Various local global principles for abelian groups

Peschke, George; Symonds, Peter

doi:10.5565/publmat_38294_06

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…(r) = G. Thus if T £ J^ , then the groups G e ^ which belong to the same Jz?-genus as T are certain extensions of the Q'-torsion free quotient of T by Fg- [T]. If G is abelian, it is possible to describe the isomorphism classes of abelian groups in the Jz?-genus of G by means of homological algebra; see [16].…”

Section: Lqg -J-*-lqhmentioning

confidence: 99%

Localization and genus in group theory

Peschke¹

1995

Trans. Amer. Math. Soc.

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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

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Section: Lqg -J-*-lqhmentioning

confidence: 99%

Localization and genus in group theory

Peschke¹

1995

Trans. Amer. Math. Soc.

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“…. If G is abelian, it is possible to describe the isomorphism classes of abelian groups in the Jz?-genus of G by means of homological algebra; see [16].…”

Section: The -S'-genus: Basic Factsmentioning

confidence: 99%

Localization and Genus in Group Theory

Peschke¹

1995

Transactions of the American Mathematical Society

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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.

show abstract

Various local global principles for abelian groups

Cited by 2 publications

References 6 publications

Localization and genus in group theory

Localization and genus in group theory

Localization and Genus in Group Theory

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