Height reduction for local uniformization of varieties and non-archimedean spaces

TL;DR

This paper demonstrates that local uniformization problems for schemes and non-archimedean spaces can be reduced to the case of valuations of height one, simplifying the overall approach.

Contribution

It proves that local uniformization for schemes and non-archimedean spaces follows from the height one valuation case, unifying the approach across different structures.

Findings

Reduces local uniformization to height one valuations

Connects non-archimedean spaces to smooth Berkovich spaces

Simplifies the proof of local uniformization

Abstract

It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean analytic spaces rigorously follows from the case of valuations of height one. For non-archimedean spaces this result reduces the problem to studying local structure of smooth Berkovich spaces.