Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod.
I'm curious if you have proven that for every maximal ideal $m\lhd A$, $\operatorname{res}_{A_m}^A(f_m)$ is an isomorphism .. then can you conclude that $f$ is an isomorphism?
(I know of a short proof relying on an argument via the annihilator of a non-zero element of $\ker(f)$ and $\operatorname{coker}(f)$ to prove that $f$ is an iso. if each $f_m$ is but I'm curious if this alternative approach yields any conclusion)