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Every place admits local uniformization in a finite extension of the function field

Hagen Knaf·Franz-Viktor Kuhlmann
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Abstract

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension E of F. We show that E|F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension EP|FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R of K and yield similar results if R is regular and of dimension smaller than 3.

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Hagen Knaf, Franz-Viktor Kuhlmann. 2007-02-28. Every place admits local uniformization in a finite extension of the function field. https://arxiv.org/abs/math/0702856

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