Dualities for self–small groups

Abstract

We construct a family of dualities on some subcategories of the quasi-category S of self-small groups of finite torsion-free rank which cover the class S. These dualities extend several of those in the literature. As an application, we show that a group A ∈ S is determined up to quasi-isomorphism by the Q-algebras {Q Hom(C, A) : C ∈ S} and {Q Hom(A, C) : C ∈ S}. We also generalize Butler's Theorem to self-small mixed groups of finite torsion-free rank.

On a theorem of Stelzer for some classes of mixed groups

“…The class of self-small groups of finite torsion-free rank is denoted by S. The following characterizations for the groups from S are presented in [1,Theorem 2.1]. For other characterizations we refer to [1, Section 3] and [13,Theorem 3.1].…”

“…Self-small groups as pushouts. We will describe (up to a finite summand) the groups in S ⋆ in a manner which is similar, but not identical, to that presented in [13,Proposition 3.2], as pushouts of quotient-divisible groups and torsion-free groups.…”

A mixed version for a Fuchs' Lemma

“…Moreover, every group in S has a unique, up to quasi-isomorphism, quasi-decomposition as a direct sum of strongly indecomposable self-small groups, [6]. We refer to [1] and to [8] for other properties of self-small groups.…”

Commuting Properties of Ext

“…A related problem concerns the 'self' specialization: describe the class of groups G for which Hom(G, −) or Hom(−, G) preserve or invert sums or products of copies of G. For example, G is self-small [4] if for all cardinals κ, Hom(G, κ G) is naturally isomorphic to κ Hom(G, G) and self-slender if Hom( κ G, G) κ Hom(G, G). These properties are considered in [3,8,11,15]. Concerning corresponding properties of the extension functors, Göbel and Trlifaj proved in [14, Example 3.1.8], using a result of Salce [18], that despite the results in [9] quoted above, there are important differences between the cases of Hom and Ext.…”

“…κ Hom(G, G). These properties are considered in [3,8,11,15]. Concerning corresponding properties of the extension functors, Göbel and Trlifaj proved in [14, Example 3.1.8], using a result of Salce [18], that despite the results in [9] quoted above, there are important differences between the cases of Hom and Ext.…”