Points of a fibre product via residue-field tensors

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

For every field $K$ and scheme $X$, morphisms $\mathrm{Spec}K\to X$ correspond bijectively to pairs $(x,\iota )$ with $x\in X$ and a field embedding $\iota :\kappa (x)\to K$. The identity embedding gives a canonical morphism $\mathrm{Spec}\kappa (x)\to X$, compatible with all scheme morphisms. More generally, for a nonzero local ring $(R,\mathfrak{m})$, morphisms $\mathrm{Spec}R\to X$ correspond to pairs $(x,\phi )$ with a local homomorphism $\phi :{\mathcal{O}}_{X,x}\to R$. Assuming Choice, two field-valued points have the same image in $X$ if and only if they are dominated by a common field-valued point, by compatible embeddings of their fields into a third field. (Field-valued points and local-ring points)

For $A\to B,A\to C$, put $D=B{\otimes }_{A}C$. Given $\mathfrak{r}\in \mathrm{Spec}D$, its projections are $\mathfrak{q}=\{b:b\otimes 1\in \mathfrak{r}\}$ and ${\mathfrak{q}}^{\mathrm{\prime }}=\{c:1\otimes c\in \mathfrak{r}\}$. Their contractions to $A$ coincide at $\mathfrak{p}$. The stalk maps are $B_{\mathfrak{q}}⟶D_{\mathfrak{r}},b\mathrm{/}s⟼(b\otimes 1)\mathrm{/}(s\otimes 1),C_{{\mathfrak{q}}^{\mathrm{\prime }}}⟶D_{\mathfrak{r}},c\mathrm{/}t⟼(1\otimes c)\mathrm{/}(1\otimes t)\mathrm{.}$ They are local and induce embeddings $\kappa (\mathfrak{q}),\kappa ({\mathfrak{q}}^{\mathrm{\prime }})\to \kappa (\mathfrak{r})$ agreeing on $\kappa (\mathfrak{p})$. (Projections on primes, stalks and residue fields)

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)

Every diagram $X\to S\leftarrow Y$ of schemes has a fibre product. Given an affine cover $S={\bigcup }_i\mathrm{Spec}{A}_i$ and affine covers $f^{-1}(\mathrm{Spec}{A}_i)={\bigcup }_j\mathrm{Spec}{B}_{ij}$ and $g^{-1}(\mathrm{Spec}{A}_i)={\bigcup }_k\mathrm{Spec}{C}_{ik}$, the product has open affine cover $\mathrm{Spec}(B_{ij}{\otimes }_{A_i}{C}_{ik})\mathrm{.}$ (Existence of all scheme fibre products)