Schemes and morphisms over a base

An $S$-scheme is a scheme $X$ equipped with a morphism $X\to S$. An $S$-morphism $X\to Y$ is a scheme morphism commuting with the maps to $S$. For $n\ge 0$, choose an affine open cover $S={\bigcup }_i\mathrm{Spec}{A}_i$. Over each chart take $\mathrm{Spec}{A}_i[t_1,\dots ,t_n]$. On overlaps, localization in the coefficients gives canonical isomorphisms that fix the variables; these satisfy the cocycle condition and glue by thm-gluing-affine-schemes. The result, independent of the cover up to the unique $S$-isomorphism respecting the coefficient maps and the ordered coordinate functions $t_1,\dots ,t_n$, is the relative affine space ${\mathbf{A}}_S^n$. Its structure morphism is affine, although its total scheme need not be affine when $S$ is not. For $n=0$ it is $S$; for $S=\mathrm{\varnothing }$ the construction gives the empty scheme. The uniqueness assertion concerns these coordinate-compatible identifications, not arbitrary $S$-isomorphisms.