Revistes Catalanes amb Accés Obert (RACO)

Various local global principles for abelian groups

G. Peschke, P. Symonds


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.

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