Auslander-Buchweitz approximation theory for triangulated categories

TL;DR

This paper extends Auslander-Buchweitz approximation theory to triangulated categories using relative homology, establishing foundational results on subcategory behavior, resolutions, and dimensions in this context.

Contribution

It introduces a triangulated category analogue of Auslander-Buchweitz approximation theory, exploring subcategory structures, resolutions, and connections to Rouquier's dimension.

Findings

Existence of preenvelopes and precovers in triangulated subcategories.

Development of relative homological algebra in triangulated categories.

Insights into projective/injective dimensions and orthogonal subcategories.

Abstract

We introduce and develop an analogous of the Auslander-Buchweitz approximation theory (see \cite{AB}) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category $\T,$ which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of $\T.$ The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category $\T$ equipped with a pair $(\X,\omega),$ where $\X$ is closed under extensions and $\omega$ is a weak-cogenerator in $\X,$ usually under additional conditions. This reduces, among other things, to the existence of distinguished triangles enjoying special properties, and the behavior of (suitably defined) (co)resolutions, projective or injective dimension of objects of $\T$ and the formation of orthogonal subcategories. Finally, some relationships with the Rouquier's dimension in triangulated categories is discussed.