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2017 The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation
Steven Cutkosky
Algebra Number Theory 11(6): 1461-1488 (2017). DOI: 10.2140/ant.2017.11.1461

Abstract

Suppose that R is a 2-dimensional excellent local domain with quotient field K, K is a finite separable extension of K and S is a 2-dimensional local domain with quotient field K such that S dominates R. Suppose that ν is a valuation of K such that ν dominates S. Let ν be the restriction of ν to K. The associated graded ring grν(R) was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension (K,ν) (K,ν) of valued fields is without defect if and only if there exist regular local rings R1 and S1 such that R1 is a local ring of a blowup of R, S1 is a local ring of a blowup of S, ν dominates S1, S1 dominates R1 and the associated graded ring grν(S1) is a finitely generated grν(R1)-algebra.

We also investigate the role of splitting of the valuation ν in K in finite generation of the extensions of associated graded rings along the valuation. We say that ν does not split in S if ν is the unique extension of ν to K which dominates S. We show that if R and S are regular local rings, ν has rational rank 1 and is not discrete and grν(S) is a finitely generated grν(R)-algebra, then S is a localization of the integral closure of R in K, the extension (K,ν) (K,ν) is without defect and ν does not split in S. We give examples showing that such a strong statement is not true when ν does not satisfy these assumptions. As a consequence, we deduce that if ν has rational rank 1 and is not discrete and if R R is a nontrivial sequence of quadratic transforms along ν, then grν(R) is not a finitely generated grν(R)-algebra.

Citation

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Steven Cutkosky. "The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation." Algebra Number Theory 11 (6) 1461 - 1488, 2017. https://doi.org/10.2140/ant.2017.11.1461

Information

Received: 4 January 2017; Revised: 15 March 2017; Accepted: 17 April 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06763332
MathSciNet: MR3687103
Digital Object Identifier: 10.2140/ant.2017.11.1461

Subjects:
Primary: 14B05
Secondary: 11S15, 13B10, 14E22

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.11 • No. 6 • 2017
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