They start with $f: A \rightarrow B$, a ring homomorphism, $\mathfrak{p}$ a prime ideal of $A$, and $S = A \setminus \mathfrak{p}$. Then they ask to deduce that the subspace $f^{*-1}(\mathfrak{p})$ of $\mathrm{Spec}(B)$ is homeomorphic to $\mathrm{Spec}(B_{\mathfrak{p}}/\mathfrak{p} B_{\mathfrak{p}}) = \mathrm{Spec}(k(\mathfrak{p}) \otimes_A B)$, where $k(\mathfrak{p})$ is the residue field of the local ring $A_{\mathfrak{p}}$.
Now I am confused about two things:
What is $B_{\mathfrak{p}}$ and $\mathfrak{p} B_{\mathfrak{p}}$? Is $B_{\mathfrak{p}}$ = $f(S)^{-1}B$, and $\mathfrak{p} B_{\mathfrak{p}}$ = $(S^{-1}\mathfrak{p})^e$ through $S^{-1}A \rightarrow f(S)^{-1}B$? Is there a standard convention on how to write these that I am not aware of? I am just starting out in graduate algebra.
Should I show that $B_{\mathfrak{p}}/\mathfrak{p} B_{\mathfrak{p}}$ and $k(\mathfrak{p}) \otimes_A B$ are isomorphic as rings? How do i put a ring structure on $k(\mathfrak{p}) \otimes_A B$?
$B_p:=f(S)^{−1}B$ and $pB_{p}:=f(p)B_{p}=f(S)^{−1}p^e$
$k(p)⊗_{A}B≅A_{p}/pA_{p}⊗_{A}B≅A_{p}⊗_{A}A/p⊗_{A}B≅A_{p}⊗_{A}B/pB≅B_p/pB_p$ as $A$-modules and rings (For the isomorphisms, we used some properties of tensor products, e.g. exercise 2 on page 31, Proposition 3.5 on page 39 of Introduction to Commutative Algebra by M.F. Atiyah, I.G. MacDonald.)