A finite normal generically étale cover of a regular scheme is étale if unramified in codimension one

Work over a Noetherian affine open of $X$; the finite normal algebra $B$ of $Y$ is the integral closure of its normal base domain $A$ in its generic product of separable fields. Indeed $B$ is integral over $A$, and an element of the generic algebra integral over $A$ is integral over $B$ and therefore belongs to $B$ by normality. By [F3] the image of the nonétale locus is a closed subset $Z$ of $\mathrm{Spec}A$. If it is nonempty, choose the generic point $\mathfrak{p}$ of one of its irreducible components. It is neither generic nor height one by the hypotheses, so $d=\dim A_{\mathfrak{p}}\ge 2$. Over the punctured spectrum of $A_{\mathfrak{p}}$ the cover is finite étale: no proper subprime lies in $Z$ by the minimality of $\mathfrak{p}$.

By [F2] this punctured cover extends to a finite étale algebra $D$ over $A_{\mathfrak{p}}$. The generic algebra is the same as that of $B_{\mathfrak{p}}$. Both algebras are its integral closure of $A_{\mathfrak{p}}$: this was proved for $B$ in step 1.1 and follows for $D$ from integral-closure compatibility in [F2] applied to the normal ring $A_{\mathfrak{p}}$ and its fraction field. Hence they are canonically isomorphic, so $B_{\mathfrak{p}}$ is étale, contradicting $\mathfrak{p}\in Z$. Thus $Z$ is empty on every affine open, and the finite cover is étale. This proves the full codimension-one criterion.