For \(X \to S\) and a morphism \(T \to S\), the base change of \(X\) to \(T\) is \(X_T \da \fiberprod{X}{S}{T}\), with its projection \(X_T \to T\).
A property \(P\) of morphisms is stable under base change if \(X \to S\) having \(P\) implies that \(X_T \to T\) has \(P\) for every \(T \to S\). It is local on the base if \(f\colon X \to S\) has \(P\) whenever there is an open cover \(\ts{V_i}\) of \(S\) with each \(\inverseof{f}(V_i) \to V_i\) having \(P\).
Extension of the ground field is base change along \(\Spec L\to\Spec k\) for a field extension \(L/k\); a \(k\)-scheme \(X\) is geometrically connected, or geometrically integral, if \(X\times_kL\) is connected, or integral, for every field extension \(L/k\). Connectedness is not stable under this base change: \(\Spec \CC\) is connected, and \(\Spec\CC\times_{\RR}\Spec\CC=\Spec(\CC\tensor_\RR\CC)\cong\Spec\CC\sqcup\Spec\CC\) is not. The fibre of \(X\to Y\) over \(y\) is the base change along \(\Spec \kappa(y) \to Y\).
Closed immersion, separated, proper, finite, flat, smooth, and affine are stable under base change and local on the base. Surjectivity is stable under base change. Dominance is not: the open immersion \(D(x)\to\AA^1_k\) is dominant, and its base change along the origin \(\Spec k\to\AA^1_k\) is \(\emptyset\to\Spec k\).
Dependencies
D-SCHFPRThe fibre product of schemesuses
Backlinks
D-MORLOCALClasses of morphisms stable under base change and local on the source or targetusesT-BASECHANGEFlat, proper and smooth base changeuses