An S-rational map defined after a faithfully flat smooth base change is defined

Given: AC, a locally Noetherian base $S$, a separated $S$-scheme $Y$, smooth finite-type $S$-schemes $X,X^{\mathrm{\prime }}$, an $S$-rational map $u:X⇢Y$ and a faithfully flat $S$-morphism $f:X^{\mathrm{\prime }}\to X$ such that $u\circ f$ is defined everywhere and equal to a morphism $g:X^{\mathrm{\prime }}\to Y$.

An $S$-rational map is an equivalence class of $S$-morphisms on $S$-dense opens, with domain of definition $\mathrm{dom}(u)$; base change preserves these notions, and for separated smooth finite-type targets the domain commutes with flat base change (S-dense open subschemes and S-rational maps).

A faithfully flat, quasi-compact, locally finitely presented morphism is a cover for fppf descent of morphisms: a morphism whose two pullbacks to $X^{\mathrm{\prime }}{\times }_X{X}^{\mathrm{\prime }}$ agree descends uniquely (Scheme morphisms satisfy fppf descent, assuming AC). Faithfully flat morphisms are surjective (Faithfully flat scheme morphism).

For a separated target, morphisms from any source that agree on a schematically dense open are equal (Agreement on a schematically dense open, assuming AC). A smooth morphism is flat with geometrically reduced fibres (Smooth morphism of schemes). On affine charts $\mathrm{Spec}B\to \mathrm{Spec}A$ of a smooth finite-type map with $A$ Noetherian, a finite prime filtration of the $A$-module $A$ tensors exactly with the flat $A$-algebra $B$ and filters $B$ by the rings $B\mathrm{/}{\mathfrak{p}}_{i}B$. Each is flat over the domain $A\mathrm{/}{\mathfrak{p}}_i$, hence injects into its reduced generic fibre and is reduced. In a reduced Noetherian ring every associated prime is minimal: the ring injects into the finite product of its minimal-prime domain quotients, so the annihilator of a nonzero element is the intersection of those minimal primes where its image is nonzero; if this annihilator is prime, it equals one of those minimal primes. Flatness over $A\mathrm{/}{\mathfrak{p}}_i$ makes every nonzero base element a nonzerodivisor, so these minimal primes contract to ${\mathfrak{p}}_i$. The associated-prime theorem for a finite filtration then shows every associated prime of $B$ is the generic point of a component of some fibre. If an open $U$ meets every fibre densely but $\ker (B\to \mathrm{\Gamma }(U,\mathcal{O}))\mathrm{\ne }0$, this finite ideal has an associated prime; it is also associated in $B$, so its point lies in $U$, where the restriction kernel has zero stalk, a contradiction. Hence $U$ is schematically dense. These uses are supplied by Finite modules over Noetherian rings admit prime filtrations, A nonzero module over a Noetherian ring has an associated prime, and Associated primes in a short exact sequence.

If $U\subseteq X$ is schematically dense and $q:T\to X$ is flat, then $q^{-1}(U)$ is schematically dense in $T$. Locally take $V=\mathrm{Spec}A\subseteq X$ and an affine $W=\mathrm{Spec}B\subseteq q^{-1}(V)$; flatness makes $B$ flat over $A$. The open $U\cap V$ is quasi-compact since $X$ is locally Noetherian. A finite principal-open cover computes $\mathrm{\Gamma }(U\cap V,\mathcal{O})$ as a finite equalizer of localizations of $A$; tensoring this equalizer with flat $B$ computes $\mathrm{\Gamma }(W{\times }_{X}U,\mathcal{O})$ and preserves the injection $A\hookrightarrow \mathrm{\Gamma }(U\cap V,\mathcal{O})$. Thus $B\hookrightarrow \mathrm{\Gamma }(W{\times }_{X}U,\mathcal{O})$, which is the required schematic density. Flatness is stable under base change (Flatness is stable under arbitrary base change).

The given hypotheses make $f$ faithfully flat, quasi-compact, and locally of finite presentation. To see the latter two properties, work locally on affine Noetherian opens $S_0=\mathrm{Spec}A\subseteq S$. For affine charts $W=\mathrm{Spec}C\subseteq X^{\mathrm{\prime }}$ and $V=\mathrm{Spec}B\subseteq X$ with $f(W)\subseteq V\subseteq X_{S_0}$, both $B$ and $C$ are finite-type $A$-algebras; generators of $C$ over $A$ also generate it over $B$, so $C$ is finite type over $B$. The ring $B$ is Noetherian because it is of finite type over the Noetherian ring $A$, and a finite-type algebra over $B$ is finitely presented, proving local finite presentation (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Locally finite presentation morphisms, Every algebra of finite type over a Noetherian ring is a Noetherian ring, If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$, Finite type is affine-local on source and target). For quasi-compactness, cover any quasi-compact open $V_0\subseteq X$ by finitely many affine opens $V_i$ each lying over an affine Noetherian open $S_i\subseteq S$. Each $f^{-1}(V_i)$ is open in the Noetherian finite-type $S_i$-scheme $X_{S_i}^{\mathrm{\prime }}$, hence is quasi-compact; therefore $f^{-1}(V_0)$ is quasi-compact. Faithful flatness is given (Faithfully flat scheme morphism).