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

By H. Knaf, Franz-Viktor Kuhlmann. 2007

51 citations recorded by the source corpus.

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F of F. We show that F|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 FP|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⊂K and yield similar results if R is regular and of dimension smaller than 3.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e8952b2450f9d922c2c94276117d077a75952d5d

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop (2012, background): “Kuhlmann refined in [11] this result for local uniformization, proving that”
Resolution of singularities of arithmetical threefolds (2014, background): “Its local variant for valuations is also considered in [53][60][69][78][79][80] but remains equally unsolved.”