When I'm reading Matsumura's Commutative Algebra, I find the following statement on the 7th page.
When $M$ is of finite presentation, i.e. when there is an exact sequence of the form $A^m\to A^n\to M\to 0$, we have also
$S^{-1}(Hom_A(M,N))=Hom_{S^{-1}A}(S^{-1}M,S^{-1}N)$.
I think it for hours, but still don't understand why we need the assumption that $M$ need to be of finite presentation. What will happen if we only assume $M$ is finitely generated? (I think it's enough when $M$ is f.g.)
