关于Matlis自反模
On the Matlis Reflexive Modules
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中文总结 AI 辅助
本文系统研究Matlis自反模，证明其构成Krull-Schmidt范畴，并在完备Noetherian局部环上等价于minimax模，建立了完全闭包性质，综述了相关应用与开放问题。
中文摘要 AI 辅助
Matlis对偶，由Eben Matlis于1958年首次引入，是交换代数中最优雅且最强大的工具之一，在完备局部环上建立了Artinian模与Noetherian模之间引人注目的反变等价。本文对Matlis自反模——即典范求值同态到双重Matlis对偶为同构的那些模——进行了彻底而系统的研究。我们的论述始于详细的历史叙述，追溯从经典Pontryagin对偶经Grothendieck的对偶复形到Matlis对偶现代表述的思想轨迹。然后我们从基本原理出发发展核心理论，证明Matlis自反模类构成一个Krull-Schmidt范畴——这一结果推广了有限长度模的经典分解定理。在完备Noetherian局部环上，我们证明Matlis自反模恰好与minimax模类（即具有Noetherian子模且其商为Artinian的模）一致。我们建立了完全的闭包性质，包括在子模、商模、扩张和有限直和下的稳定性，并给出了严谨且自足的证明。本文还综述了在广义局部上同调、换环结果以及与纯内射模和线性紧致性联系方面的近期应用，最后讨论了开放问题和未来研究方向。
英文摘要
Matlis duality, first introduced by Eben Matlis in 1958, stands as one of the most elegant and powerful tools in commutative algebra, establishing a striking contravariant equivalence between Artinian and Noetherian modules over complete local rings. In this paper, we undertake a thorough and systematic investigation of Matlis reflexive modules --- those modules for which the canonical evaluation homomorphism into the double Matlis dual is an isomorphism. Our exposition begins with a detailed historical narrative, tracing the intellectual trajectory from classical Pontryagin duality through Grothendieck's dualizing complexes to the modern formulation of Matlis duality. We then develop the core theory from first principles, demonstrating that the class of Matlis reflexive modules constitutes a Krull-Schmidt category --- a result that generalizes the classical decomposition theorems for modules of finite length. Over complete Noetherian local rings, we prove that Matlis reflexive modules coincide precisely with the class of minimax modules (those possessing a Noetherian submodule whose quotient is Artinian). We establish full closure properties, including stability under submodules, quotients, extensions, and finite direct sums, with rigorous and self-contained proofs. The paper also surveys recent applications to generalized local cohomology, change-of-rings results, and connections to pure-injective modules and linear compactness, and concludes with a discussion of open problems and future research directions.
发表机构
- Department of Mathematics, Mara.C., Islamic Azad University Marand, Iran（马拉盖伊斯兰阿扎德大学数学系）
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