A Study on Approximations of Totally Acyclic Complexes

Abstract

Let $R$ be a commutative local ring to which we associate the subcategory $\Ktac(R)$ of the homotopy category of $R$-complexes, consisting of totally acyclic complexes. Further suppose there exists a surjection of Gorenstein local rings $Q \xrightarrowdbl{\varphi} R$ such that $R$ can be viewed as a $Q$-module with finite projective dimension. Under these assumptions, Bergh, Jorgensen, and Moore define the notion of approximations of totally acyclic complexes. In this dissertation we make extensive use of these approximations and define several novel applications. In particular, we extend Auslander-Reiten theory from the category of $R$-modules over a Henselian Gorenstein ring and show that under the same assumptions, the triangulated category $\Ktac(R)$ has only finitely many distinct indecomposable totally acyclic complexes. We then present a classification scheme for this category based upon the decomposition into indecomposable complexes. Furthermore, we prove the existence of minimal approximations in the category. The authors above also apply the idea of right approximations to create resolutions of totally acyclic complexes. We provide further results with respect to these resolutions and introduce a minimality condition. Lastly, we prove the uniqueness of such minimal resolutions and show several more properties which extend nicely from the module category.