Vol. 11, No. 6, 2017

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The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation

Steven Dale Cutkosky

Vol. 11 (2017), No. 6, 1461–1488
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.

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Keywords
valuation, local uniformization
Mathematical Subject Classification 2010
Primary: 14B05
Secondary: 11S15, 13B10, 14E22
Milestones
Received: 4 January 2017
Revised: 15 March 2017
Accepted: 17 April 2017
Published: 16 August 2017
Authors
Steven Dale Cutkosky
Department of Mathematics
University of Missouri
Columbia, MO
United States