When is the canonical conductor minimal?

TL;DR

The paper investigates one-dimensional analytically unramified Cohen-Macaulay local rings $R$ through the blowup algebra $B( ext{canonical})$ of the canonical ideal and the conductor chain $\text{co}(R)\subseteq b(\omega)\subseteq \text{tr}(\omega)\subseteq \mathfrak{m}$, introducing the minimal canonical conductor condition $b(\omega)=\text{co}(R)$. It situates this condition relative to far-flung Gorenstein rings (where $\text{tr}(\omega)=\text{co}(R)$) and to almost Gorenstein cases, linking the property to canonical reduction numbers and to reflexive/Gorenstein birational extensions via endomorphism rings and Ulrich theory. The authors prove that, for rings with minimal canonical conductor, any Gorenstein birational extension that is reflexive must be the normalization $\overline{R}$, and they connect these structural results to Arf theory and endomorphism rings. They also classify numerical semigroup rings with minimal canonical conductor, giving explicit families such as $N=\langle n,n+1,n^2-n-1\rangle$ ($n\ge3$), which are nearly Gorenstein but not almost Gorenstein for $n>3$, thereby illustrating the separation between the various Gorenstein-related regimes.

Abstract

For a one dimensional analytically unramified Cohen-Macaulay local ring $R$, the blowup algebra of the canonical ideal is a module finite birational extension. The conductor of this extension always contains the conductor of $R$. We study the case when there is equality. This is the case where $R$ is far from being almost Gorenstein. We study this property within the landscape of numerical semigroup rings and local Arf rings.

Theorems & Definitions (27)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Theorem 2.9
• Definition 2.10