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

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 ${\rm gr}_ν(R)$ was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension $(K,ν)\rightarrow (K^*,ν^*)$ 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 $S$, $ν^*$ dominates $S_1$, $S_1$ dominates $R_1$ and the associated graded ring ${\rm gr}_{ν^*}(S_1)$ is a finitely generated ${\rm gr}_ν(R_1)$-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 will 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 ${\rm gr}_{ν^*}(S)$ is a finitely generated ${\rm gr}_ν(R)$-algebra, then $ν$ does not split in $S$. We give examples showing that such a strong statement is not true when $ν$ does not satisfy these assumptions. We deduce that if $ν$ has rational rank 1 and is not discrete and if $R\rightarrow R'$ is a nontrivial sequence of quadratic transforms along $ν$, then ${\rm gr}_ν(R')$ is not a finitely generated ${\rm gr}_ν(R)$-algebra.