On quasi-purity of the branch locus

Abstract

Let $k$ be a field, $K/k$ finitely generated and $L/K$ a finite, separable extension. We show that the existence of a $k$-valuation on $L$ which ramifies in $L/K$ implies the existence of a normal model $X$ of $K$ and a prime divisor $D$ on the normalization $X_L$ of $X$ in $L$ which ramifies in the scheme morphism $X_L \rightarrow X$. Assuming the existence of a regular, proper model $X$ of $K$, this is a straight-forward consequence of the Zariski-Nagata theorem on the purity of the branch locus. We avoid assumptions on resolution of singularities by using M. Temkin's inseparable local uniformization theorem.