Algebra & Number Theory

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

Steven Cutkosky

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.

Article information

Source
Algebra Number Theory, Volume 11, Number 6 (2017), 1461-1488.

Dates
Revised: 15 March 2017
Accepted: 17 April 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842807

Digital Object Identifier
doi:10.2140/ant.2017.11.1461

Mathematical Reviews number (MathSciNet)
MR3687103

Zentralblatt MATH identifier
06763332

Keywords
valuation local uniformization

Citation

Cutkosky, Steven. The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation. Algebra Number Theory 11 (2017), no. 6, 1461--1488. doi:10.2140/ant.2017.11.1461. https://projecteuclid.org/euclid.ant/1510842807

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