Fiber product of schemes

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 $T$ ,

\operatorname{Hom}(T,X\times_S Y) \cong \operatorname{Hom}(T,X)\times_{\operatorname{Hom}(T,S)}\operatorname{Hom}(T,Y).

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

\operatorname{Spec}A\times_{\operatorname{Spec}R}\operatorname{Spec}B \cong \operatorname{Spec}(A\otimes_R B),

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

Remarks

The underlying topological space of $X\times_S Y$ is not, in general, the naive point-set fiber product. Scheme points carry residue-field data, so several scheme points can lie over one compatible pair of topological points.