Abstract
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.
Highlights
Approximation theory is the main part of relative homological algebra and representation theory of algebras, and its starting point is to approximate arbitrary objects by a class of suitable subcategories
As an important example of resolving subcategories, Auslander and Buchweitz [4] studied the approximation theory of the subcategory consisting of maximal Cohen-Macaulay modules over an artin algebra, and Hernández et al [5] developed an analogous theory for triangulated categories
In analogy to relative homological algebra in abelian categories, Beligiannis [11] developed a relative version of homological algebra in a triangulated category, that is, a pair (, ξ ), in which ξ is a proper class of triangles
Summary
Approximation theory is the main part of relative homological algebra and representation theory of algebras, and its starting point is to approximate arbitrary objects by a class of suitable subcategories. In analogy to relative homological algebra in abelian categories, Beligiannis [11] developed a relative version of homological algebra in a triangulated category , that is, a pair ( , ξ ), in which ξ is a proper class of triangles (see Definition 2.4). There are lots of non-trivial cases, for example, let be a compactly generated triangulated category, the class ξ consisting of pure triangles is a proper class ([12]), and the pair ( , ξ ) is no longer triangulated in general Later on, this theory has been paid more attentions and developed (e.g., [13,14,15,16,17]). We obtain Auslander-Buchweitz approximation triangles (see Proposition 3.10) for objects having finite resolving resolution dimensions. Throughout this paper, all subcategories are full, additive, and closed under isomorphisms
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