Localization of algebras

Question

Let $f:A\rightarrow B$ be a ring homomorphism. Let $p$ be a prime ideal in $B$. Is it true that $B \otimes A_{f^{-1}(p)}$ is isomorphic to $B_{p}$ as $A$-algebras. If so, why?

Answer 1

The answer is no:
Take $A=k$ (any field), $B=k[X], p=\langle 0 \rangle$ and let $f:k\to k[X]$ be the inclusion.
Then $q=f^{-1}(p)=\langle 0 \rangle$ so that $A_q=k$ and $B \otimes_A A_q=k[X]\otimes _k k=k[X]$ .
On the other hand $B_p=k[X]_{\langle 0 \rangle}=k(X)$.