Stalks of the scheme-theoretic fibre

Given: The objects, hypotheses and conventions in the statement above.

For $f:X\to S$ and $s\in S$, the projection $X_s\to X$ is a homeomorphism onto $f^{-1}(s)$ with the subspace topology and preserves the residue field at every point. In compatible affine charts $A\to B$, $s=\mathfrak{p}$, its points correspond exactly to primes $\mathfrak{q}\subset B$ contracting to $\mathfrak{p}$; no extra embedding choice occurs. Also $X{\times }_S\mathrm{Spec}{\mathcal{O}}_{S,s}\to X$ is a homeomorphism onto the inverse image of the set of generalizations of $s$. (Points and topology of a fibre)

Let $A\to B$ be a ring map, $\mathfrak{p}\in \mathrm{Spec}A$, and $M=A\setminus \mathfrak{p}$, acting on $B$ through the ring map. The fibre over $\mathfrak{p}$ is canonically $\mathrm{Spec}(B{\otimes }_A\kappa (\mathfrak{p}))\cong \mathrm{Spec}(M^{-1}B\mathrm{/}\mathfrak{p}{M}^{-1}B)\mathrm{.}$ The residue field is $\kappa (\mathfrak{p})=A_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{A}_{\mathfrak{p}}$. No reduction of the tensor ring is taken. (Coordinate ring of an affine fibre)

For $\mathfrak{p}\in \mathrm{Spec}A$, there is a canonical isomorphism ${\mathcal{O}}_{\mathrm{Spec}A,\mathfrak{p}}\cong A_{\mathfrak{p}}$. (The stalk of the affine structure sheaf at a prime is A_p)