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Weak normality, Gorenstein and Serre's conditions

Mohsen Asgharzadeh

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.

Weak normality, Gorenstein and Serre's conditions

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.

Paper Structure

This paper contains 5 sections, 26 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. (Higher) associated primes of $\frac{\overline{R}}{R}$
  3. weak versions of normality and being $(S_2)$
  4. Duality and $(G_{n})$
  5. The h-reduced problem

Key Result

Theorem 1.3

Let $R$ be an integral domain and suppose a nonzero ideal is reflexive if and only if all of its associated primes are of height one. Then $R$ is quasi-normal.

Theorems & Definitions (69)

  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Conjecture 1.7
  • Example 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • ...and 59 more