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arXiv · 2204.12132

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

Naoki Endo·Shiro Goto·Shin-ichiro Iai·Naoyuki Matsuoka
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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.

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Naoki Endo, Shiro Goto, Shin-ichiro Iai, Naoyuki Matsuoka. 2022-04-26. On the weakly Arf $(S_2)$-ifications of Noetherian rings. https://arxiv.org/abs/2204.12132

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