Resolving resolution dimensions in triangulated categories

Let T be a triangulated category with a proper class xi of triangles and chi be a subcategory of T. We first introduce the notion of chi-resolution dimensions for a resolving subcategory of T and then give some descriptions of objects having finite chi-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 chi-resolution dimensions in terms of a notion of xi-cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

Automated summary

In this paper, the concept of chi-resolution dimensions for a resolving subcategory chi of a triangulated category T is introduced, with descriptions of objects having finite chi-resolution dimensions and the derivation of Auslander-Buchweitz approximations for these objects. The construction of adjoint pairs for inclusion functors and the characterization of objects with finite chi-resolution dimensions using xi-cellular towers are discussed. Additionally, a new resolving subcategory is constructed from a given one with the reformulation of some known results.