Weak normality, Gorenstein and Serre's conditions

TL;DR

This paper investigates properties of normalization in commutative algebra, establishing new characterizations and connections between various generalizations, with applications to quasi-normality and homological reduction.

Contribution

It introduces new characterizations of quasi-normality and explores the homological properties of normalization modulo the ring, addressing conjectures by Vasconcelos and Matlis.

Findings

Characterization of quasi-normality

Normalization modulo the ring can be homologically reduced

Connections between generalizations and specializations of normalization

Abstract

We compute the associated prime ideals of the normalization modulo the ring, and establish connections between different types of generalizations (resp. specializations) of the normalization. This has some applications. For example, we present a characterization of quasi-normality, in the format that was conjectured by Vasconcelos. Also, we show in some cases, the normalization modulo the ring, is homologically reduced. This provides a partial answer to the conjecture of Matlis.