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

TL;DR

This paper characterizes when extensions of associated graded rings along valuations are finitely generated, linking it to defect absence and valuation splitting, with implications for local uniformization.

Contribution

It establishes a criterion connecting finite generation of graded ring extensions to defect and splitting of valuations, providing new insights into valuation theory and local uniformization.

Findings

Extension without defect iff finite generation of associated graded rings after blowups.

If graded ring extension is finitely generated, then valuation does not split under certain conditions.

Counterexamples show the necessity of assumptions for finite generation results.

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 $\nu^*$ is a valuation of $K^*$ such that $\nu^*$ dominates $S$. Let $\nu$ be the restriction of $\nu^*$ to $K$. The associated graded ring ${\rm gr}_{\nu}(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,\nu)\rightarrow (K^*,\nu^*)$ of valued fields is without defect if and only if there exist regular local rings $R_1$ and $S_1$ such that $R_1$ is a local ring of a blow up of $R$, $S_1$ is a local ring of a blowup of $S$, $\nu^*$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${\rm gr}_{\nu^*}(S_1)$ is a finitely generated ${\rm gr}_{\nu}(R_1)$-algebra. We also investigate the role of splitting of the valuation $\nu$ in $K^*$ in finite generation of the extensions of associated graded rings along the valuation. We will say that $\nu$ does not split in $S$ if $\nu^*$ is the unique extension of $\nu$ to $K^*$ which dominates $S$. We show that if $R$ and $S$ are regular local rings, $\nu^*$ has rational rank 1 and is not discrete and ${\rm gr}_{\nu^*}(S)$ is a finitely generated ${\rm gr}_{\nu}(R)$-algebra, then $\nu$ does not split in $S$. We give examples showing that such a strong statement is not true when $\nu$ does not satisfy these assumptions. We deduce that if $\nu$ has rational rank 1 and is not discrete and if $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $\nu$, then ${\rm gr}_{\nu}(R')$ is not a finitely generated ${\rm gr}_{\nu}(R)$-algebra.