## 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