Finite Birational extension with stable conductor

TL;DR

This paper investigates conditions under which the conductor of a module-finite birational extension of a 1-dimensional Cohen--Macaulay ring is stable, especially when the ring is generically Gorenstein, and relates this to reflexivity of the extension.

Contribution

It establishes that the conductor is stable in generically Gorenstein cases and characterizes reflexivity of the extension via the intersection of certain Cohen--Macaulay modules.

Findings

Conductor is stable when the ring is generically Gorenstein.

Reflexivity of the extension is characterized by a specific intersection condition.

Provides criteria for stability of conductors in module-finite extensions.

Abstract

Let $S$ be a module finite birational extension of a $1$-dimensional local Cohen--Macaulay ring $R$. When is the conductor of $S$ in $R$ a stable ideal? If $R$ is also generically Gorenstein, then we show that the conductor of $S$ in $R$ is a stable ideal, and $S$ is a reflexive $R$-module if and only if $\Omega \operatorname{CM}(S)=\operatorname{CM}(S)\cap \Omega \operatorname{CM}(R)$.