Integral closure of 1-dimensional rings
Mohsen Asgharzadeh
TL;DR
This work studies how the conductor, its order, and multiplicity interact in 1-dimensional local domains, connecting these invariants to reflexivity properties of colength-two ideals and the freeness problem for absolute integral closures in positive characteristic. It provides counterexamples showing that the expected value $\\operatorname{ord}(\\frak{C})=2$ does not hold universally, while establishing conditions under which $\\operatorname{ord}(\\frak{C})=e_{\\frak{m}}(R)-1$, particularly for certain 1-dimensional complete hypersurfaces and Gorenstein rings. The paper also investigates when colength-two ideals are reflexive, linking this to hypersurface structure and trace properties, with results highlighting both exceptions and structural implications in almost Gorenstein and minimal multiplicity contexts. In prime characteristic, the authors show $R^+$ can fail to be free (\\operatorname{pd}_R(R^+)=1$ for a DVR) and summarize when $R^{\\frac{1}{p^n}}$ is reflexive under various hypotheses, emphasizing the interplay between Gorensteinness, completion, and finiteness conditions. Collectively, these findings advance understanding of how conductor-related invariants govern the reflexivity and freeness phenomena that arise in dimension-one singularities and their integral closures.
Abstract
We study certain properties of modules over 1-dimensional local integral domains. First, we examine the order of the conductor ideal and its expected relationship with multiplicity. Next, we investigate the reflexivity of certain colength-two ideals. Finally, we consider the freeness problem of the absolute integral closure of a DVR, and connect this to the reflexivity problem of $R^{\frac{1}{p^n}}$.