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A generalisation of the Abhyankar Jung theorem to associated graded rings of valuations

Published online by Cambridge University Press:  10 December 2015

STEVEN DALE CUTKOSKY
STEVEN DALE CUTKOSKY*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. email: [email protected]
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Abstract

Suppose that R → S is an extension of local domains and ν* is a valuation dominating S. We consider the natural extension of associated graded rings along the valuation grν*(R) → grν*(S). We give examples showing that in general, this extension does not share good properties of the extension R → S, but after enough blow ups above the valuations, good properties of the extension R → S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalisation of the Abhyankar–Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 2 , March 2016 , pp. 233 - 255
Copyright
Copyright © Cambridge Philosophical Society 2015 

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