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?



how to prove that localisation preserves Hom's
Yes. You can read a proof of this fact at:


<https://math.stackexchange.com/questions/75812/does-localisation-commute-with-hom-for-finitely-generated-modules>


