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.
@article{urn:eudml:doc:41185,
title = {Various local global principles for abelian groups.},
journal = {Publicacions Matem\`atiques},
volume = {38},
year = {1994},
pages = {353-370},
mrnumber = {MR1316632},
zbl = {0839.20070},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41185}
}
Peschke, George; Symonds, Peter. Various local global principles for abelian groups.. Publicacions Matemàtiques, Tome 38 (1994) pp. 353-370. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41185/