Skew pure projective abelian groups

Weakly injective and weakly projective modules, which are extensions of the classes of quasi-injective modules and quasi-projective modules respectively, are studied. Descriptions are obtained for weakly injective and weakly projective Abelian groups, as well as for some classes of Abelian groups closely related to them.

We define the class of almost <TEX>${\omega}_1-p^{\omega+n}$</TEX>-projective abelian p-primary groups and investigate their basic properties. The established results extend classical achievements due to Hill (Comment. Math. Univ. Carol., 1995), Hill-Ullery (Czech. Math. J., 1996) and Keef (J. Alg. Numb. Th. Acad., 2010).

The computation of odd dimensional projective surgery groups of finite groups

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

Let p be a prime, R a commutative ring of characteristic p , G a p -primary abelian group, and V(RG) the group of normalized units in the group algebra RG . It is largely determined when V(RG) is totally projective. This generalizes the well-known special case when R is a perfect field. Moreover, the current proof has the advantage of greater clarity.

We provide descriptions for the moduli spaces Rep(Γ, PU(m)), where Γ is any finitely generated Abelian group and PU(m) is the group of m × m projective unitary matrices. As an application, we show that for any connected CW-complex X with π1(X)≅Zn, the natural map π0(Rep(π1(X), PU(m))) → [X, BPU(m)] is injective, hence providing a complete enumeration of the isomorphism classes of flat principal PU(m)-bundles over X.

Quasi-pure projective and injective torsion free abelian groups of rank 2

The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the rank of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Let G-ZFIDCW denote the category of G-spaces having the G-homotopy type of a finitely dominated G-CW complex for a compact Lie group G. Luck [8] has introduced a functor Wa from G-EF^C^ into the category of abelian groups and has realized the equivariant finiteness obstriction as the element w(X) in Wa(X). That is, a finitely dominated G-CW complex X is G-homotopy equivalent to a finite G-CW complex if and only if w(X)=Q. When G is the trivial group, there is an isomorphism from Wa(X) to the reduced projective group KQ(Z[πι(X)]) which sends the element w (X) to the WalΓs finiteness obstruction ([14]). Anderson [1] and Ehrlich [4] have studied a sufficient condition for w(E)= 0 for some fibration E-^B with fiber S. Munkholm, Pedresen [11], Luck [6, 7, 9] and others have studied the transfer map KQ(Z[^(B)])-^K:Q(Z[^(E)]). The purpose of this paper is to get a sufficient condition for w(X)=0 for a S^space X and a finite cyclic group L. We call G-maps /0: YQ-^X and /0: Y4-*X equivalent if there exists a commutative diagram

Balanced projectives and Axiom 3

In this paper a class of Abelian groups which includes the torsion totally projective groups, S-groups, and balanced projectives is studied. It is shown that this class of groups has a complete set of invariants.

In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the M theory on noncommutative tori. This turns out to be an interesting field of applications, since the space [Formula: see text] of the equivalence classes of the vector unitary irreducible representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the algebraic integration is equivalent to integrate over [Formula: see text]. However, its very definition is related only at the structural properties of the group algebra, therefore it is well defined also in the projective case, where the space [Formula: see text] has no classical meaning. This allows a generalization of the usual group harmonic analysis. Particular attention is given to Abelian groups, which are the relevant ones in the compactification problem, since it is possible, from the previous results, to establish a simple generalization of the ordinary calculus to the associated noncommutative spaces.