In this paper we will study the properties of locally compact Abelian Hausdorff topological groups (hereafter known as LCA groups) by means of their mapping properties. The results contained herein are an outgrowth of work done by Professor Armacost [Al] on "sufficiency classes" of LCA groups. The sufficiency class S\textunderscore(H) of an LCA group H is the class of all LCA groups G such that there are sufficiently many continuous homomorphisms from G to H to separate the points of G. This condition is easily seen to be equivalent to the requirement that ∩ker(f)=0, where f ranges over all elements of (G,H), the set of continuous homomorphisms from G to H. This suggests consideration of the subgroup R_H(G)=(∩ over f∈(G,H))ker(f) in any LCA group G. Then S\textunderscore(H) is just the class of groups G such that R_H(G)=0. The subgroups R_H(G) are canonical in the following sense: if a:G_1→G_2 is a continuous homomorphism, then a(R_H(G_1)) is contained in R_H(G_2). This means that R_H can be considered a subfunctor of the identity functor on ℒ, the category consisting of LCA groups as objects and continuous homomorphisms as morphisms. The obvious generalization then is to consider arbitrary subfunctors of the identity on ℒ. Now we cannot say very much about something this general. However, given certain natural restrictions we can prove quite a lot. Namely, if we assume that the subfunctor of the identity r:ℒ→ℒ is idempotent (r(r(G))=r(G) for all LCA groups G) and a radical (r(G/r(G))=0 for all LCA groups G) then it can be shown that r(G)=(∩ over (f∈(G,H) over H∈(C\textunderscore)))ker(f) for a well defined class C of LCA groups. Dualizing the above considerations we can also show that any idempotent radical is of the form r(G)=(Σim(f))\textoverscore) over (f∈(H,G) over H∈(C\textunderscore)), so that every idempotent radical can be explicitly constructed in two essentially distinct ways. Returning to our remarks about the sufficiency class S\textunderscore(H) in connection with R_H(G), we note that S\textunderscore(H) is only one of two classes of LCA groups distinguished by R_H. The other is the class of all LCA groups G such that R_H(G)=G. Obviously this condition is equivalent to the assertion that there are no nontrivial continuous homomorphisms from G to H. Let us denote this latter class by T\textunderscore(H). The pair (T\textunderscore(H),S\textunderscore(H)) has some interesting properties. There are no nontrivial homomorphisms from a member of T\textunderscore(H) to a member of S\textunderscore(H), and T\textunderscore(H) is maximal with respect to this property. If R_H is idempotent then S\textunderscore(H) is also maximal with respect to this property, and conversely. Abstracting, we define a torsion theory for LCA groups to be a pair ((T,F)\textunderscore) of classes of LCA groups such that there are no continuous homomorphisms from a member of T\textunderscore to a member of F\textunderscore and such that T\textunderscore and F\textunderscore are maximal with respect to this property. Examples of torsion theories abound: T\textunderscore contains all connected groups and F\textunderscore contains all totally disconnected groups, or T\textunderscore contains all densely divisible groups and F\textunderscore contains all reduced groups, or T\textunderscore contains all groups with every element compact and F\textunderscore contains all groups with no compact elements. In fact every idempotent radical gives rise to a torsion theory. Now, a torsion theory also yields in a natural way an idempotent radical and, remarkably enough, these correspondences are inverse to each other. Thus we have a 1-1 correspondence between torsion theories and idempotent radicals. This correspondence and the representation of any idempotent radical as the intersection of kernels of continuous homomorphisms and as the closure of the sum of images of continuous homomorphisms are the primary results of the first section. In the second section we devote our attention to specific radicals, sufficiency classes, torsion theories and their duals. In so doing we characterize many important classes of LCA groups by their mapping properties. We also characterize several important canonical subgroups= of LCA groups, some of which, such as the component of the identity and the subgroup of all compact elements, are well known, others of which are not well known but have sufficiently important properties to be worthy of attention. It is the author's hope that these investigations will prove helpful in elucidating the structure of LCA groups. In the final section we turn our attention to problems of this nature. This section is the least complete and the most open-ended. we attempt to say as much as can be said at present about these problems and to indicate possible approaches and plausible conjectures, at least a few of which we hope will be proved at some point in the future.
Algebraic Geometry | Geometry and Topology
Bruner, R. R. Radicals and Torsion Theories in Locally Compact Groups [undergraduate thesis]. Amherst, MA: Amherst College. 1972.