Various local global principles for abelian groups (Q1896473)

An adjoint functor pair is examined in connection with localizations for abelian groups. The left adjoint functor sends an abelian group \(A\) to an inverse local diagram \({\mathcal L}(A):= \{\mathbb{Z}_{(p)} \otimes A\to \mathbb{Q}\otimes A\}\) for \(p\) primes and the right adjoint applies the inverse limit to such diagrams. The author shows that the corresponding natural map \(c:A\to \varprojlim {\mathcal L} (A)\) is an isomorphism iff \(A\) has torsion at only finitely many primes. An analogue study is done for arithmetic systems \({\mathcal S}(A):= \{\mathbb{Q}\otimes A\to \mathbb{Q}\otimes A^\wedge\leftarrow A^\wedge\}\) and for local systems \({\mathcal {LS}} (A):= \{\mathbb{Q}\otimes A\to \mathbb{Q}\otimes (\prod \mathbb{Z}_{(p)} \otimes A)\leftarrow \prod (\mathbb{Z}_{(p)} \otimes A)\}\). For a fixed abelian group \(A\), the author gives an answer to the genus problem of identifying all those groups \(B\) for which the local diagrams are isomorphic.