The Milnor Number of One Dimensional Local Rings

TL;DR

The paper introduces a Milnor-number analogue $\mu(R)$ for one-dimensional local rings and develops two semigroups, $\Gamma(R)$ and $\nu(R)$, to capture value and intersection data across components. It proves $\mu(R)=2\delta(R)-r(R)+1$ with finiteness and completion-invariance, and establishes inequalities $\mu(R)\geq \delta(R)\geq r(R)-1$ and $\mu(R)\geq (r(R)-1)^2$, including the sharp cases $\mu(R)=0$ (DVR) and $\mu(R)=1$ (double point) and a Morse-type criterion relating to equisingularity. A Hironaka-type decomposition expresses $\delta(R)$ in terms of component deltas and pairwise intersection multiplicities, linking the global and branch data. The semigroups $\Gamma(R)$ and $\nu(R)$ provide a structured, per-branch view of values, with $\Gamma(R)$ being a numerical semigroup and $\nu(R)$ encoding a vector of branch-wise invariants; in the Gorenstein case, $\vec{\mu}(R)$ is the minimal element of $\nu(R)$ generating the conductor. Finally, the authors show that one-dimensional Cohen–Macaulay local rings of finite CM type are equisingular to ADE-type singularities, connecting their invariants to classical ADE classifications and extending Milnor-type ideas to a broader algebraic setting.

Abstract

In this paper we present an analogue of the Milnor number for one dimensional local ring, and we show that it satisfies analogous properties to those of the Milnor number of plane curves over a field. In addition, we present two analogues of the semi-group of values for a one dimensional ring and show how they relate to our Milnor number. Finally, we use these tools and techniques to show we can relate these semigroups of one dimensional rings of finite Cohen-Macaulay type to those of the classical ADE singularities.

Figures (5)

• Figure 1: The semigroup $\Gamma(R)$ of an $A_{2n}$ singularity
• Figure 2: The semigroup $\nu(R)$ of an $A_{2n+1}$ singularity.
• Figure 3: The semigroup $\nu(R)$ of an $D_{2n+1}$ singularity.
• Figure 4: The semigroup $\Gamma(R)$ of an $E_6$ and an $E_8$ singularity.
• Figure 5: The semigroup $\nu(R)$ of an $E_7$ singularity.

Theorems & Definitions (107)

• Theorem
• Remark 2.2
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof