A closed-immersion fibre is one residue point or empty

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

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)

Let $A\to C$ be a unital ring map. For any set of variables $(t_i)$ and any ideal $I\subseteq A[t_i]$, $(A[t_i]\mathrm{/}I){\otimes }_{A}C\cong C[t_i]\mathrm{/}IC[t_i]\mathrm{.}$ Here the extended ideal is generated by the coefficient images of all elements of $I$. For a multiplicative subset $M\subseteq A$, $(M^{-1}A){\otimes }_{A}C\cong {\bar{M}}^{-1}C\mathrm{.}$ These are ring isomorphisms; no flatness, finite-generation or nonzero-ring hypothesis is required. (Presentations and localization under base extension)