 Solution

The prime ideal correspondence for localization identifies primes of $B_{\mathfrak{p}}$ with primes $\mathfrak{q}$ of $B$ disjoint from $A\setminus \mathfrak{p}$. Passing to the quotient by $\mathfrak{p}{B}_{\mathfrak{p}}$ retains exactly those containing $\mathfrak{p}{B}_{\mathfrak{p}}$. Thus the image is the fiber of the map on spectra: