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Every place admits local uniformization in a finite extension of the function field. (English) Zbl 1221.14016

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

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14G27 Other nonalgebraically closed ground fields in algebraic geometry
11R58 Arithmetic theory of algebraic function fields
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References:

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