* Resolving resolution dimensions in triangulated categories

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


Introduction
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 particular, resolving subcategories play important roles in approximation theory (e.g., [1][2][3]). 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. Using the approximation triangles established by Hernández et al. [5,Theorem 5.4], Di and Wang [6] constructed additive functors (adjoint pairs) between additive quotient categories. On the other hand, Zhu [7] studied the resolution dimension with respect to a resolving subcategory in an abelian category, and Huang [8] introduced relative preresolving subcategories in an abelian category and defined homological dimensions relative to these subcategories, which generalized many known results (see [4,9,10]).
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). Under this notion, a triangulated category is just equipped with a proper class consisting of all triangles. However, there are lots of non-trivial cases, for example, let be a compactly generated triangulated category, then 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]). It is natural to ask how the approximation theory acts on this relative setting of triangulated categories. In [18], Ma et al., introduced the notions of (pre)resolving subcategories and homological dimensions relative to these subcategories in this relative setting, which gives a parallel theory analogy to that of abelian categories [8]. In this paper, we devote to further studying relative homological dimensions in triangulated categories with respect to a resolving subcategory. The paper is organized as follows: In Section 2, we give some terminology and some preliminary results. In Section 3, some homological properties of resolving subcategories are obtained. In particular, we obtain Auslander-Buchweitz approximation triangles (see Proposition 3.10) for objects having finite resolving resolution dimensions. Our main result is the following: Theorem. Let be a resolving subcategory of and , a ξxt-injective ξ -cogenerator of . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. For any ∈ M , if  ∈ M , then the following statements are equivalent: for all ≥ n m. , where φ is ξ -proper epic, such that = K Hoker φ satisfying -≤ − K m res.dim 1. In Section 4, we will further study objects having finite resolution dimensions with respect to a resolving subcategory . We first construct adjoint pairs for two kinds of inclusion functors. Then we characterize objects having finite resolution dimensions in terms of a notion of ξ -cellular towers.
Throughout this paper, all subcategories are full, additive, and closed under isomorphisms.

Preliminaries
Let be an additive category and → Σ : an additive functor. One defines the category ( ) Diag , Σ as follows: , , of morphisms in such that the following diagram:

commutes.
A triangulated category is a triple ( ) , Σ, Δ , where is an additive category and → Σ : is an autoequivalence of (called suspension functor), and Δ is a full subcategory of ( ) Diag , Σ which is closed under isomorphisms and satisfies the axioms ( ) T 1 -( ) T 4 in [11, Section 2.1] (also see [19]), where the axiom ( ) T 4 is called the octahedral axiom. The elements in Δ are called triangles.
The following result is well known, which is an efficient tool in studying triangulated categories. an autoequivalence of , and Δ a full subcategory of ( ) Diag , Σ which is closed under isomorphisms. Suppose that the triple ( ) , Σ, Δ satisfies all the axioms of a triangulated category except possibly of the octahedral axiom. Then, the following statements are equivalent: (1) Octahedral axiom. For any two morphisms ⟶ u X Y : and , there exists a commutative diagram in which all rows and the third column are triangles in Δ.
(2) Base change. For any triangle , there exists the following commutative diagram: in which all rows and columns are triangles in Δ.
, there exists the following commutative diagram: in which all rows and columns are triangles in Δ.
We use Δ 0 to denote the full subcategory of Δ consisting of all split triangles.
Let ξ be a class of triangles in .
(1) ξ is said to be closed under base change (resp. cobase change) if for any triangle in ξ and any morphism ′ ⟶ α Z Z : (resp. ⟶ ′ β X X : ) as in Remark 2.1(2) (resp. Remark 2.1(3)), the triangle in ξ and any ∈ i (the set of all integers), the triangle Definition 2.4. [11] A class ξ of triangles in is called proper if the following conditions are satisfied.
(2) ξ is closed under suspensions and is saturated.
Throughout this paper, we always assume that ξ is a proper class of triangles in .
be a triangle in ξ . Then, the morphism u (resp. v) is called ξ -proper monic (resp. ξ -proper epic), and u (resp. v) is called the hokernel of v (resp. the hocokernel of u).
We say that has enough ξ -projective objects if for any object ∈ M , there exists a triangle ⟶ Dually, we say that has enough ξ -injective objects if for any object Remark 2.7. ( ) ξ is closed under direct summands, hokernels of ξ -proper epimorphisms, and ξ -extensions. Dually, ( ) ξ is closed under direct summands, hocokernels of ξ -proper monomorphisms, and ξ -extensions. in ξ and the differential d n is defined as , -exact (resp. (− ) Hom , -exact) for any ∈ n .
Definition 2.9. [13, Definition 3.6] Let be a triangulated category with enough ξ -projective objects and X an object in .
in with all P ξ i -projective objects. The objects K n as in (2.2) are called ξ -Gorenstein projective objects. We use ( ) ξ to denote the full subcategory of consisting of all ξ -Gorenstein projective objects.
Throughout this paper, we always assume that is a triangulated category with enough ξ -projective objects and ξ -injective objects.
Let M be an object in . Beligiannis [11] defined the ξ -extension groups , , that is, be a triangle in ξ . By [11,Corollary 4.12], there exists a long exact sequence of "ξxt" functor. If has enough ξ -injective objects and N is an object in , then there exists a long exact sequence Following Remark 2.10, we usually use the strategy of "dimension shifting," which is an important tool in relative homological theory of triangulated categories. Now, we set For two subcategories and of , we say ⊥ if ⊆ ⊥ (equivalently, ⊆ ⊥ ).
3 Resolution dimensions with respect to a resolving subcategory res.dim inf 0 there exists a exact complex 0 0 in with all objects in . [13] as ξ -Gorenstein projective dimension. We use  to denote the full subcategory of whose objects have finite -resolution dimension.
1. Consider the following triangle: As a similar argument to that of [11,Proposition 4.11], we get the following ξ -exact complex Similarly, we have the following ξ -exact complex Since is resolving, we have that X and Y are objects in . Consider the following triangles: . But from the following triangles in ξ . Then, the following statements are equivalent: Proof. Apply Lemma 3.2. □ Now we can compare resolution dimensions in a given triangle in ξ as follows.
Proposition 3.4. Let be a resolving subcategory of , and let be a triangle in ξ . Then, we have the following statements: and -= C n res.dim . We proceed it by induction on m and n. The case = = m n 0 is trivial. Without loss of generality, we assume ≤ m n, then we can let As a similar argument to that of [11,Proposition 4.11], we get the following ξ -exact complex: and the desired assertion are obtained.
(2) Assume -= B m res.dim and -= C n res.dim . We proceed it by induction on m and n. The case = = m n 0 is trivial. Without loss of generality, we assume ≤ − m n 1, then we can let and a triangle in ξ , it follows that ∈ ( ) K ξ by Remark 2.7. Thus, -≤ − A n res.dim 1 and the desired assertion is obtained.
(3) Assume -= A m res.dim and -= B n res.dim . We proceed it by induction on m and n. The case = = m n 0 is trivial. Without loss of generality, we assume + ≤ m n 1 , then we can let = P 0 i A for > i m. By [18, Theorem 3.8], we have the following ξ -exact complex and the desired assertion is obtained. □ As direct results, we have the following closure properties for the subcategory  .   (1)   ⊆ .
(2) If is resolving, then for any res.dim if and only if  ∩ = . In particular, if ⊥ , and is closed under hokernels of ξ -proper epimorphisms or closed under direct summands, then  ∩ = . Proof.
. By the assumption, we have - . Clearly, ≤ m n. Consider the following ξ -exact complexes: Then, , and thus, -≤ M m res.dim and the desired equality is obtained. Now, we assume that ⊥ and is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. Clearly, We need the following easy and useful observation. (1) If ⊥ , then  ⊥ . In particular, if ⊥ , then  ⊥ .
Proposition 3.10. Let be a subcategory of closed under ξ -extensions, and let be a subcategory of such that is a ξ -cogenerator of . Then, for each ∈ M with -= <∞ M n res.dim , there exist two triangles and In particular, if ⊥ , then the ξ -proper epimorphism ⟶ X M is a right -approximation of M.
Proof. We proceed by induction on n. The case for = n 0 is trivial. If = n 1, there exists a triangle in ξ with ∈ H and ′ ∈ X 1 . Applying cobase change for the triangle (4) along the morphism ⟶ X H 1 , we get the following commutative diagram: Since ξ is closed under cobase changes, we obtain that the triangle is in ξ with -= H res.dim 0. Note that ′ = α u α is ξ -proper epic, so we have that ′ α is ξ -proper epic by [16, Proposition 2.7]; hence, the triangle and ″ ∈ X 0 . Applying cobase change for the triangle (5) along the morphism ′ ⟶ X H 0 0 , we get the following commutative diagram: . For ′ K , by the induction hypothesis, we get a triangle . Applying cobase change for the triangle (7) along the morphism ′ ⟶ K K , we get the following commutative diagram: is in ξ . It follows that ∈ X from the assumption that is closed under ξ -extensions. Since ξ is closed under cobase changes, we obtain the first desired triangle in ξ with -≤ − K n res.dim 1 and ∈ X . For X, since is a ξ -cogenerator of , we get the following triangle and ′ ∈ X . Applying cobase change for the triangle (8) along the morphism ⟶ X H 1 , we get the following commutative diagram: As a similar argument to that of the diagram (6), we obtain that the triangles are in ξ . Thus, (9) is the second desired triangle in ξ with -≤ W n res.dim and ′ ∈ X .
In particular, suppose ⊥ , by Lemma 3.9, we have  (2) and (3), we have -= − K n res.dim 1 and - and is resolving, then there is a triangle Proof.
(1) Suppose is resolving. Applying Corollary 3.6(2) to the triangle (2)  (2) Since ⊥ , we have  ⊥ by Lemma 3.9, and so the result immediately follows from (1).  (10) in ξ with ∈ 0 and -= − K n res.dim 1. By (2), there is a triangle . Applying cobase change for the triangle (10) along the morphism ⟶ ″ K K , we get the following commutative diagram: One can see that the triangle and so, Hom , 0 is exact. Thus, the ξ -proper epimorphism ′ ⟶ X Mis a right -approximation of M and -″ = − K n res.dim 1 in the triangle (11). Note that ″ ∈ ⊥ K , so we have - Let be a subcategory of with ⊥ . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. Then, . Consider the following ξ -exact complex:  Our main result is the following.
Theorem 3.14. Let be a resolving subcategory of and a ξxt-injective ξ -cogenerator of . Assume that is closed under hokernels of ξ -proper epimorphisms or closed under direct summands. For any , then the following statements are equivalent: , where φ is ξ -proper epic, such that = K φ Hoker satisfying -

Additive quotient categories and ξ-cellular towers with respect to a resolving subcategory
In this section, we will further study objects having finite resolution dimension with respect to a resolving subcategory . We first construct adjoint pairs for two kinds of inclusion functors. Then, we characterize objects having finite resolution dimension in terms of a notion of ξ -cellular towers.

Adjoint pairs
Suppose that and are two subcategories of . Denote by [ ] the ideal of consisting of morphisms factoring through some object in . Thus, we have a quotient category /[ ], which is also an additive category. is a morphism in with ∈ X and  ∈ M , then the following statements are equivalent: (1) f factors through an object in .
(2) f factors through an object in  .
Proof. It suffices to show that ( ) ⇒ ( ) 2 1. Suppose that f factors through an object and → g L M : . Consider the following triangle which we call the ξ -cellular tower of M with respect to . According to the above construction, one can obtain the following result by Proposition 3.3.

Applications
In this section, we will construct a new resolving subcategory from a given resolving subcategory, which generalizes the notion of ξ -Gorenstein projective objects given by Asadollahi and Salarian [13]. By applying the previous results to this subcategory, we obtain some known results in [13][14][15]. Proof. Let P be a ξ -projective object. Consider the following ξ -exact complex: , -exact. In particular, ⟶ ⟶ ⟶ ⟶ ⟶ ⟶ P P P P P 0 0 a n d 0 Σ 0 i d 0 i d 0 0 P P are corresponding triangles in ξ . Since ∈ ∩ ⊥ P by Remark 5.2(1). we have ( ) ⊆ ( ) ξ ξ. As a similar argument to the proof of [18, Theorem 4.3(1)], we obtain that ( ) ξ is closed under ξ -extensions and hokernels of ξ -proper epimorphisms. Thus, ( ) ξ is a resolving subcategory of . □