On quasi-purity of the branch locus

TL;DR

This paper investigates the conditions under which ramification in field extensions implies ramification in associated geometric models, extending classical purity results without relying on resolution of singularities.

Contribution

It demonstrates that ramification in field extensions leads to ramification in models using Temkin's inseparable local uniformization, bypassing the need for resolution assumptions.

Findings

Ramification in field extensions implies ramification in models.

Uses Temkin's inseparable local uniformization to avoid resolution assumptions.

Connects valuation ramification to geometric ramification in models.

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.