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On quasi-purity of the branch locus

  • Alexander Schmidt
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Abstract

Let k be a field, K/k finitely generated and L/K a finite, separable extension. We show that the existence of a k-valuation on L which ramifies in L/K implies the existence of a normal model X of K and a prime divisor D on the normalization \(X\_L\) of X in L which ramifies in the scheme morphism \(X\_L\rightarrow X\). Assuming the existence of a regular, proper model X of K, this is a straight-forward consequence of the Zariski–Nagata theorem on the purity of the branch locus. We avoid assumptions on resolution of singularities by using M. Temkin’s inseparable local uniformization theorem.

Mathematics Subject Classification

14F20 13A18 

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Notes

Acknowledgements

The author thanks A. Holschbach for helpful discussions and for providing Example 2.2. Moreover, we thank K. Hübner for her comments on a preliminary version of this article.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelbergDeutschland

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