Fiber product of schemes

Remarks

Example: for ring maps $A \to B$ and $A \to C$, the fiber product of $\operatorname{Spec} B$ and $\operatorname{Spec} C$ over $\operatorname{Spec} A$ is $\operatorname{Spec}(B \otimes_A C)$. The projections come from $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$, and clause 2 follows from the universal property of the tensor product of algebras. Non-example: the underlying set of $X \times_S Y$ is in general not the fiber product of the underlying sets. For instance, $\operatorname{Spec} \C \times_{\operatorname{Spec} \R} \operatorname{Spec} \C = \operatorname{Spec}(\C \otimes_\R \C) \cong \operatorname{Spec}(\C \times \C)$ has two points, while the set product has one. When $S = \operatorname{Spec} \Z$, one writes $X \times Y$ for $X \times_S Y$.