On the weakly Arf $(S_2)$-ifications of Noetherian rings

TL;DR

This paper introduces the weakly Arf $(S_2)$-ification, a minimal module-finite birational extension of a Noetherian ring satisfying the weakly Arf property and Serre's $(S_2)$ condition, expanding the theory of such extensions.

Contribution

It constructs the smallest weakly Arf $(S_2)$-ification for rings meeting mild conditions and develops foundational theory and existence results for these extensions.

Findings

Constructed the minimal weakly Arf $(S_2)$-ification for certain Noetherian rings.

Established existence theorems for these extensions.

Developed basic properties and theory of weakly Arf $(S_2)$-ifications.

Abstract

The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman with his famous paper [12], and recently rediscovered by [4], being closely explored with further developments. The present paper aims at constructing, for a given Noetherian ring $R$ which satisfies certain mild conditions, the smallest module-finite birational extension of $R$ which satisfies WAP and the condition $(S_2)$ of Serre. We shall call this extension the weakly Arf $(S_2)$-ification, and develop the basic theory, including some existence theorems.