Resolving resolution dimensions in triangulated categories

Abstract

Let T{\mathcal{T}} be a triangulated category with a proper class ξ\xi of triangles and X{\mathcal{X}} be a subcategory of T{\mathcal{T}}. We first introduce the notion of X{\mathcal{X}}-resolution dimensions for a resolving subcategory of T{\mathcal{T}} and then give some descriptions of objects having finite X{\mathcal{X}}-resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X{\mathcal{X}}-resolution dimensions in terms of a notion of ξ\xi -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

Keywords

• triangulated categories
• a proper class of triangles
• resolving resolution dimensions
• resolving subcategories
• auslander-buchweitz approximations
• 18g20
• 18g25
• 18g10