On the degree function coefficient of a simple complete ideal in dimension two

TL;DR

This paper provides a concise proof that in a two-dimensional regular local ring with an algebraically closed residue field, the degree function coefficient of a simple complete ideal with respect to its unique Rees valuation is always 1.

Contribution

The paper offers a simplified proof of a known result regarding the degree function coefficient for simple complete ideals in two-dimensional regular local rings.

Findings

The degree function coefficient d(I,w) equals 1 for simple complete ideals.

The proof applies to regular local rings with algebraically closed residue fields.

The result confirms the unique valuation property of simple complete ideals in this setting.

Abstract

Let $(R,\mathfrak{M})$ be a two-dimensional regular local ring with algebraically closed residue field. Let $I$ be a simple complete $\mathfrak{M}$-primary ideal of $R$ and let $w$ denote its unique Rees valuation. Then the degree function coefficient $d(I,w)=1$. In this note a short proof of this result is given.