The smooth locus of a morphism

Let $f:X\to S$ be a morphism locally of finite presentation (Locally finite presentation morphisms). The smooth locus of $f$ is $\mathrm{Sm}(f)=\{x\in X:\text{the morphism }f\text{ is smooth at }x\}\subseteq X,$ the set of points at which the three pointwise conditions of Smooth morphism of schemes hold. It is a subset of $X$ with no scheme structure imposed; smoothness at a point is a condition on the germ of $f$ there, so membership of $x$ depends only on an arbitrarily small open neighbourhood of $x$ and of $f(x)$. The morphism $f$ is smooth exactly when $\mathrm{Sm}(f)=X$, and $\mathrm{Sm}(f)$ meets no fibre of $f$ in the empty-set case trivially: $X$ empty gives $\mathrm{Sm}(f)=\mathrm{\varnothing }$.

The locus is defined for the locally finitely presented maps to which the pointwise definition applies; for $f={\mathrm{id}}_X$ it is all of $X$, and for an open immersion it is all of the source. The openness of $\mathrm{Sm}(f)$ is proved on this page and is not assumed here.