S-dense open subschemes and S-rational maps

Let $S$ be a locally Noetherian scheme (Locally Noetherian and Noetherian schemes) and let $X$ and $Y$ be smooth $S$-schemes (Smooth morphism of schemes). An open subscheme $U\subseteq X$ (Open immersions of schemes) is $S$-dense if for every $s\in S$ the fibre $U_s=U{\times }_S\mathrm{Spec}k(s)$ is Zariski dense in the fibre $X_s=X{\times }_S\mathrm{Spec}k(s)$ (Scheme-theoretic fibre). Fiberwise, $(U\cap V{)}_s=U_s\cap V_s$ and the intersection of two dense open subsets of a topological space is dense; hence finite intersections of $S$-dense open subschemes of $X$ are again $S$-dense in $X$. Similarly, if $U$ is $S$-dense and open in $X$ and $V\subseteq X$ is open, then $U\cap V$ is $S$-dense in $V$, since $(U\cap V{)}_s=U_s\cap V_s$ is dense in $V_s$.

An $S$-rational map $u:X⇢Y$ is an equivalence class of $S$-morphisms $U\to Y$ defined on $S$-dense open subschemes $U\subseteq X$, where two such morphisms $U\to Y$ and $U^{\mathrm{\prime }}\to Y$ are equivalent if they coincide on an $S$-dense open subscheme of $U\cap U^{\mathrm{\prime }}$. We say $u$ is defined at a point $x\in X$ if some representative is defined on an open subscheme containing $x$. The union of the domains of all representatives is an $S$-dense open subscheme $\mathrm{dom}(u)$, the domain of definition of $u$. When $Y$ is separated the representatives agree on their intersections and glue to a morphism on $\mathrm{dom}(u)$; without separatedness such a global representative need not exist. This is the relative version of Rational maps of integral finite-type schemes.

Base change. The notions $S$-dense and $S$-rational are preserved by arbitrary base change $S^{\mathrm{\prime }}\to S$. The domain of definition is compatible with flat base change in a sharp sense: if $X$ and $Y$ are smooth of finite type over $S$ and $Y$ is separated over $S$, if $u:X⇢Y$ is an $S$-rational map and $S^{\mathrm{\prime }}\to S$ is flat, then the base-changed $S^{\mathrm{\prime }}$-rational map $u_{S^{\mathrm{\prime }}}$ satisfies $\mathrm{dom}(u_{S^{\mathrm{\prime }}})=\mathrm{dom}(u){\times }_S{S}^{\mathrm{\prime }}$ (BLR 2.5/6, Proposition 6). Flatness is essential: over $S=\mathrm{Spec}\mathbb{Z}$, the $S$-rational map ${\mathbb{A}}_S^1⇢{\mathbb{A}}_S^1$ given by $2\mathrm{/}T$ on the $S$-dense open $D(T)$ has domain exactly $D(T)$, since $2\mathrm{/}T$ is not regular at any prime containing $T$. After base change to $\mathrm{Spec}{\mathbb{F}}_2$ it is the zero rational map, which extends over the whole affine line. Thus the domain of definition does not commute with this non-flat base change.

A collection of fibrewise rational maps with informal specialization compatibility is not used on this page as an equivalent definition of an $S$-rational map: an actual representative on an $S$-dense open subscheme is required, and all extension arguments below produce such representatives.