A Categorical Approach to Linkage

• Published in: Applied categorical structures Vol. 29; no. 3; pp. 447 - 483
• Main Author: York, Alexander
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2021; Springer Nature B.V
• Subjects: Convex and Discrete Geometry; Geometry; Homology; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Modules; Theory of Computation; Homological dimensions; 18G10; 18G20; Category theory; Ideal linkage; Duality; Primary: 13C40; Secondary: 13D02; 18G25; Module linkage
• ISSN: 0927-2852; 1572-9095
• DOI: 10.1007/s10485-020-09623-9

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.