where v(p) is either an ordinal or the symbol °°, and the product is taken over all primes p. If G is an Abelian group, we define hG = C\ pv(-p^G. A short exact sequence 0 -■*■ A -*■ B -> C -> 0 is balanced if for every height h, the sequence 0 -* hA -► hB -* hC -> 0 is exact. A group P is a balanced projective if it is projective with respect to all such sequences-i.e., for any such sequence, the induced map Hom(P, B) -► Hom(/>, C) is surjective. A torsion-free group is balanced projective if and and only if it is a direct sum of groups of rank one. A p-group is balanced projective if and only if it is the direct sum of a divisible group and a totally projective group (defined below or in [17], [23], [6]). We will prove a classification theorem for the balanced projectives which will include, as special cases, Baer's 1937 classification of the torsion-free balanced projectives [2], and Hill's classification of totally projective p-groups [7], [23]. We also prove that any balanced projective is a direct sum of groups of torsion-free rank at most one. (Most of these rank one groups are not direct sums of a torsion group and a torsion-free group, so the invariants we need cannot be obtained directly from Baer's invariants and the Ulm invariants.)An important tool in developing the theory is to work first in the local case-with modules over a discrete valuation ring. One then proves many of the results for an Abelian group M by referring to the localization M = M ® Zp

Abelian p-groups with minimal characteristic inertia

“…We now describe the balanced-projective groups with bounded p-torsion. Imitating [10], recall that a generalized height h means the formal direct product p∈P p v(p) (where P is the set of primes), and if G is a group, then h(G) := p∈P p v(p) G, where v(p) is either an ordinal or the symbol ∞. A short-exact sequence…”

“…In Section 2 we introduce some examples of groups with bounded p-torsion. Two of the most important classes of mixed abelian groups are the Warfield groups, and its subclass consisting of the balanced-projective groups, introduced in [10] and [6], respectively (we define these classes later; see also [7] and [8]). We completely characterize the groups in these classes which have bounded p-torsion (Propositions 2.3 and 2.5).…”

A projective characterization for SKT-modules

“…Similarly if c Gpac(C) then there is a homomorphism/: M-» c(C) such that f(m) = c. Let 5 be the torsion submodule of M from which it follows that f(S) G C. Thus the restriction of / to S is a homomorphism from the 5-group S to C. There is a homomorphism g: S -» B such that -ng = /: S^*C. Since A//S1 is torsion-free and divisible, g lifts to a homomorphism g*: M -^c(B); furthermore, ■ng* = /: M-»c(C), [1]. Since g*(m) G pac(B) and ir(g(m)) = c, the map sr: pac(B) -»pac(C) is surjective.…”

“…The family of vector spaces K(px, G) is defined for every group G to be pxc(G/pxG)/px+xc(G/pxG) where X is any ordinal andp is a prime. Warfield [1] and [2] introduced two classes of groups, balanced projectives and 5-groups, which will play an important role in this paper. For the limit ordinal X, a Zp-module M is a X-elementary balanced projective if and only if pxM « Zp and M/pxM is a totally projective p-group.…”

“…Proof. If a sequence is ch-pure then it is balanced and by [1], divisible modules and balanced projectives are ch-projective.…”