Faithfully flat scheme morphism

Let $f:X\to S$ be a morphism of schemes (Schemes and morphisms over a base). Its underlying map of topological spaces is $\mathrm{\mid }f\mathrm{\mid }:X\to S$, and $f$ is flat when it is flat at every point of $X$ (Flat morphism of schemes).

The morphism $f$ is faithfully flat if it is flat and the underlying map $\mathrm{\mid }f\mathrm{\mid }$ is surjective. It is an fpqc morphism if it is faithfully flat and quasi-compact; this is the single-morphism form of the fpqc covering convention of Fpqc covering morphisms, and the two descriptions agree there.

Faithful flatness is therefore flatness plus a single topological condition. It is local on the target: a morphism is faithfully flat exactly when its restriction to the members of an open cover of the target is, since flatness is stalkwise and surjectivity is local on the target. A faithfully flat morphism is surjective by definition, so a morphism to a nonempty target has nonempty source; dually, the empty morphism $\mathrm{\varnothing }\to S$ is faithfully flat exactly when $S=\mathrm{\varnothing }$. The identity and every open covering morphism with surjective underlying map are faithfully flat, and the empty-source case imposes no pointwise flatness condition, exactly as for flatness.