The étale locus of a morphism

Let $f:X\to S$ be a morphism locally of finite presentation (Locally finite presentation morphisms). The étale locus of $f$ is $\mathrm{Et}(f)=\{x\in X:\text{the morphism }f\text{ is }\acute{\text{e}}\text{tale at }x\}\subseteq X,$ the set of points at which the conditions of Étale morphism of schemes hold.

The locus is a subset of $X$ and carries no scheme structure of its own. Étaleness at $x$ is a condition on the germ of $f$ at $x$, so membership of $x$ depends only on arbitrarily small open neighbourhoods of $x$ and of $f(x)$; in particular $\mathrm{Et}(f)$ meets an open subscheme $U\subseteq X$ in $\mathrm{Et}(f{\mathrm{\mid }}_U)$ for the restricted morphism. Since étale at a point means smooth at that point, $\mathrm{Et}(f)\subseteq \mathrm{Sm}(f)$ is contained in the smooth locus of The smooth locus of a morphism, and $f$ is étale exactly when $\mathrm{Et}(f)=X$, equivalently when $\mathrm{Sm}(f)=X$ and the relative dimension of $f$ is zero at every point. If $X=\mathrm{\varnothing }$ then $\mathrm{Et}(f)=\mathrm{\varnothing }$. The openness of $\mathrm{Et}(f)$ for $f$ locally of finite presentation is proved later on this page and is not assumed here.