Finite Birational extension with stable conductor

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 $\Omega \mathrm{CM}(S)=\mathrm{CM}(S)\cap \Omega \mathrm{CM}(R)$.

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