localization over different rings

Question

I guess this is a bit of a soft question. Suppose you have a hom of commutative rings with identity $R\to S$, and a hom of $S$ modules $f: M\to N$. Somehow it shouldn't matter whether you localize $f$ in the category of $S$ modules or in the category of $R$ modules (I realize I'm leaving vague the set you're using to localize). What is the correct, most general or most categorical statement of this?

Answer 1

Maybe this will do: My assumption is that the localization in $S$-MOD is over a set, $U\subset S$ which is the image of a multiplicative set, $W$, in $R$. I don't know how essential this assumption is. Then if we call $\phi: R\to S$, and if $\phi_W^\ast$ is the restriction of scalars functor from $S[U^{-1}]$-MOD to $R[W^{-1}]$-MOD, $\phi^\ast$ is restriction of scalars from S-MOD to R-MOD, and $loc_W$ and $loc_U$ are localization functors from $R$-MOD to $R[W^{-1}]$-MOD and from $S$-MOD to $S[U^{-1}]$-MOD, respectively, then the claim is we have a natural isomorphism of functors:

$$\phi_W^\ast \circ loc_U \cong loc_W \circ \phi^\ast$$

defined in an obvious way.

Edit: the assumption can obviously be weakened to "$\phi(U)\subset W$ and every element of $W$ divides an element of $\phi(U)$." It seems the only obstruction is possible non-surjectivity of the obvious abelian group hom $M[W^{-1}] \to M[U^{-1}]$.