Localization over commutative Noetherian rings Let $S$ be a multiplicatively closed subset of a commutative noetherian ring $A$. Let $M$ and $N$ be finitely generated $A$-modules. If $M_S$ is isomorphic to $N_S$, show that $M_t$ is isomorphic to $N_t$ for some $t \in S.$
 A: Since $M$ is finitely presented, we have that $$\text{Hom}_A(M,N)_S \cong \text{Hom}_{A_S}(M_S, N_S) \text{ (*)}$$
via the map which takes $f/s$ to the product of the constant map $1/s$ and the map from $M_S$ to $N_S$ induced by $f$, by THM 7.11 in Matsumura's "Commutative Ring Theory".  Take $f \in \text{Hom}_{A_S}(M_S, N_S)$ to be an isomorphism.  Suppose $g/s \in \text{Hom}_R(M,N)_S$ maps to $f$ under isomorphism (*).  Then $g$ must induce an isomorphism from $M_S$ to $N_S$.  
We have an exact sequence 
$$0 \rightarrow \ker g \rightarrow M \rightarrow N \rightarrow \text{coker }g \rightarrow 0.$$  
Notice that $(\ker g)_S=0=(\text{coker } g)_S$.  Since $\ker g$ and $\text{coker } g$ are finitely generated, we may choose $a \in \text{Ann}_A(\ker g) \cap S$ and $b \in \text{Ann}_A(\text{coker } g) \cap S$.  Now, take $t:=ab$.  
A: There is a more general geometric statement whose proof I find a little bit clearer.
Let $F,G$ sheaves of modules over some ringed space $X$. If $F$ is of finite presentation, then $\underline{\hom}_{\mathcal{O}_X}(F,G)_x \to \hom_{\mathcal{O}_{X,x}}(F_x,G_x)$ is an isomorphism (one easily reduces to the case $F=\mathcal{O}_X$). Thus, if $F,G$ are of finite presentation, and $F_x \cong G_x$, then there are local sections of $\underline{\hom}(F,G)$ and $\underline{\hom}(G,F)$ at $x$ which compose to the identity in $\underline{\hom}(F,F)_x$ and $\underline{\hom}(G,G)_x$. Both equalities hold actually as local sections of $\underline{\hom}(F,F)$ and $\underline{\hom}(G,G)$ for some small open neighborhood $U$ of $x$. This shows $F|_U \cong G|_U$.
If we apply this to quasi-coherent sheaves on an affine scheme, we get that if $M_{\mathfrak{p}} \cong N_{\mathfrak{p}}$ for finitely presented modules over some commutative ring $A$ (not assumed to be noetherian) for some prime ideal $\mathfrak{p}$, then there is some $f \notin \mathfrak{p}$ with $M_f \cong N_f$. If more generally $M_S \cong N_S$ for some multiplicative subset $S$, then we may assume $0 \notin S$ and localize at a prime ideal $\mathfrak{p}$ disjoint from $S$. Then we are in the situation before.
