# how to prove that localisation preserves Hom's [closed]

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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## closed as too localized by Martin Brandenburg, Steven Landsburg, unknown (google), Karl Schwede, Andy PutmanMar 15 '12 at 5:19

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This is a) an easy exercise, b) standard material, for example included in Bourbaki's book on commutative algebra. In any case it is not appropriate for mathoverflow. –  Martin Brandenburg Mar 13 '12 at 14:31

## 1 Answer

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

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

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@Ramsey: When easy question are answered, other questions of this type will occur again. –  Martin Brandenburg Mar 13 '12 at 14:32
I was kind of hoping that the terse reference to MSE would be taken as a bit of a hint that this question would be better asked there. –  Ramsey Mar 13 '12 at 14:43
Ramsey: I see what you were thinking (and can imagine thinking the same way myself), but I suspect this sort of subtlety is likely to be lost on anybody who is too intellectually lazy to do his own homework. –  Steven Landsburg Mar 13 '12 at 16:07
Steven, that's pretty harsh. This could be a research mathematician who is asking a question outside his/her field, and not necessarily an "intellectually lazy" student. The OP might not have realized that it's regarded as an easy question. –  Todd Trimble Mar 13 '12 at 17:01