Younes El Haddaoui, Hwankoo Kim, Najib Mahdou

On the weak global dimension of a subclass of Pr\"ufer non-coherent rings

Introduction

On the weak global dimension of a subclass of pr\"ufer non-coherent rings. Explore the weak global dimension of non-coherent Pr"ufer rings. This study employs diverse homological techniques, extending classical frameworks for a unified understanding.

Abstract

It is known that if $R$ is a coherent Pr\"ufer ring, which is necessarily a Gaussian ring, then its weak global dimension w.gl.dim($R$) must be $0$, $1$, or $\infty$. In this paper, we investigate the possible values of the weak global dimension for a broader class of Pr\"ufer rings that are not necessarily coherent. Our analysis employs four conceptually distinct proofs, each relying on different homological techniques, including localization at the nilradical, finitistic projective dimension, and flatness properties. The results extend the classical framework to a non-coherent setting by incorporating the effective $\mathcal{H}^D$ framework, which serves as a surrogate for coherence in controlling homological dimensions. This work aims to deepen the understanding of the weak global dimension in the context of non-coherent Pr\"ufer rings and provide a unified perspective on its behavior.

Review

The paper "On the weak global dimension of a subclass of Pr\"ufer non-coherent rings" addresses a fundamental question in commutative algebra by investigating the weak global dimension of Pr\"ufer rings that are not necessarily coherent. It expands upon the known result that coherent Pr\"ufer rings possess a weak global dimension restricted to $0$, $1$, or $\infty$. This extension to non-coherent settings is particularly valuable, as coherence often simplifies homological arguments, and exploring its absence provides deeper insights into the intrinsic properties of these ring structures. The study promises to fill a significant gap in the literature, moving beyond the well-understood coherent case. The methodology employed appears to be both rigorous and innovative. The abstract mentions the use of four conceptually distinct proofs, which suggests a comprehensive and robust approach to the problem. The authors leverage a sophisticated array of homological techniques, including localization at the nilradical, finitistic projective dimension, and flatness properties. A particularly noteworthy aspect is the incorporation of the "effective $\mathcal{H}^D$ framework" as a surrogate for coherence. This demonstrates a creative solution to managing homological dimensions in a non-coherent environment, potentially offering new avenues for research in similar challenging contexts within ring theory. The findings of this research are poised to significantly advance our understanding of weak global dimensions in a more generalized algebraic setting. By pushing the boundaries beyond the classical coherent framework, the paper contributes to a more complete and nuanced picture of how this key homological invariant behaves in non-coherent Pr\"ufer rings. The promise of providing a "unified perspective on its behavior" suggests that the work will not merely present new values but also offer a coherent theoretical framework. This research is likely to stimulate further investigations into the homological characteristics of non-coherent rings and the broader application of the $\mathcal{H}^D$ framework, thus making a substantial contribution to the field.

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