Flat morphism of schemes

Let $f:X\to S$ be a morphism of schemes (Schemes and morphisms over a base) and let $x\in X$ with image $s=f(x)$. The morphism induces a homomorphism of local rings ${\mathcal{O}}_{S,s}⟶{\mathcal{O}}_{X,x},$ making ${\mathcal{O}}_{X,x}$ an ${\mathcal{O}}_{S,s}$-module (Flat and faithfully flat modules and ring homomorphisms).

The morphism $f$ is flat at $x$ if ${\mathcal{O}}_{X,x}$ is a flat ${\mathcal{O}}_{S,s}$-module for this module structure, and $f$ is flat if it is flat at every point of $X$. A morphism with empty source is flat vacuously, and a morphism whose target is empty has empty source and is flat.

Flatness at $x$ is a condition on the local ring map alone, so a flat morphism is flat at every point of every open subscheme through which it factors, and restricting $f$ to an open subscheme of the source preserves flatness. The affine-local reformulation in terms of an algebra map $A\to B$ is proved separately on this page and is not assumed here.