Fiber product of schemes

Definition

In the Galois example, $\operatorname{Spec}K\times_{\operatorname{Spec}F}\operatorname{Spec}K$ records two copies of $\operatorname{Spec}K$ constrained to lie over the same base. It is the scheme-theoretic version of “pairs in one fiber.”

Given morphisms of schemes $X\to S\leftarrow Y$, their fiber product is a scheme $X\times_S Y$ with projections to $X$ and $Y$ whose composites to $S$ agree, and which is universal with that property: for every scheme $T$,

On affine schemes the construction is concrete. Ring maps $R\to A$ and $R\to B$ give

where $A\otimes_R B$ is the tensor product. This affine calculation is why $K\otimes_F K$ appears in the Galois torsor identity.