Can anyone tell me where I can read a proof that the natural map
$Hom_{A}(M,N)[S^{1}]\rightarrow Hom_{A[S^{1}]}(M[S^{1}],N[S^{1}])$
is an isomorphism if $M$ is finitely presented?
Can anyone tell me where I can read a proof that the natural map $Hom_{A}(M,N)[S^{1}]\rightarrow Hom_{A[S^{1}]}(M[S^{1}],N[S^{1}])$ is an isomorphism if $M$ is finitely presented? 

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Yes. You can read a proof of this fact at: 

