arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2602.13809math.AC
收藏

关于均匀 $S$-版本环、模及其同调理论的综述

A survey on the uniform $S$-version of rings, modules and their homological theories

Xiaolei Zhang, Wei Qi

备注：
This is a survey on the uniform $S$-version of rings, modules and their homological theories
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文综述了均匀 $S$-版本环、模及其同调理论的最新进展，介绍了 $u$-$S$ 模类及其相关同调维数。

摘要：

本文综述了近年来在交换环理论中‘均匀 $S$’-代数结构理论的最新进展。起源于Noetherian、coherent、von Neumann正则和semisimple环的经典概念，引入乘法子集 $S$ 后，发展出了 $S$-Noetherian、$S$-coherent 等 $S$-类比。然而，原始定义中的元素 $s \in S$ 通常依赖于所考虑的理想或模。为克服这一限制并实现更深入的模理论刻画，引入了‘均匀 $S$’（简称 $u$-$S$）的概念。本文系统地介绍了 $u$-$S$- torsion 模、$u$-$S$- exact 序列以及后续的基本模类的均匀类比：$u$-$S$- finitely presented、$u$-$S$- Noetherian、$u$-$S$- coherent、$u$-$S$- flat、$u$-$S$- projective、$u$-$S$- injective 和 $u$-$S$- absolutely pure 模。随后探讨了相关的均匀同调维数，包括 $u$-$S$- weak 全局维数、$u$-$S$- 全局维数及其与多项式环和局部化的关系。本文还涵盖了结构环类，如 $u$-$S$- von Neumann 正则、$u$-$S$- semisimple、$u$-$S$- Artinian、$u$-$S$- multiplication 环以及具有 $u$-$S$- Noetherian 谱的环。

英文摘要：

This survey provides a comprehensive overview of the recent advancements in the theory of ``uniformly $S$''-algebraic structures in commutative ring theory. Originating from the classical concepts of Noetherian, coherent, von Neumann regular, and semisimple rings, the introduction of a multiplicative subset $S$ has led to the development of $S$-Noetherian, $S$-coherent, and other $S$-analogues. However, the element $s \in S$ in the original definitions often depends on the ideal or module under consideration. To overcome this limitation and enable deeper module-theoretic characterizations, the notion of "uniformly $S$" (abbreviated as $u$-$S$) was introduced. This survey systematically presents the definitions, characterizations, and properties of $u$-$S$-torsion modules, $u$-$S$-exact sequences, and the subsequent uniform analogues of fundamental module classes: $u$-$S$-finitely presented, $u$-$S$-Noetherian, $u$-$S$-coherent, $u$-$S$-flat, $u$-$S$-projective, $u$-$S$-injective, and $u$-$S$-absolutely pure modules. We then explore the associated uniform homological dimensions, including the $u$-$S$-weak global dimension, the $u$-$S$-global dimension, and their interplay with polynomial rings and localizations. The survey also covers structural ring classes such as $u$-$S$-von Neumann regular, $u$-$S$-semisimple, $u$-$S$-Artinian, $u$-$S$-multiplication rings, and rings with $u$-$S$-Noetherian spectrum.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑