Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 113 / 2 / ii

Work first over an affine neighbourhood $\mathrm{Spec}A$ of $y=\mathfrak{p}$, with an affine chart $\mathrm{Spec}B$ in its inverse image. The scheme-theoretic fibre on this chart has ringwhere $\kappa (\mathfrak{p})$ is the residue field. The prime ideal correspondence for localization and the correspondence for a quotient identify its points with primes $\mathfrak{q}\subset B$ containing $\mathfrak{p}B$ and disjoint from $S$. These two conditions are exactly $\mathfrak{q}\cap A=\mathfrak{p}$.

Under this bijection, each principal open $D(b\otimes 1)$ corresponds to $D(b)\cap f^{-1}\{y\}$. These are bases for the two topologies: every element of the fibre ring is represented by $b/s$, with the image of $s$ invertible. Thus the bijection is a homeomorphism. The identifications agree on overlapping charts and glue, provingThe scheme-theoretic fibre may nevertheless carry nilpotents, so this topological statement does not identify its structure sheaf with a naive restriction of ${\mathcal{O}}_X$.