Linkage of ideals is a very well-studied topic in algebra. It has lead to the development of module linkage which looks to extend the ideas and results of the former. Although linkage has been used extensively to find many interesting and impactful results, it has only been extended to schemes and modules. This paper builds a framework in which to perform linkage from a categorical perspective. This allows a generalization of many theories of linkage including complete intersection ideal linkage, Gorenstein ideal linkage, linkage of schemes and module linkage. Moreover, this construction brings together many different robust fields of homological algebra including linkage, homological dimensions, and duality. After defining linkage and showing results concerning linkage directly, we explore the connection between linkage, homological dimensions, and duality. Applications of this new framework are sprinkled throughout the paper investigating topics including module linkage, horizontal linkage, module theoretic invariants, and Auslander and Bass classes.

中文翻译：

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链接的分类方法

理想的联系是代数中一个非常深入研究的话题。它导致了模块链接的发展，它希望扩展前者的思想和结果。尽管链接已被广泛用于发现许多有趣且有影响力的结果，但它仅扩展到方案和模块。本文构建了一个框架，在其中从分类的角度执行链接。这允许对许多链接理论进行推广，包括完全交叉理想链接、Gorenstein 理想链接、方案链接和模块链接。此外，这种构造汇集了许多不同的强大同调代数领域，包括链接、同调维度和对偶性。在定义连锁并直接显示有关连锁的结果后，我们探索连锁、同源维度、和二元性。这个新框架的应用遍布整篇论文，研究主题包括模块链接、水平链接、模块理论不变量以及 Auslander 和 Bass 类。