A functorial approach to linkage and the asymptotic stabilization of the tensor product

This thesis consists of three projects.;The first project deals with generalizing the definition of zeroth derived functors to work for any abelian category. The classical definitions of zeroth derived functors require existence of injectives or projectives. In this project, we give definitions of the zeroth derived functors that do not require the existence of injectives or projectives. The new definitions result in generalized definitions of projective and injective stabilization of functors. The category of coherent functors is shown to admit a zeroth right derived functor. An interesting result of this fact is a counterpart to the Yoneda lemma for coherent functors. Moreover, zeroth derived functors are seen more appropriately as approximations of functors by left exact or right exact functors. Under certain reasonable conditions, the category of coherent functors is shown to have enough injectives. This result was first shown by Ron Gentle. We give an alternate proof of this fact.;The second project deals with extending the definition of horizontally linked modules over semiperfect Noetherian rings to the category of finitely presented functors over arbitrary Noetherian rings. Linkage of modules can be defined using the syzygy and transpose operation. Auslander established the merit of studying modules by studying the functors from the module category into the category of abelian groups. It turns out that the module theoretic notion of linkage can be extended to a functorial notion of linkage and the satellite endofunctors are crucial to this extension.;The third project deals with finding alternate ways of recovering Vogel homology. There are three known ways of generalizing Tate cohomology to a cohomology theory that works over arbitrary rings, that given by Vogel, that given by Mislin, and that given by Buchweitz. Vogel also provided a homological counterpart to his generalization of Tate cohomology. Yoshino attempted to recover this homology theory using an approach similar to Mislin's approach to recovering Tate cohomology; however, he only produced Vogel homology in positive degrees. This is fixed by returning to Mislin's construction and observing that it can be dualized. Completely missing from the picture was an approach similar to Buchweitz's approach to generalizing Tate cohomology. The asymptotic stabilization of the tensor product is introduced to fill this gap.