Let $A,B$ be (noetherian) domains and consider $\phi: A \rightarrow B$ a morphism of rings. This map endows $B$ with an $A$-algebra structure. Consider the induced map $\psi: \mathrm{Spec}\,B \rightarrow \mathrm{Spec}\,A$ between the prime spectra: I can study $\mathrm{Spec}\,B$ by considering the fibers of this map. Let $\mathfrak{p}\subset A$ be a prime ideal and take $\mathfrak{q} \in \psi^{-1}(\mathfrak{p})$, i.e. $\mathfrak{q}\subset B$ is a prime ideal such that $\phi^{-1}(\mathfrak{q}) = \mathfrak{p}$.
I can localize $B$:
- As an $A$-module, obtaining $B_\mathfrak{p} := A_\mathfrak{p} \otimes_A B\,$;
- As a ring, obtaining $B_\mathfrak{q}\,$.
I'm trying to understand if there's some relation between these two rings. In particular I'm interested in a relation, if any, betweeen the residue field $\mathcal{k}(\mathfrak{q}) := \dfrac{B_\mathfrak{q}}{\mathfrak{q}B_\mathfrak{q}}$ and the quotient $\dfrac{B_\mathfrak{p}}{\mathfrak{p}B_\mathfrak{p}}\cong B \otimes_A \dfrac{A_\mathfrak{p}}{\mathfrak{p}A_\mathfrak{p}}$.
If there's no general relation between these two localizations, is there any particular case in which there is some relation? I think I have proven that if $\mathfrak{p},\mathfrak{q}$ are maximal then the two $A$-modules are isomorphic since: $$\dfrac{B_\mathfrak{q}}{\mathfrak{q}B_\mathfrak{q}} \cong \dfrac{B}{\mathfrak{q}} \cong B \otimes_A \dfrac{A}{\mathfrak{p}} \cong B \otimes_A \dfrac{A_\mathfrak{p}}{\mathfrak{p}A_\mathfrak{p}}$$ but I'm not sure if they're isomorphic also as rings.