Reduction of Local Uniformization to the rank one case

TL;DR

This paper shows that proving local uniformization for rank one valuations suffices for all cases, simplifying the approach to this important problem in algebraic geometry.

Contribution

It reduces the general local uniformization problem to the rank one case, independent of the class of local rings involved.

Findings

Reduction of local uniformization to rank one valuations

Proofs do not depend on specific local ring classes

Includes reductions for various versions of the theorem

Abstract

The main result of this paper is that in order to prove the local uniformization theorem for local rings it is enough to prove it for rank one valuations. Our proof does not depend on the nature of the class of local rings for which we want to prove local uniformization. We prove also the reductions for different versions of the local uniformization theorem.