On quasi-purity of the branch locus - Schmidt, Alexander (Manuscripta mathematica, 2020)
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 → 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.