Localization and genus in group theory

Abstract

We provide a unifying category theoretical framework to discuss various kinds of local global phenomena. Specializing to localization of groups at sets of primes P , we identify a large class of groups for which localization supports a passage from local information to global information. Local global principles for groups in this class are established and used to calculate certain homomorphism sets as well as splittings of epimorphisms and monomorphisms from local data. A. If SC is a family of localizing functors on a category ^, then the members of 5? are related by natural transformations coming from the following *£intrinsically defined partial ordering: LM > Lv if the class of L^-local objects (objects for which the localizing map X-» L^X is an isomorphism) contains all L"-local objects. We form the category 5? ^ whose objects are diagrams in f modeled on the partial order of 7777?. In particular, applying the functors in 5f to an object X e *W yields an object £?(X) e S? %>. This process is a functor-2": f-Jg'g'. B. The extent to which .2Mocal data actually determine objects in f is measured by the "size" of the fibers of the functor 5C: f-»ff? i?. Thus we define the genus of X_ € 5f ^ to be the collection of all isomorphism classes of objects

Various local global principles for abelian groups

“…We associate to an abelian group A the inverse local diagram .C(A) :---{rp : A (p ) -> A é , where A (p) is the p-localization of A an d rp rationalizes . Specializing a result of [7], we see that the natural ma p c : A ---} l lim ,C(A) is an isomorphism if and only if A has torsion at onl y finitely many primes .…”

“…Proof: All claims pertaining to the top two rows of the diagram ar e special cases of 7 .2 in [7] . We sketch the argument for completeness ' sake .…”