Various local global principles for abelian groups

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 $\Cal L(A)=\{\Bbb Z_{(p)}\otimes A\rightarrow \Bbb Q\otimes A\}$ and (right adjoint) applying the inverse limit functor to such diagrams; $p$ runs through all integer primes. We show that the natural map $A\rightarrow \varprojlim \Cal L(A)$ 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 $\Cal L(A)$ and $\Cal L(B)$ are isomorphic. A similar analysis is carried out for the arithmetic systems $\Cal S(A)=\{\Bbb Q\otimes A\rightarrow\Bbb Q\otimes A^{\wedge}\leftarrow A^{\wedge}\}$ and the local systems $\{\Bbb Q\otimes A\rightarrow \Bbb Q\otimes (\Pi\Bbb Z_{(p)}\otimes A)\leftarrow\Pi (\Bbb Z_{(p)}\otimes A)\}$. The delicate relationship between the various adjoint functor pairs described above is explained.