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  "body": "\n## Definition\n\nLet $S$, $X$ and $Y$ be [[schemes|5A35TYZD02]], and let $f \\colon X \\to S$ and $g \\colon Y \\to S$ be [[morphisms of schemes|JXRKH6P1Z4]].\nA *fiber product* of $X$ and $Y$ over $S$ is a scheme $P$ with morphisms $p \\colon P \\to X$ and $q \\colon P \\to Y$ such that:\n\n1. $f \\circ p = g \\circ q$.\n2. For every scheme $T$ and morphisms $a \\colon T \\to X$ and $b \\colon T \\to Y$ with $f \\circ a = g \\circ b$, there is a unique morphism $h \\colon T \\to P$ with $p \\circ h = a$ and $q \\circ h = b$.\n\nThe fiber product exists for all such data and is unique up to unique isomorphism.\nIt is written $X \\times_S Y$; it is the [[pullback|Q05K04QJ71]] in the category of schemes.\n\n## Remarks\n\nExample: 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 $\\operatorname{Spec}(B \\otimes_A C)$.\nThe 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|1735VGRT2P]].\nNon-example: the underlying set of $X \\times_S Y$ is in general not the fiber product of the underlying sets.\nFor 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.\nWhen $S = \\operatorname{Spec} \\Z$, one writes $X \\times Y$ for $X \\times_S Y$.\n",
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