All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-algebras C}$-algebras if it helps.] Suppose that $B' = A'\otimes_A B$ is a localization of $A'$, A'$, i.e. there is a multiplicatively closed subset $S$ of $A'$ such that $B' = S^{-1}A'$. Must $B$ be a localization of $A$? A$?
I find it hard to believe that the answer is "yes." But I'm having a mental block coming up with an example to show that it's "no."
