# Various local global principles for abelian groups.

• Volume: 38, Issue: 2, page 353-370
• ISSN: 0214-1493

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## Abstract

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We discuss local global principles for abelian groups by examining the adjoint functor pair obtained by (left adjoint) sending an abelian group A to the local diagram L(A) = {Z(p) ⊗ A → Q ⊗ A} and (right adjoint) applying the inverse limit functor to such diagrams; p runs through all integer primes. We show that the natural map A → lim L(A) is an isomorphism if A has torsion at only finitely many primes. If A is fixed we answer the genus problem of identifying all those groups B for which the local diagrams L(A) and L(B) are isomorphic. A similar analysis is carried out for the arithmetic systems S(A) = {Q⊗A → Q⊗A^ ← A^} and the local systems {Q⊗A → Q⊗(ΠZ(p)⊗A) ← Π(Z(p)⊗A)}. The delicate relationship between the various adjoint functor pairs described above is explained.

## How to cite

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Peschke, George, and Symonds, Peter. "Various local global principles for abelian groups.." Publicacions Matemàtiques 38.2 (1994): 353-370. <http://eudml.org/doc/41185>.

@article{Peschke1994,
abstract = {We discuss local global principles for abelian groups by examining the adjoint functor pair obtained by (left adjoint) sending an abelian group A to the local diagram L(A) = \{Z(p) ⊗ A → Q ⊗ A\} and (right adjoint) applying the inverse limit functor to such diagrams; p runs through all integer primes. We show that the natural map A → lim L(A) is an isomorphism if A has torsion at only finitely many primes. If A is fixed we answer the genus problem of identifying all those groups B for which the local diagrams L(A) and L(B) are isomorphic. A similar analysis is carried out for the arithmetic systems S(A) = \{Q⊗A → Q⊗A^ ← A^\} and the local systems \{Q⊗A → Q⊗(ΠZ(p)⊗A) ← Π(Z(p)⊗A)\}. The delicate relationship between the various adjoint functor pairs described above is explained.},
author = {Peschke, George, Symonds, Peter},
journal = {Publicacions Matemàtiques},
keywords = {Grupos abelianos; Grupo nilpotente; adjoint functor pairs; localizations for Abelian groups; genus problem; local diagrams},
language = {eng},
number = {2},
pages = {353-370},
title = {Various local global principles for abelian groups.},
url = {http://eudml.org/doc/41185},
volume = {38},
year = {1994},
}

TY - JOUR
AU - Peschke, George
AU - Symonds, Peter
TI - Various local global principles for abelian groups.
JO - Publicacions Matemàtiques
PY - 1994
VL - 38
IS - 2
SP - 353
EP - 370
AB - We discuss local global principles for abelian groups by examining the adjoint functor pair obtained by (left adjoint) sending an abelian group A to the local diagram L(A) = {Z(p) ⊗ A → Q ⊗ A} and (right adjoint) applying the inverse limit functor to such diagrams; p runs through all integer primes. We show that the natural map A → lim L(A) is an isomorphism if A has torsion at only finitely many primes. If A is fixed we answer the genus problem of identifying all those groups B for which the local diagrams L(A) and L(B) are isomorphic. A similar analysis is carried out for the arithmetic systems S(A) = {Q⊗A → Q⊗A^ ← A^} and the local systems {Q⊗A → Q⊗(ΠZ(p)⊗A) ← Π(Z(p)⊗A)}. The delicate relationship between the various adjoint functor pairs described above is explained.
LA - eng
KW - Grupos abelianos; Grupo nilpotente; adjoint functor pairs; localizations for Abelian groups; genus problem; local diagrams
UR - http://eudml.org/doc/41185
ER -

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