Weak Normality and Seminormality

TL;DR

This survey reviews the historical development and recent advances in the theories of weak normality and seminormality for commutative rings and algebraic varieties, emphasizing their applications and consequences.

Contribution

It provides a comprehensive overview of the evolution and recent progress in weak normality and seminormality theories, especially for reduced Noetherian rings.

Findings

Recent developments have expanded the understanding of weak normality and seminormality.

Theories have been generalized to broader classes of commutative rings.

The survey highlights key consequences and applications of these theories.

Abstract

In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen years. We develop the theories for general commutative rings, but specialize to reduced Noetherian rings when necessary. We hope to acquaint the reader with many of the consequences of the theories.