For morphisms \(X \to S\) and \(Y \to S\), the fibre product \(\fiberprod{X}{S}{Y}\) is a scheme over \(S\) with projections to \(X\) and \(Y\), universal among schemes mapping compatibly to both: \[ \Hom_S(T, \fiberprod{X}{S}{Y}) \cong \Hom_S(T, X) \times \Hom_S(T, Y) . \]
(Existence)
Fibre products exist in \(\Sch\). For affines the anti-equivalence turns the limit into a colimit of rings: \[ \fiberprod{\Spec A}{\Spec R}{\Spec B} \cong \Spec (A \tensor_R B) . \] The general case is obtained by covering \(X\), \(Y\), \(S\) by affines, building the affine pieces, and gluing.
The underlying set of \(\fiberprod{X}{S}{Y}\) is in general not the fibre product of the underlying sets. \(\fiberprod{\Spec \CC}{\Spec \RR}{\Spec \CC} \cong \Spec (\CC \tensor_\RR \CC) \cong \Spec (\CC \times \CC)\) has two points, while the fibre product of the underlying sets has one. For \(K = \FF_p(t)\) over \(L = \FF_p(t^p)\), \(\fiberprod{\Spec K}{\Spec L}{\Spec K} \cong \Spec K[x]/(x-t)^p\) has one point and a nonreduced structure sheaf.
Dependencies
D-AN662Schemes, and morphisms of schemesuses
Backlinks
D-SCHBCBase change, and properties stable under itusesD-SCHFIBRESGeneric, special and central fibres; fat points; scheme-theoretic intersectionsusesPR-SCHINTScheme-theoretic intersection as a fibre productusesPR-SCHFOPThe functor of points, and what small test schemes detectusesPR-SCHFIBThe scheme-theoretic fibre of a morphismuses