Separated S-scheme

Let $S$ be a scheme and let $X$ be an $S$-scheme, that is, a scheme equipped with a morphism $X\to S$ (Schemes and morphisms over a base). We say that $X$ is separated over $S$, or that $X$ is a separated $S$-scheme, if the structure morphism $X\to S$ is separated (Separated morphism of schemes). We say that a scheme $X$ is absolutely separated, or simply separated, if it is separated over $\mathrm{Spec}\mathbb{Z}$.

Relative and absolute statements must name their base. Absolute separatedness is the case $S=\mathrm{Spec}\mathbb{Z}$ and concerns the diagonal $X\to X{\times }_{\mathrm{Spec}\mathbb{Z}}X$. Absolute separatedness implies separatedness over every base $S$ equipped with a morphism $X\to S$: the diagonal ${\mathrm{\Delta }}_{X\mathrm{/}S}$ is the pullback of ${\mathrm{\Delta }}_{X\mathrm{/}\mathbb{Z}}$ along $X{\times }_{S}X\to X{\times }_{\mathbb{Z}}X$. Indeed, a pair of maps to $X$ in this pullback must coincide, which identifies the pullback with $X$. Closed immersions survive this base change (Base change of immersions). Conversely separatedness over $S$ is a property of the structure morphism and constrains only the diagonal $X\to X{\times }_{S}X$. The effect of changing the base and of composing structure morphisms is established by the lemmas on this page rather than assumed here.