Horizontal Linkage of Coherent Functors

TL;DR

This paper extends the concept of linkage from ideals to finitely presented functors using satellite endofunctors, broadening applicability and connecting to duality theories in module categories.

Contribution

It introduces a new notion of linkage for totally finitely presented functors, generalizes duality results, and develops derived functors without needing projective or injective objects.

Findings

Recovered Auslander-Gruson-Jensen duality via functor resolutions

Provided a general formula for the defect of finitely presented functors

Defined derived functors without projective or injective assumptions

Abstract

The satellite endofunctors are used to extend the definition of linkage of ideals to the linkage of totally finitely presented functors. The new notion for linkage works over a larger class of rings and is consistent with the functorial approach of encoding information about modules into the category of finitely presented functors. In the process of extending linkage, we recover the Auslander-Gruson-Jensen duality using injective resolutions of finitely presented functors. Using the satellite endofunctors we give general definitions of derived functors which do not require the existence of projective or injective objects. A general formula for calculating the defect of a totally finitely presented functor is given.