Relative tilting theory in abelian categories I: Auslander–Buchweitz–Reiten approximations theory in subcategories and cotorsion-like pairs

Abstract

In this paper, we introduce a special kind of relative (co)resolutions associated with a pair of classes of objects in an abelian category $\mathcal {C}.$ We will see that, by studying this relative (co)resolutions, we get a possible generalization of a part of the Auslander–Buchweitz approximation theory that is useful for developing n-$\mathcal {X}$-tilting theory in Argudin Monroy and Mendoza Hernández (relative tilting theory in abelian categories II: n-$\mathcal {X}$ tilting theory. arXiv:2112.14873, 2021). With this goal, new concepts as $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category (Argudin Monroy and Mendoza Hernández in categories of quiver representations and relative cotorsion pairs. arXiv:2311.12774v1, 2023). Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of n-$\mathcal {X}$-tilting theory in Argudin Monroy and Mendoza Hernández (2021).

1 Introduction

Broadly speaking, tilting modules over a ring were born with the purpose of answering questions, about the ring we are working on, through the endomorphisms ring of such tilting module. With this idea, tilting modules have become an important tool in different areas like representation theory, homological algebra, and theory of categories [1].

In the last 40 years, tilting theory has been generalized in different ways and contexts. Namely, it has been generalized from finitely generated [14] to infinitely generated modules [2]; from finite [20] to infinite projective dimension [22]; from categories of modules to abstract categories [23]. All these generalizations give rise to a family of different tilting definitions with different properties and objectives.

In this long history of exploring and expanding tilting theory, we would like to mention the contribution made by Auslander and Reiten in [7, 8]. They were interested in the study of covariantly and contravariantly finite subcategories in the category $\textrm{mod}(\Lambda )$ of finitely generated left $\Lambda$-modules, where $\Lambda$ is an Artin algebra. One of the main results of such papers is that they found a bijection between tilting $\Lambda$-modules and covariantly finite categories: the Auslander–Reiten correspondence [8, Thm. 4.4]. Using this bijection, Auslander and Reiten showed the utility of the tilting objects for studying covariantly finite subcategories and vice versa.

This manuscript is the first of two forthcoming papers and is devoted to develop certain foundational aspects needed for the settle and the study of a relative tilting theory on abelian categories following the same philosophy of Auslander and Reiten in [7, 8]. We shall be based mainly in the first part of the homological algebra presented by Auslander and Buchweitz in [6]. We can describe such theory as the study of the (co)resolution dimension over a class $\mathcal {X},$ the relative projective (injective) dimension on $\mathcal {X},$ some closure properties of certain classes, and the existence of $\mathcal {X}$-precovers or $\mathcal {X}$-preenvelopes. In this paper, we will introduce a subtle modification on the studied (co)resolutions which will lead us to new definitions and results on relative homological algebra that are useful for the development of n-$\mathcal {X}$-tilting theory in [4] and the study of representations of quivers in abelian categories [5].

In the second forcoming paper [4], using the fundaments developed in this first work, we set and study a theory of relative tilting classes in abelian categories. The idea of this theory is that, if we have a class $\mathcal {X}$ in an abelian category $\mathcal {C}$, then we can use the structure of $\mathcal {C}$ to define a tilting subcategory $\mathcal {T}$ in $\mathcal {C}$ which is related with $\mathcal {X},$ that is the so-called n-$\mathcal {X}$-tilting subcategory. Notice that, in general, $\mathcal {T}$ has not to be contained in $\mathcal {X}$, and thus, this leads us to study cotorsion-like pairs introduced in Sect. 3 of this first paper. We will show, in the second paper, that this work offers a unified framework of different previous notions of tilting, ranging from Auslander–Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. With this new approach, we will review Bazzoni’s tilting characterization, relative homological dimensions on the induced tilting classes, and parametrise certain cotorsion-like pairs. As an example, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently by Zhu and Zhuang [26]. It is worth mentioning that a relative tilting theory was recently presented by Moradifar and Yassemi in [21] with the goal of studying infinitely generated Gorenstein tilting objects. We believe that our work will be a complementary tool for this research line.

Let us describe briefly the contents and main results of this first paper. Section 2 is devoted to introduce some categorical and homological preliminaries in an arbitrary abelian category $\mathcal {C}$. In particular, we shall recall the definition of homological dimensions relative to a class $\mathcal {X}\subseteq \mathcal {C}$. Namely, for a class $\mathcal {T}\subseteq \mathcal {C},$ $\textrm{pd}_\mathcal {X}(\mathcal {T})$ denotes the $\mathcal {X}$-projective dimension of $\mathcal {T}$ and $\textrm{id}_\mathcal {X}(\mathcal {T})$ denotes the $\mathcal {X}$-injective dimension of $\mathcal {T}.$ Furthermore, in Proposition 2.6, we characterize the classes $\mathcal {T}\subseteq \mathcal {C}$, such that $\textrm{pd}_\mathcal {X}(\mathcal {T})\le n$ through a property called closed by n-quotients in $\mathcal {X}$. We also present the definition of n-$\mathcal {X}$-cluster tilting in the abelian category $\mathcal {C},$ which is a generalization of the n-cluster tilting category given by O. Iyama in [17], and study such categories from the point of view of n-quotients in $\mathcal {X}\subseteq \mathcal {C}.$

In Sect. 3, we present and study new notions that will help us to state a relative Auslander-Reiten correspondence in the second forcoming paper; see [4, Cors. 3.47 and 3.48]. Namely, we shall introduce the notions of $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pairs. These concepts are generalizations of the complete and hereditary cotorsion pairs with the difference that the above-mentioned pairs do not need to be cotorsion pairs. We point out that $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pairs appear in a natural way in [4] and are of the form $({}^\perp (\mathcal {T}^\perp ), \mathcal {T}^\perp )$ for $\mathcal {T}$ an n-$\mathcal {X}$-tilting subcategory of $\mathcal {C}.$ Notice that we do not know if the pair $({}^\perp (\mathcal {T}^\perp ), \mathcal {T}^\perp )$ is a cotorsion pair and a particular case when this occur can be seen in Corollary 3.11. Other context where the $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pairs appear in a natural way is in the study of quiver representations in abelian categories; see [5, Thms. 5.16 and 5.17 and Cor. 5.18].

Section 4 is devoted to develop our main goals. We introduce the type of relative (co)resolutions that interests us and that will be useful for the development of the theory related with some special classes, their closure properties, and the relationship between different relative homological dimensions. Namely, let $\mathcal {X},\mathcal {Y}\subseteq \mathcal {C}$ be classes of objects in an abelian category $\mathcal {C}.$ We introduce, see Definition 4.1, $\mathcal {Y}_\mathcal {X}$-(co)resolutions and the respective (co)resolution dimension associated with the pair $(\mathcal {X},\mathcal {Y})$. These relative (co)resolutions give rise to the relative coresolution classes $(\mathcal {X},\mathcal {Y})^\vee _{\infty }$ and $(\mathcal {X},\mathcal {Y})^\vee ,$ and to the relative resolution classes $(\mathcal {X},\mathcal {Y})^\wedge _{\infty }$ and $(\mathcal {X},\mathcal {Y})^\wedge .$ We point out that these classes already appeared as particular cases in classical tilting theory and Gorenstein homological algebra. Of course, in these particular situations, some closure properties have been studied. In the general case, we studied in this paper their closure properties in Theorems 4.19, 4.20 and Corollary 4.21. These closure properties of the aforementioned relative classes play an important role in the study and development of n-$\mathcal {X}$-tilting theory in the second forthcoming paper [4]. On the other hand, Theorem 4.4 shows the existence of the main approximations that we will be using in the case of relative coresolution classes. This theorem is a possible generalization of the dual result of [6, Thm 1.1] and will play an important role in the development of n-$\mathcal {X}$-tilting theory in [4]. For example, it is fundamental in the proof that the pair $({}^\perp (\mathcal {T}^\perp ), \mathcal {T}^\perp )$ is $\mathcal {X}$-complete if $\mathcal {T}$ is an n-$\mathcal {X}$-tilting class in $\mathcal {C}.$ Finally, since the aforementioned pair is $\mathcal {X}$-complete and $\mathcal {X}$-hereditary, it has sense to study the relationship between the different relative homological dimensions related to any $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pair $(\mathcal {A},\mathcal {B})$ in $\mathcal {C}.$ The main results obtained in this direction are Proposition 4.23 and Theorem 4.24 that will play an important role in the development of the n-$\mathcal {X}$-tilting theory in [4].

Let $\mathcal {X}$ and $\mathcal {T}$ be classes of objects in an abelian category $\mathcal {C}.$ In Sect. 5, we study a new class of objects $C\in \mathcal {C}$ admitting an exact sequence

$$\begin{aligned} 0\rightarrow K\rightarrow T_n\xrightarrow {f_n} T_{n-1}\rightarrow \cdots \rightarrow T_2\xrightarrow {f_2} T_1\xrightarrow {f_1}C\rightarrow 0, \end{aligned}$$

with $T_{i}\in \mathcal {T}\cap \mathcal {X}$ and $\textrm{Ker}(f_{i})\in \mathcal {X}$ $\forall i\in [1,n]$. This is a generalization of the class of modules n-generated by a given module, which were introduced by Bazzoni [9] and Wei [25] as a tool in the characterization of tilting modules. The goal of this section is to review some basic properties of such class that will be used in [4] to characterize when a class $\mathcal {T}$ in $\mathcal {C}$ is n-$\mathcal {X}$-tilting.

It is worth to point out that we will be working in abstract abelian categories without assuming the existence of enough projectives or injectives. Furthermore, we will incorporate examples where our theory can be exploited. Namely, we shall include examples on n-cluster tilting categories, $\mathcal{F}\mathcal{P}_{n}$ objects and relative Gorenstein objects among others.

2 Preliminaries

2.1 Notation

Throughout the paper, $\mathcal {C}$ denotes an abelian category. We will use the Grothendieck’s notation [13] to distinguish abelian categories with further structure. Namely, AB3 (if it has coproducts), AB4 (if it is AB3 and the coproduct functor is exact), and AB5 (if it is AB3 and the direct limit functor is exact).

Given $X,Y\in \mathcal {C}$ and $n\ge 0,$ we will consider the nth Yoneda extensions group $\textrm{Ext}^n_{\mathcal {C}}(X,Y)$ [19, Chap. VII] and the bi-functor $\textrm{Ext}^n_{\mathcal {C}}(-,-):\mathcal {C}^{op}\times \mathcal {C}\rightarrow \text{ Ab }.$ In particular, if $\mathcal {C}$ has enough projectives or injectives, this bi-functor coincides with the n-derived functor of the Hom functor. For any class of objects $\mathcal {X}\subseteq \mathcal {C}$ and any $i\ge 1,$ we define the right ith orthogonal class $\mathcal {X}^{\perp _i}:=\{C\in \mathcal {C}:\;\textrm{Ext}^i_{\mathcal {C}}(-,C)|_{\mathcal {X}}=0\}$ and the right orthogonal class $\mathcal {X}^{\perp }:=\cap _{i>0}\;\mathcal {X}^{\perp _i}$ of $\mathcal {X}.$ Dually, we have the left ith orthogonal class ${}^{\perp _i}\mathcal {X}$ and the left orthogonal class ${}^{\perp }\mathcal {X}$ of $\mathcal {X}.$

With respect to inclusions of classes and objects, $\mathcal {M}\subseteq \mathcal {C}$ means that $\mathcal {M}$ is a class of objects of $\mathcal {C},$ and $C\in \mathcal {C}$ means that C is an object of $\mathcal {C}$. In the case we are given another class of objects $\mathcal {N\subseteq \mathcal {C}}$, then $\mathcal {N}\subseteq \mathcal {M}$ and $C\in \mathcal {M}$ have similar meanings. Finally, the term subcategory means full subcategory.

Associated with some $\mathcal {M}\subseteq \mathcal {C},$ we have the following classes of objects in $\mathcal {C}.$ Namely: the class $\textrm{smd}(\mathcal {M})$ whose objects are all the direct summands of objects in $\mathcal {M};$ the class $\mathcal {M}^{\oplus }$ ($\mathcal {M}^{\oplus _{<\infty }}$) of all the (finite) coproducts of objects in $\mathcal {M};$ $\textrm{Add}(\mathcal {M}):=\textrm{smd}(\mathcal {M}^{\oplus })$ and $\textrm{add}(\mathcal {M}):=\textrm{smd}(\mathcal {M}^{\oplus _{<\infty }}).$ In case $\mathcal {M}=\{M\},$ we have $M^{\oplus }:=\mathcal {M}^{\oplus },$ $M^{\oplus _{<\infty }}:=\mathcal {M}^{\oplus _{<\infty }},$ $\textrm{smd}(M):=\textrm{smd}(\mathcal {M}),$ $\textrm{Add}(M):=\textrm{Add}(\mathcal {M}),$ $\textrm{add}(M):=\textrm{add}(\mathcal {M}),$ $M^{\bot }:=\mathcal {M^{\bot }},$ and $^{\bot }M:={}^{\bot }\mathcal {M}.$

Throughout this paper, we will work with a variety of concepts along with its dual notions. We will omit writing down dual results and notions, but we will be using both of them.

2.2 Relative homological dimensions

We will be using the following known results in homological algebra and its duals.

Theorem 2.1

[19, Chap. VI, Thm. 5.1] Let $0\rightarrow N\rightarrow M\rightarrow K\rightarrow 0$ be an exact sequence in the abelian category $\mathcal {C}$. Then, for any $X\in \mathcal {C},$ there is a long exact sequence of abelian groups induced by $\textrm{Hom}_\mathcal {C}(X,-)$

$$\begin{aligned}&0 \rightarrow \textrm{Hom}_\mathcal {C}(X,N)\rightarrow \textrm{Hom}_\mathcal {C}(X,M)\rightarrow \textrm{Hom}_\mathcal {C}(X,K)\rightarrow \textrm{Ext}^1_\mathcal {C}(X,N)\rightarrow \cdots \\&\quad \cdots \rightarrow \textrm{Ext}^k_\mathcal {C}(X,N)\rightarrow \textrm{Ext}^k_\mathcal {C}(X,M)\rightarrow \textrm{Ext}^k_\mathcal {C}(X,K)\rightarrow \textrm{Ext}^{k+1}_\mathcal {C}(X,N)\rightarrow \cdots . \end{aligned}$$

Lemma 2.2

(Shifting Lemma) Let $0\rightarrow K \rightarrow C_{n-1}\rightarrow \cdots \rightarrow C_{1}\rightarrow C_{0}\rightarrow A\rightarrow 0$ be an exact sequence in the abelian category $\mathcal {C}$, such that $C_{i}\in {}^{\bot }Y$ $\forall i\in [0,n-1],$ where $Y\in \mathcal {C}.$ Then, $\textrm{Ext}^k_\mathcal {C}(K,Y)\cong \textrm{Ext}^{k+n}_\mathcal {C}(A,Y)\,\forall k\ge 1.$

Proof

It can be proved in a similar way as in [19, Chap. VII, Lem. 6.3]. $\square$

Theorem 2.3

[3, Prop. 4.2] Let $\mathcal {C}$ be an AB4 category, $\{A_{i}\}_{i\in I}$ be a family of objects in $\mathcal {C}$ and $B\in \mathcal {C}.$ Then, for any $n\ge 1,$ there is a natural isomorphism

$$\begin{aligned} \Psi _{n}:\textrm{Ext}^n_\mathcal {C}(\bigoplus _{i\in I}\, A_i, B)\rightarrow \prod _{i\in I}\textrm{Ext}^n_\mathcal {C}(A_i,B). \end{aligned}$$

For $\mathcal {B},\mathcal {A}\subseteq \mathcal {C}$ and $C\in \mathcal {C},$ we recall the following notions [6]. The $\mathcal {A}$-projective dimension of C is $\textrm{pd}_\mathcal {A}(C):=\min \left\{ n\in \mathbb {N}\;:\; \textrm{Ext}^k_\mathcal {C}(C,-)|_\mathcal {A}=0\,\forall k>n\right\} ;$ the $\mathcal {A}$-projective dimension of $\mathcal {B}$ is $\textrm{pd}_\mathcal {A}(\mathcal {B}):=\sup \left\{ \textrm{pd}_\mathcal {A}(B):\; B\in \mathcal {B}\right\} .$ The $\mathcal {A}$-injective dimension $\textrm{id}_\mathcal {A}(C)$ of C and $\mathcal {A}$-injective dimension $\textrm{id}_\mathcal {A}(\mathcal {B})$ of $\mathcal {B}$ are defined dually. For a pair $(\mathcal {X},\omega )\subseteq \mathcal {C}^2,$ it is said that $\omega$ is a relative cogenerator in $\mathcal {X}$ if $\omega \subseteq \mathcal {X}$ and any $X\in \mathcal {X}$ admits an exact sequence $0\rightarrow X\rightarrow W\rightarrow X'\rightarrow 0$ in $\mathcal {C},$ with $W\in \omega$ and $X'\in \mathcal {X}.$ The class $\omega$ is $\mathcal {X}$-injective if $\textrm{id}_\mathcal {X}(\omega )=0.$ Dually, we have the notions of relative generator in $\mathcal {X}$ and $\mathcal {X}$-projective class.

2.3 Relative n-quotients and n-subobjects

Definition 2.4

Let $\mathcal {C}$ be an abelian category, $\mathcal {Y}\subseteq \mathcal {X}\subseteq \mathcal {C}$ and $n\ge 1.$

• (a) $\mathcal {Y}$ is closed by n-quotients in $\mathcal {X}$ if for any exact sequence in $\mathcal {C}$

  $$\begin{aligned} 0\rightarrow A\rightarrow Y_{n}\xrightarrow {\varphi _{n}} Y_{n-1}\rightarrow \cdots \rightarrow Y_{1}\xrightarrow {\varphi _{1}}B\rightarrow 0, \end{aligned}$$

  with $Y_{i}\in \mathcal {Y},$ $\textrm{Ker}(\varphi _{i})\in \mathcal {X}$ $\forall i\in [1,n]$ and $B\in \mathcal {X},$ we have that $B\in \mathcal {Y}.$

• (b) $\mathcal {Y}$ is closed by n-subobjects in $\mathcal {X}$ if for any exact sequence in $\mathcal {C}$

  $$\begin{aligned} 0\rightarrow A\rightarrow Y_{1}\xrightarrow {\varphi _{1}}Y_2\rightarrow \cdots \rightarrow Y_{n}\xrightarrow {\varphi _{n}}B\rightarrow 0, \end{aligned}$$

  with $Y_{i}\in \mathcal {Y},$ $\textrm{Im}(\varphi _{i})\in \mathcal {X}$ $\forall i\in [1,n]$ and $A\in \mathcal {X}$, we have that $A\in \mathcal {Y}.$

Example 2.5

Let n be a positive integer, and $\mathcal {C}$ be an AB5 category admitting a set $\omega$, such that $\omega ^{\oplus }$ is a relative generator in $\mathcal {C}.$

• (1) Denote by $\mathcal{F}\mathcal{P}_{n}$ the class of all the objects $X\in \mathcal {C}$, such that the functor $\textrm{Ext}^i_\mathcal {C}(X,-)$ preserves direct limits $\forall i\in [0,n-1].$ By [12, Lem. 2.11], $\mathcal{F}\mathcal{P}_{n}$ is closed by $(n+1)$-quotients in $\mathcal {C}$ if $\omega \subseteq \mathcal {F}\mathcal {P}_{1}.$

• (2) Denote by $\mathcal {C}_{n}$ the class of all the objects $X\in \mathcal{F}\mathcal{P}_{n}$, such that every subobject $Y\subseteq X$ in the class $\mathcal{F}\mathcal{P}_{n-1}$ is in fact in $\mathcal{F}\mathcal{P}_{n}.$ Then, by [12, Cor. 4.5], $\mathcal {C}_{n}$ is closed by 1-quotients in $\mathcal{F}\mathcal{P}_{n-1}.$

The following connection between “closed under n-quotients” and the relative projective dimension will be very useful in our paper [4], where we develop the theory of n-$\mathcal {X}$-tilting classes.

Proposition 2.6

Let $\mathcal {X},\mathcal {T}\subseteq \mathcal {C}$ and $\alpha \subseteq \mathcal {T}^{\bot }\cap \mathcal {X}^\perp$ be a relative cogenerator in $\mathcal {X}.$ Then, for $n\ge 1,$ $\mathcal {X}\cap \mathcal {T}^{\bot }$ is closed by n-quotients in $\mathcal {X}$ if, and only if, $\textrm{pd}_\mathcal {X}(\mathcal {T})\le n.$

Proof

$(\Rightarrow )$ Note that $\alpha \subseteq \mathcal {X}\cap \mathcal {T}^{\bot }.$ Let $X\in \mathcal {X}$ and $M\in \mathcal {T}.$ Since $\alpha$ is a relative cogenerator in $\mathcal {X},$ there is an exact sequence $0\rightarrow X\rightarrow I_{0}\rightarrow \cdots \rightarrow I_{n-1}\xrightarrow {f}V\rightarrow 0,$ with $K:=\textrm{Ker}(f)\in \mathcal {X},$ $V\in \mathcal {X}$ and $I_{i}\in \alpha$ $\forall i\in [0,n-1]$. By applying the functor $\textrm{Hom}_\mathcal {C}(M,-)$ to the exact sequence $0\rightarrow K\rightarrow I_{n-1}\rightarrow V\rightarrow 0$ and since $\textrm{Ext}^i_\mathcal {C}(M,I_{n-1})=0=\textrm{Ext}^{i+1}_\mathcal {C}(M,I_{n-1}),$ it follows that $\textrm{Ext}^i_\mathcal {C}(M,V)\cong \textrm{Ext}^{i+1}_\mathcal {C}(M,K)$ $\forall \;i\ge 1.$ By the dual of Lemma 2.2, $\textrm{Ext}^{i+1}_\mathcal {C}(M,K)\cong \textrm{Ext}^{n+i}_\mathcal {C}(M,X).$ Now, since $\mathcal {T}^{\bot }\cap \mathcal {X}$ is closed by n-quotients in $\mathcal {X}$ and $V\in \mathcal {T}^{\bot }\cap \mathcal {X},$ we get

$$\begin{aligned} 0=\textrm{Ext}^i_\mathcal {C}(M,V)\cong \textrm{Ext}^{i+1}_\mathcal {C}(M,K)\cong \textrm{Ext}^{n+i}_\mathcal {C}(M,X)\;\forall i\ge 1, \end{aligned}$$

and thus, $\textrm{pd}_\mathcal {X}(M)\le n.$

$(\Leftarrow )$ Let $0\rightarrow A\rightarrow X_{n}\xrightarrow {\varphi _n}X_{n-1}\rightarrow \cdots \rightarrow X_{1}\xrightarrow {\varphi _1} B\rightarrow 0$ be an exact sequence, where $X_{1},\cdots ,X_{n}\in \mathcal {X}\cap \mathcal {T}^{\bot },$ $B\in \mathcal {X}$ and $\textrm{Ker}(\varphi _{i})\in \mathcal {X}$ $\forall i\in [1,n]$. By the dual of Lemma 2.2, $\textrm{Ext}^k_\mathcal {C}(M,B)\cong \textrm{Ext}^{n+k}_\mathcal {C}(M,A)\;\forall k\ge 1,$ where $\textrm{Ext}^{n+k}_\mathcal {C}(M,A)=0\;\forall k\ge 1$, since $A\in \mathcal {X}$ and $\textrm{pd}_\mathcal {X}(\mathcal {T})\le n.$ Therefore, $B\in \mathcal {T}^{\bot }\cap \mathcal {X}.$ $\square$

We finish this section with an application in the n-cluster tilting theory. To do that, we generalize the notion of n-cluster tilting category, given by O. Iyama in [17].

Definition 2.7

Let $\mathcal {X},\mathcal {T}\subseteq \mathcal {C},$ $n\ge 1.$ We say that $\mathcal {T}$ is n-$\mathcal {X}$-cluster tilting in $\mathcal {C}$ if the following conditions hold true.

• (a) $\mathcal {T}=\textrm{add}(\mathcal {T}).$

• (b) There exists $\alpha \subseteq \mathcal {X}^\perp \cap \mathcal {T}^\perp$ which is a relative cogenerator in $\mathcal {X}.$

• (c) There exists $\beta \subseteq {}^\perp \mathcal {X}\cap {}^\perp \mathcal {T}$ which is a relative generator in $\mathcal {X}.$

• (d) $\mathcal {T}$ is functorially finite.

• (e) $\mathcal {X}\cap (\cap _{i=1}^{n-1}{}^{\perp _i}\mathcal {T})=\mathcal {T}=\mathcal {X}\cap ( \cap _{i=1}^{n-1}\mathcal {T}^{\perp _i}).$

Let $\Lambda$ be a finite-dimensional k-algebra and $\textrm{mod}(\Lambda )$ be the category of finitely generated left $\Lambda$-modules. The notion of n-cluster tilting subcategory in $\textrm{mod}(\Lambda )$ was introduced by Osamu Iyama in the study of a higher analogue of the classical Auslander correspondence between representation finite algebras and Auslander algebras [17]. In this case, the term n-$\textrm{mod}(\Lambda )$-cluster tilting matches with Iyama’s definition of n-cluster tilting subcategory in $\textrm{mod}(\Lambda ).$

The following corollary has been a key result in [16] to relate n-$\mathcal {X}$-cluster tilting with m-periodic relative Gorenstein projective objects.

Corollary 2.8

For an n-$\mathcal {X}$-cluster tilting class $\mathcal {T}$ in the abelian category $\mathcal {C},$ the following statements hold true.

• (a) $\alpha ,\beta \subseteq \mathcal {T}\subseteq \mathcal {X}.$

• (b) If $\textrm{pd}_\mathcal {X}(\mathcal {T})\le n-1,$ then $\mathcal {T}=\mathcal {X}\cap \mathcal {T}^\perp =\mathcal {X}.$

• (c) If $\textrm{id}_\mathcal {X}(\mathcal {T})\le n-1,$ then $\mathcal {T}=\mathcal {X}\cap {}^\perp \mathcal {T}=\mathcal {X}$.

• (d) If $\textrm{pd}_\mathcal {X}(\mathcal {X})\le n-1,$ then $\mathcal {X}\cap \mathcal {T}^\perp =\mathcal {T}=\mathcal {X}\cap {}^\perp \mathcal {T}=\mathcal {X}$.

Proof

(a) If $n=1,$ then by Definition 2.7 (e) $\mathcal {T}=\mathcal {X}$ and there is nothing to prove. Let $n\ge 2.$ Then, by Definition 2.7 (b,e), $\alpha \subseteq \mathcal {X}^\perp \cap \mathcal {X}\cap \mathcal {T}^\perp \subseteq \mathcal {X}\cap \cap _{i=1}^{n-1} \mathcal {T}^{\perp _i}= \mathcal {T}.$ Similarly, we also get that $\beta \subseteq \mathcal {T}.$

(b) Let $\textrm{pd}_\mathcal {X}(\mathcal {T})\le n-1.$ If $n=1$, then $\mathcal {T}\subseteq \mathcal {T}^\perp$, and thus, $\mathcal {T}=\mathcal {T}^\perp \cap \mathcal {T}.$ Moreover, by Definition 2.7 (e) $\mathcal {T}=\mathcal {X}$, and hence, (b) holds true in this case.

Suppose that $n\ge 2.$ Since $\textrm{pd}_\mathcal {X}(\mathcal {T})\le n-1,$ it follows from Definition 2.7 (e) that $\mathcal {X}\cap \mathcal {T}^\perp =\mathcal {X}\cap \cap _{i=1}^{n-1}\mathcal {T}^{\perp _i}=\mathcal {T}.$ Furthermore, by Proposition 2.6, $\mathcal {T}^{\bot }\cap \mathcal {X}$ is closed by $(n-1)$-quotients in $\mathcal {X}.$ Then, using that $\beta \subseteq \mathcal {T}$ is a relative generator in $\mathcal {X},$ we can conclude that $\mathcal {X}\subseteq \mathcal {T}.$

(c) It follows as in the proof of (b) using the dual of Proposition 2.6.

(d) It follows from (b) and (c) since $\textrm{pd}_\mathcal {X}(\mathcal {T})\le \textrm{pd}_\mathcal {X}(\mathcal {X})$ and $\textrm{id}_\mathcal {X}(\mathcal {T})\le \textrm{id}_\mathcal {X}(\mathcal {X})=\textrm{pd}_\mathcal {X}(\mathcal {X}).$ $\square$

3 Cotorsion-like pairs and related notions

A main tool in the study of relative homological dimensions is the notion of cotorsion pair which was introduced by Salce in [24]. Namely, a pair $(\mathcal {A},\mathcal {B})$ of classes of objects in an abelian category $\mathcal {C}$ is a cotorsion pair if $\mathcal {A}={}^{\perp _1}\mathcal {B}$ and $\mathcal {B}=\mathcal {A}^{\perp _1}.$

In this section, we introduce a more general notion of relative cotorsion pair which will be useful for the development of n-$\mathcal {X}$-tilting theory in [4].

Definition 3.1

Let $\mathcal {A},\mathcal {B},\mathcal {X}\subseteq \mathcal {C}.$ The pair $(\mathcal {A},\mathcal {B})$ is left (resp. right) cotorsion pair in $\mathcal {X}$ if $\mathcal {A}\cap \mathcal {X}={}^{\perp _1}\mathcal {B}\cap \mathcal {X}$ (resp. $\mathcal {B}\cap \mathcal {X}=\mathcal {A}^{\perp _1}\cap \mathcal {X}).$ In case both conditions hold true, we call $(\mathcal {A},\mathcal {B})$ cotorsion pair in $\mathcal {X}.$

Note that, if $(\mathcal {A}, \mathcal {B})$ is a cotorsion pair in $\mathcal {C},$ then $(\mathcal {A}, \mathcal {B})$ is a cotorsion pair in $\mathcal {X}$ for any $\mathcal {X}\subseteq \mathcal {C}$. In case $\mathcal {X}=\mathcal {C},$ we simply say that $(\mathcal {A}, \mathcal {B})$ is a left (right) cotorsion pair if it is a left (resp. right) cotorsion pair in $\mathcal {X}.$

Example 3.2

Let $\mathcal {C}$ be an abelian category and $\mathcal {X}\subseteq \mathcal {C}.$

• (1) Let $\mathcal {X}$ be closed under extensions. The cotorsion pairs $(\mathcal {A}, \mathcal {B})$ in $\mathcal {X}$ satisfying that $\mathcal {A},\mathcal {B}\subseteq \mathcal {X}$ were amply studied in [10] and called $\mathcal {X}$-cotorsion pairs. In particular, the authors proved that, for a left thick class $\mathcal {X}\subseteq \mathcal {C}$ (see Definition 4.13), and an $\mathcal {X}$-injective relative cogenerator $\omega =\textrm{smd}(\omega )$ in $\mathcal {X},$ the pair $(\mathcal {X},\omega ^{\wedge })$ is a cotorsion pair in $\mathcal {X}^{\wedge }$ [10, Thm. 3.6].

• (2) The cotorsion pairs $(\mathcal {A}, \mathcal {B})$ in $\mathcal {X}$ satisfying that $\mathcal {A}=\textrm{smd}(\mathcal {A})$ and $\mathcal {B}=\textrm{smd}(\mathcal {B})$ were amply studied in [15] and called cotorsion pairs cut along $\mathcal {X}.$ In this paper, we can find several examples and applications. Namely, examples in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves; and as applications there are some results on the finitistic dimension conjecture, on the existence of right adjoints of quotient functors by Serre categories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

3.1 Relative hereditary cotorsion-like pairs

Let $\mathcal {C}$ be an abelian category, and we recall that a pair $(\mathcal {A},\mathcal {B})$ in $\mathcal {C}$ is hereditary if $\textrm{id}_{\mathcal {A}}(\mathcal {B})=0.$ Examples of such pairs are $({}{}^{\bot }\mathcal {Y},({}^{\bot }{\mathcal {Y}})^{\bot })$ and $({}^{\bot }\left( {\mathcal {Y}}^{\bot }\right) ,{\mathcal {Y}}^{\bot }),$ for any $\mathcal {Y}\subseteq \mathcal {C}.$ Notice that such pairs are not necessarily cotorsion pairs. However, if $\mathcal {C}$ has enough projectives and injectives (see Corollary 3.11), we have that the pairs $({}{}^{\bot }\mathcal {Y},({}^{\bot }{\mathcal {Y}})^{\bot })$ and $({}^{\bot }\left( {\mathcal {Y}}^{\bot }\right) ,{\mathcal {Y}}^{\bot })$ are cotorsion pairs. Hereditary cotorsion pairs are an important tool for building resolutions and coresolutions in homological algebra. Hence, it is of our interest to use this kind of pairs in more general contexts. Unfortunately, if $\mathcal {C}$ is an abelian category without enough projectives or injectives, we do not know if $(\mathcal {A}, \mathcal {B})$ is a cotorsion pair. Nonetheless, we will see in the following lines that, under the proper hypotheses, $(\mathcal {A}, \mathcal {B})$ can be seen as a cotorsion pair inside a class $\mathcal {X}\subseteq \mathcal {C}$ that admits an $\mathcal {X}$-injective cogenerator and an $\mathcal {X}$-projective generator. In a way, this will represent a relative notion of cotorsion pairs.

Definition 3.3

Let $\mathcal {C}$ be an abelian category and $\mathcal {M},\mathcal {X}\subseteq \mathcal {C}$.

• (a) $\mathcal {M}$ is closed under mono-cokernels in $\mathcal {M}\cap \mathcal {X}$ if, for any exact sequence

  $$\begin{aligned} 0\rightarrow M \rightarrow M' \rightarrow M''\rightarrow 0 \text{ with } M,M'\in \mathcal {M}\cap \mathcal {X}\text{, } \end{aligned}$$

  we have that $M''\in \mathcal {M}$. In case $\mathcal {M}\subseteq \mathcal {X}$, we will simply say that $\mathcal {M}$ is closed under mono-cokernels. The classes which are closed under epi-kernels are defined dually.

• (b) $\mathcal {M}$ is $\mathcal {X}$-resolving if $\mathcal {M}$ contains an $\mathcal {X}$-projective relative generator in $\mathcal {X}$, and $\mathcal {M}$ is closed under epi-kernels in $\mathcal {M}\cap \mathcal {X}$ and under extensions. The $\mathcal {X}$-coresolving classes are defined dually.

Lemma 3.4

Let $\mathcal {X},\mathcal {B}\subseteq \mathcal {C}$ be classes, such that $\mathcal {B}$ is $\mathcal {X}$-coresolving. Then

$$\begin{aligned} ^{\bot _{1}}\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}={}{}^{\bot }\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}\text{. } \end{aligned}$$

Proof

It is enough to prove that $^{\bot _{1}}\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}\subseteq {}{}^{\bot }\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}$. With this goal, we will show that $^{\bot _{k}}\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}\subseteq {}{}^{\bot _{k+1}}\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}\,\forall k>0\text{. }$ Let $C\in {}{}^{\bot _{k}}\left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {X}$ and $\alpha \subseteq \mathcal {B}\cap \mathcal {X}$ be an $\mathcal {X}$-injective relative cogenerator in $\mathcal {X}$. Hence, any $M\in \mathcal {B}\cap \mathcal {X}$ admits an exact sequence $\eta :\quad 0 \rightarrow M \rightarrow A \rightarrow K \rightarrow 0,$ with $A\in \alpha \subseteq \mathcal {B}\cap \mathcal {X}$ and $K\in \mathcal {X}$. Observe that $K\in {\mathcal {B}}\cap \mathcal {X}$, since ${\mathcal {B}}$ is closed under mono-cokernels in $\mathcal {B}\cap \mathcal {X}$. Hence, in the exact sequence $\textrm{Ext}^{k}_{\mathcal {C}}(C,K)\rightarrow \textrm{Ext}^{k+1}_{\mathcal {C}}(C,M)\rightarrow \textrm{Ext}^{k+1}_{\mathcal {C}}(C,A),$ we have $\textrm{Ext}^{k+1}_{\mathcal {C}}(C,A)=0$ and $\textrm{Ext}^{k}_{\mathcal {C}}(C,K)=0.$ Thus, $\textrm{Ext}^{k+1}_{\mathcal {C}}(C,\mathcal {B}\cap \mathcal {X})=0.$ $\square$

As a direct consequence of Lemma 3.4, we get the following corollary.

Corollary 3.5

Let $\mathcal {A},\mathcal {B},\mathcal {X}\subseteq \mathcal {C}$ be a classes, such that $\textrm{Ext}^{1}_{\mathcal {C}}(\mathcal {A}\cap \mathcal {X}, \mathcal {B}\cap \mathcal {X})=0$. If $\mathcal {B}$ is $\mathcal {X}$-coresolving and $\mathcal {A}$ is closed under epi-kernels in $\mathcal {A}\cap \mathcal {X}$, then for any exact sequence $0 \rightarrow K \rightarrow A \rightarrow A'\rightarrow 0\text{, }$ with $A,A'\in \mathcal {A}\cap \mathcal {X}$ and $K\in \mathcal {X}$, we have $K\in {}{}^{\bot }\left( \mathcal {B}\cap \mathcal {X}\right)$.

Lemma 3.6

Let $\mathcal {A},\mathcal {B},\mathcal {X}\subseteq \mathcal {C}$ be classes, such that $\textrm{Ext}^{1}_{\mathcal {C}}(\mathcal {A}\cap \mathcal {X},\mathcal {B}\cap \mathcal {X})=0$ and $\mathcal {A}$ contains a class which is an $\mathcal {X}$-projective relative generator in $\mathcal {X}.$ Then, the following statements are equivalent:

• (a) For any exact sequence $0 \rightarrow K \rightarrow A \rightarrow A' \rightarrow 0$, with $A,A'\in \mathcal {A}\cap \mathcal {X}$ and $K\in \mathcal {X}$, we have that $K\in {}{}^{\bot }\left( \mathcal {B}\cap \mathcal {X}\right) .$

• (b) $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$.

Proof

It is straightforward and we left it to the reader. $\square$

Definition 3.7

Let $\mathcal {C}$ be an abelian category and $\mathcal {X}\subseteq \mathcal {C}$. A pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {C}$ is $\mathcal {X}$-hereditary if $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$. In case that $\mathcal {X}=\mathcal {C}$, we will simply say that $(\mathcal {A}, \mathcal {B})$ is hereditary.

Let $\mathcal {Y}\subseteq \mathcal {C}.$ As was mentioned before, we do not know if $(^{\bot }(\mathcal {Y}^{\bot }), \mathcal {Y}^{\bot })$ is a cotorsion pair. In the next Lemmas, we discuss this phenomenon. We will see that, under some conditions related to a given class $\mathcal {X}$, it is possible to find a cotorsion pair $(\mathcal {A}, \mathcal {B})$, such that $\mathcal {B}\cap \mathcal {X}=\mathcal {Y}^{\bot }\cap \mathcal {X}$, or such that $\mathcal {A}\cap \mathcal {X}={}^{\bot }(\mathcal {Y}^{\bot })\cap \mathcal {X}$.

Lemma 3.8

Let $\mathcal {X},\mathcal {Y}\subseteq \mathcal {C}$ be such that there is an $\mathcal {X}$-injective relative cogenerator in $\mathcal {Y}.$ Then, $^{\bot }\mathcal {Y}\cap {\mathcal {X}}={}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\text{. }$

Proof

Let $\alpha$ be an $\mathcal {X}$-injective relative cogenerator in $\mathcal {Y}$ and $M\in \mathcal {Y}$. Then, there is an exact sequence $0\rightarrow M\rightarrow A_{0}\rightarrow \cdots \rightarrow A_{n}\rightarrow A_{n+1}\rightarrow \cdots$ with $A_{i}\in \alpha$ and $M_{i+1}:={\text {Im}}\left( A_{i}\rightarrow A_{i+1}\right) \in \mathcal {Y}$ $\forall i\ge 0$. Consider the set $\mathcal {N}_{M}:=\left\{ M_{i}\right\} {}_{i=0}^{\infty }$, where $M_{0}=M$. Since $\alpha$ is $\mathcal {X}$-injective, by the Shifting Lemma, we have $\textrm{Ext}^{1}_{\mathcal {C}}(X,M_{i})\cong \textrm{Ext}^{1+i}_{\mathcal {C}}(X,M)\forall X\in \mathcal {X}\,,\forall i\ge 0\text{. }$ Therefore, $^{\bot }M\cap {\mathcal {X}}={}^{\bot _{1}}\mathcal {N}_{M}\cap \mathcal {X}$. Finally, note that $\mathcal {Y}=\bigcup _{M\in \mathcal {Y}}\mathcal {N}_{M}$, and hence, ${}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}={}^{\bot }\mathcal {Y}\cap {\mathcal {X}}$. $\square$

Proposition 3.9

Let $\mathcal {Y}\subseteq \mathcal {X}\subseteq \mathcal {C}$ be classes, such that there is an $\mathcal {X}$-injective relative cogenerator in $\mathcal {Y}$ and there is an $\mathcal {X}$-projective relative generator in $\mathcal {X}$ contained in $^{\bot }\mathcal {Y}.$ Then, $({}{}^{\bot }\mathcal {Y}\cap \mathcal {X},\left( {}{}^{\bot }\mathcal {Y}\cap \mathcal {X}\right) ^{\bot }\cap \mathcal {X})$ is a cotorsion pair in $\mathcal {X}$.

Proof

By Lemma 3.8, $^{\bot }\mathcal {Y}\cap {\mathcal {X}}={}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}.$ Since $^{\bot }\mathcal {Y}$ is $\mathcal {X}$-resolving, by the dual of Lemma 3.4, we have

$$\begin{aligned} \left( {}{}^{\bot }\mathcal {Y}\cap \mathcal {X}\right) ^{\bot }\cap \mathcal {X}=\left( {}{}^{\bot }\mathcal {Y} \cap \mathcal {X}\right) ^{\bot _{1}}\cap \mathcal {X}=\left( {}{}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\right) ^{\bot _{1}} \cap \mathcal {X}\text{. } \end{aligned}$$

Therefore, $\mathfrak {C}:=({}{}^{\bot }\mathcal {Y}\cap \mathcal {X},\left( {}{}^{\bot }\mathcal {Y}\cap \mathcal {X}\right) ^{\bot })$ is a right cotorsion pair in $\mathcal {X}$. It remains to prove that $\mathfrak {C}$ is a left cotorsion pair in $\mathcal {X}$. To do this, we must show that $^{\bot _{1}}\left( \left( {}{}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\right) ^{\bot _{1}}\cap \mathcal {X}\right) \cap \mathcal {X}={}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\text{. }$ Indeed, note that

$$\begin{aligned}{} & {} ^{\bot _{1}}\left( \left( {}{}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\right) ^{\bot _{1}}\cap \mathcal {X}\right) \cap \mathcal {X}\subseteq {}^{\bot _{1}}\left( \left( {}{}^{\bot _{1}}\mathcal {Y}\right) ^{\bot _{1}}\cap \mathcal {X}\right) \cap \mathcal {X}\subseteq {}^{\bot _{1}}\left( \mathcal {Y}\cap \mathcal {X}\right) \cap \mathcal {X}\\{} & {} \qquad \qquad ={}^{\bot _{1}}\mathcal {Y} \cap \mathcal {X}\text{, } \end{aligned}$$

and $^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\subseteq {}{}^{\bot _{1}}\left( \left( {}{}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\right) ^{\bot _{1}}\right) \cap \mathcal {X}\subseteq {}{}^{\bot _{1}}\left( \left( {}{}^{\bot _{1}}\mathcal {Y}\cap \mathcal {X}\right) ^{\bot _{1}}\cap \mathcal {X}\right) \cap \mathcal {X}.$ $\square$

In case $\mathcal {Y}=\{T\}$, we can claim the following statements. For $\omega \subseteq \mathcal {C},$ we denote by $\omega _{\infty }^{\wedge }$ the class of all the objects $C\in \mathcal {C}$ admitting an infinite exact sequence $\cdots \rightarrow W_{n}\rightarrow \cdots \rightarrow W_{1}\rightarrow W_{0}\rightarrow C\rightarrow 0\text{, }$ with $W_{i}\in \omega \cup \left\{ 0\right\}$ $\forall \, i\ge 0.$ We denote by $\omega ^{\wedge }$ the class of all the objects $C\in \mathcal {C}$ admitting a finite exact sequence $0\rightarrow W_{n}\rightarrow \cdots \rightarrow W_{1}\rightarrow W_{0}\rightarrow C\rightarrow 0\text{, }$ with $W_{i}\in \omega$ $\forall \, i\in [0,n].$ Now, we consider the following result whose idea comes from M. Auslander.

Lemma 3.10

For $T\in \mathcal {C}$ and $\omega ,\mathcal {X}\subseteq \mathcal {C}$ such that $\omega$ is $\mathcal {X}$-projective, the following statements hold true.

• (a) if $T\in \omega _{\infty }^{\wedge }$, then there is a set $\mathcal {S}\subseteq \mathcal {C}$, such that $T^{\bot }\cap {\mathcal {X}}=\mathcal {S}{}^{\bot _{1}}\cap \mathcal {X}.$

• (b) if $T\in \omega ^{\wedge }$, then there is an object $S\in \mathcal {C}$, such that $T^{\bot }\cap {\mathcal {X}}=S{}^{\bot _{1}}\cap \mathcal {X}.$

• (c) if $\mathcal {C}$ is an AB4 category and $T\in \omega _{\infty }^{\wedge }$, then there is an object $S\in \mathcal {C}$, such that $T^{\bot }\cap {\mathcal {X}}=S{}^{\bot _{1}}\cap \mathcal {X}.$

Proof

Let $T\in \omega _{\infty }^{\wedge }.$ Then, there is an exact sequence

$$\begin{aligned} \cdots \rightarrow W_{n+1}\rightarrow W_{n}\rightarrow \cdots \rightarrow W_{1}\rightarrow W_{0}\rightarrow T\rightarrow 0\text{, } \end{aligned}$$

with $W_{i}\in \omega \cup \left\{ 0\right\}$ and $T_{i+1}:={\text {Im}}\left( W_{i+1}\rightarrow W_{i}\right) \,\forall i\ge 0$. Consider $\mathcal {S}:=\left\{ T_{i}\right\} {}_{i=0}^{\infty }$, where $T_{0}:=T$. Since $\omega \subseteq {}{}^{\bot }\mathcal {X}$, by the Shifting Lemma, we have

$$\begin{aligned} \textrm{Ext}^{i}_{\mathcal {C}}(T,X)\cong \textrm{Ext}^{1}_{\mathcal {C}}(T_{i},X)\forall i\ge 0,\forall X\in \mathcal {X}, \end{aligned}$$

and thus, $\mathcal {S}^{\bot _{1}}\cap \mathcal {X}=T^{\bot }\cap \mathcal {X};$ proving (a). If $\mathcal {C}$ is AB4, by Theorem 2.3, the coproduct $S:=\bigoplus _{i=0}^{\infty }T_{i}$ satisfies that $S^{\bot _{1}}=\mathcal {S}^{\bot _{1}}$, and thus, (c) holds true. The proof of (b) is quite similar to the one given of (c), since in this case, the set $\mathcal {S}$ is finite. $\square$

The following is a particular case when the pair $({}^{\perp }(\mathcal {T}^\perp ),\mathcal {T}^\perp )$ is a cotorsion pair, for a class $\mathcal {T}\subseteq \mathcal {C}.$

Corollary 3.11

Let $\mathcal {C}$ be an abelian category with enough projectives and injectives, and let $\mathcal {T}\subseteq \mathcal {C}.$ Then, $({}^{\perp }(\mathcal {T}^\perp ),\mathcal {T}^\perp )$ is a hereditary cotorsion pair in $\mathcal {C}.$

Proof

Let $T\in \mathcal {T}.$ Since $\mathcal {C}$ has enough projectives, by Lemma 3.10 (a), there is a set $S_T\subseteq \mathcal {C}$ such that $T^\perp =S^{\perp _1}_T.$ Now, consider the class $S:=\bigcup _{T\in \mathcal {T}} S_T.$ Then, $S^{\perp _1}=\bigcap _{T\in \mathcal {T}} S^{\perp _1}_T=\bigcap _{T\in \mathcal {T}} T^\perp =\mathcal {T}^\perp .$ Finally, since $\mathcal {C}$ has enough injectives, by Lemma 3.4, we get that ${}^{\perp _1}(S^{\perp _1})={}^{\perp _1}(\mathcal {T}^{\perp })={}^{\perp }(\mathcal {T}^{\perp })$, and thus, the result follows. $\square$

3.2 Relative complete pairs

We will be using the following notation and vocabulary for approximations. We start this section by recalling the well-known notions of precovers and preenvelopes.

For a class $\mathcal {Z}$ of objects in an abelian category $\mathcal {C},$ a morphism $f:Z\rightarrow M$ in $\mathcal {C}$ is called a $\mathcal {Z}$-precover if $Z\in \mathcal {Z}$ and $\textrm{Hom}_{\mathcal {C}}(Z', f):\textrm{Hom}_{\mathcal {C}}(Z', Z)\rightarrow \textrm{Hom}_{\mathcal {C}}(Z', M)$ is surjective $\forall Z'\in \mathcal {Z}.$ A $\mathcal {Z}$-precover $Z\rightarrow M$ is called special if it fits in an exact sequence $0 \rightarrow M' \rightarrow Z \rightarrow M \rightarrow 0\text{, } \text{ where } M'\in \mathcal {Z}^{\bot _{1}}\text{. }$ It is said that $\mathcal {Z}$ is precovering if each $C\in \mathcal {C}$ admits a $\mathcal {Z}$-precover $Z\rightarrow C.$ The notions of (special) $\mathcal {Z}$-preenvelope and preenveloping class are defined dually. The class $\mathcal {Z}$ is called functorially finite if it is precovering and preenveloping. In the case of an Artin algebra $\Lambda ,$ a precovering (resp. preenveloping) class in ${\text {mod}}(\Lambda )$ is usually called contravariantly (resp. covariantly) finite.

To develop the relative n-$\mathcal {X}$tilting theory in [4], we need a relative version of special precover (preenvelope). We elaborate these ideas below.

Definition 3.12

Let $\mathcal {C}$ be an abelian category and $\mathcal {Z},\mathcal {X}\subseteq \mathcal {C}.$ We say that $\mathcal {Z}$ is special precovering in $\mathcal {X}$ if any $X\in \mathcal {X}$ admits an exact sequence

$$\begin{aligned} 0 \rightarrow B \rightarrow A \rightarrow X \rightarrow 0\text{, } \text{ with } A\in \mathcal {Z}\cap \mathcal {X} \text{ and } B\in \mathcal {Z}^{\bot _{1}}\cap \mathcal {X} \text{. } \end{aligned}$$

The notion of special preenveloping in $\mathcal {X}$ is defined dually.

The previous notions are related with the notion of complete cotorsion pair. Let us present the following definitions that generalize such notion.

Definition 3.13

Let $\mathcal {C}$ be an abelian category and $\mathcal {A},\mathcal {B},\mathcal {X}\subseteq \mathcal {C}.$ The pair $(\mathcal {A}, \mathcal {B})$ is left $\mathcal {X}$-complete if any $X\in \mathcal {X}$ admits an exact sequence $0 \rightarrow B \rightarrow A \rightarrow X \rightarrow 0$, with $A\in \mathcal {A}\cap \mathcal {X}$ and $B\in \mathcal {B}\cap \mathcal {X}$. The notion of right $\mathcal {X}$-complete pair is defined dually. We say that $(\mathcal {A}, \mathcal {B})$ is $\mathcal {X}$-complete if it is right and left $\mathcal {X}$-complete.

The Salce’s Lemma in [24] can be written, with the obvious proof, as follows.

Lemma 3.14

Let $\mathcal {X}\subseteq \mathcal {C}$ be closed under extensions and $(\mathcal {A}, \mathcal {B})$ be a left $\mathcal {X}$-complete pair in $\mathcal {C}$, such that $\mathcal {B}$ is closed under extensions and contains a relative cogenerator in $\mathcal {X}$. Then, $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete.

Example 3.15

Let $\mathcal {C}$ be an abelian category and $\mathcal {S}\subseteq \mathcal {C}.$ A left cotorsion pair $(\mathcal {A}, \mathcal {B})$ cut along $\mathcal {S}$ is called complete if, for each $S\in \mathcal {S}$, there is an exact sequence $0\rightarrow B\rightarrow A\rightarrow S \rightarrow 0 \text{, }$ with $A\in \mathcal {A}$ and $B\in \mathcal {B}$ [15, Def. 2.1]. The notions of relative completeness and cotorsion settled in [15] and ours are a little different, but they are related as follows:

• (a) Every left $\mathcal {S}$-complete and left cotorsion pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {S}$ with $\mathcal {A}=\textrm{smd}(\mathcal {A})$ is a complete left cotorsion pair cut along $\mathcal {S}.$

• (b) Let $\mathcal {A},\mathcal {B}\subseteq \mathcal {S}$. If $(\mathcal {A}, \mathcal {B})$ is a complete left cotorsion pair cut along $\mathcal {S},$ then $(\mathcal {A}, \mathcal {B})$ is left $\mathcal {S}$-complete and a left cotorsion pair in $\mathcal {S}.$

4 Relative resolutions, coresolutions, and related classes

Let us introduce the type of relative resolutions and coresolutions that interests us and that will be useful for the development of the theory related to some special classes and properties of the relative homological dimensions. This treatment will be necessary for n-$\mathcal {X}$-tilting theory in [4].

Definition 4.1

Let $\mathcal {C}$ be an abelian category, $M\in \mathcal {C}$ and $\mathcal {X},\mathcal {Y},\mathcal {Z}\subseteq \mathcal {C}.$

• (a) An exact sequence $0\rightarrow M{\mathop {\rightarrow }\limits ^{f_{0}}}Y_{0}{\mathop {\rightarrow }\limits ^{f_{1}}}Y_{1}{\mathop {\rightarrow }\limits ^{}}\cdots {\mathop {\rightarrow }\limits ^{}}Y_{n-1}{\mathop {\rightarrow }\limits ^{f_{n}}}Y_{n}{\mathop {\rightarrow }\limits ^{}}\cdots$ in $\mathcal {C},$ with $Y_{k}\in \mathcal {Y}\cup \left\{ 0\right\}$ $\forall k\ge 0$ and ${\text {Im}}\left( f_{i}\right) \in \mathcal {X}\cup \{0\}$ $\forall i\ge 1$, is called a $\mathcal {Y}_{\mathcal {X}}$-coresolution of M.

• (b) An exact sequence $0\rightarrow M{\mathop {\rightarrow }\limits ^{f_{0}}}Y_{0}{\mathop {\rightarrow }\limits ^{f_{1}}}Y_{1}{\mathop {\rightarrow }\limits ^{}}\cdots {\mathop {\rightarrow }\limits ^{}}Y_{n-1}{\mathop {\rightarrow }\limits ^{f_{n}}}Y_{n}{\mathop {\rightarrow }\limits ^{}}0$ in $\mathcal {C},$ with $Y_{n}\in \mathcal {X}\cap \mathcal {Y}$, $Y_{k}\in \mathcal {Y}$ $\forall k\in [0,n-1]$ and ${\text {Im}}\left( f_{i}\right) \in \mathcal {X},$ $\forall i\in [1,n-1]$, is called a $\mathcal {Y}_{\mathcal {X}}$-coresolution of length n of M, or simply a finite $\mathcal {Y}_{\mathcal {X}}$-coresolution of M.

• (c) We define $\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( M\right) ,$ the $\mathcal {Y}_{\mathcal {X}}$-coresolution dimension of M,  which is the smallest non-negative integer n, such that there is a $\mathcal {Y}_{\mathcal {X}}$-coresolution of length n of M. If such n does not exist, we set $\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( M\right) :=\infty \text{. }$

• (d) The $\mathcal {Y}_{\mathcal {X}}$-coresolution dimension of the class $\mathcal {Z}$ is defined as

  $$\begin{aligned} \textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( \mathcal {Z}\right) :=\sup \left\{ \textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( Z\right) \,|\,Z\in \mathcal {Z}\right\} \text{. } \end{aligned}$$

• (e) We denote by $\mathcal {Y}_{\mathcal {X},\infty }^{\vee }$ $(\text {resp. }\mathcal {Y}_{\mathcal {X}}^{\vee })$ the class of all the objects in $\mathcal {C}$ having a (resp. finite) $\mathcal {Y}_{\mathcal {X}}$-coresolution. Notice that $\mathcal {Y}_{\mathcal {X}}^{\vee }\subseteq \mathcal {Y}_{\mathcal {X},\infty }^{\vee }$.

• (f) We denote by $\mathcal {Y}_{\mathcal {X},n}^{\vee }$ the class of all the objects in $\mathcal {C}$ having a $\mathcal {Y}_{\mathcal {X}}$-coresolution of length $\le n$.

• (g) $(\mathcal {X}, \mathcal {Y})_{\infty }^{\vee }:=\mathcal {X}\cap \mathcal {Y}_{\mathcal {X},\infty }^{\vee },$ $(\mathcal {X}, \mathcal {Y})^{\vee }:=\mathcal {X}\cap \mathcal {Y}_{\mathcal {X}}^{\vee }$ and $(\mathcal {X}, \mathcal {Y})_{n}^{\vee }:=\mathcal {X}\cap \mathcal {Y}_{\mathcal {X},n}^{\vee }.$

• (h) Dually, we define the $\mathcal {Y}_{\mathcal {X}}$-resolution $(\text {of length }n),$ the $\mathcal {Y}_{\mathcal {X}}$-resolution dimension $\textrm{resdim}_\mathcal {Y}^{\mathcal {X}}(M)$ of M and the classes $\mathcal {Y}_{\mathcal {X}}^{\wedge },$ $\mathcal {Y}_{\mathcal {X},\infty }^{\wedge }$ and $\mathcal {Y}_{\mathcal {X},n}^{\wedge }.$ We also have $(\mathcal {Y}, \mathcal {X})_{\infty }^{\wedge }:=\mathcal {Y}_{\mathcal {X},\infty }^{\wedge }\cap \mathcal {X},$ $(\mathcal {Y}, \mathcal {X})^{\wedge }:=\mathcal {Y}_{\mathcal {X}}^{\wedge }\cap \mathcal {X}$ and $(\mathcal {Y}, \mathcal {X})_{n}^{\wedge }:=\mathcal {Y}_{n}^{\wedge }\cap \mathcal {X}.$

If $\mathcal {X}=\mathcal {C},$ we omit the symbol “$\mathcal {X}$” in the above notations. Note that

$$\begin{aligned} \textrm{resdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( M\right) =0\,\,\Leftrightarrow \,M\simeq \mathcal {X}\cap \mathcal {Y}\, \Leftrightarrow \,\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( M\right) =0. \end{aligned}$$

Some of the above relative resolutions have been considered before in several papers. In what follows, we consider some examples where such resolutions appear.

Example 4.2

(1) In [7, Sect. 5] Auslander and Reiten were interested in the study of the contravariantly finite subcategories in $\textrm{mod}(\Lambda )$ induced by cotilting modules over an Artin algebra $\Lambda .$ In particular, they studied the following classes $\mathcal {X}_{\omega }:=({\text {add}}(\omega )){}_{^{\bot }\omega ,\infty }^{\vee }$ and $_{\omega }\mathcal {X}:=({\text {add}}(\omega ))_{\omega ^{\bot },\infty }^{\wedge },$ for a class $\omega \subseteq {\text {mod}}(\Lambda )$ such that $\omega \subseteq \omega ^{\bot }.$

(2) In [11, Def. 3.11], the authors studied the relative Gorenstein objects in abelian categories. Namely, for an abelian category $\mathcal {C}$ and $\mathcal {A},\mathcal {B}\subseteq \mathcal {C},$ they introduced, respectively, the class of the weak $(\mathcal {A}, \mathcal {B})$-Gorenstein projectives (resp. injectives) objects $\mathcal {WGP}_{(\mathcal {A}, \mathcal {B})}:=({}^{\bot }\mathcal {B},\mathcal {A})_{\infty }^{\vee }\quad$ $\quad (\text {resp. }\mathcal {WGI}_{(\mathcal {A}, \mathcal {B})}:=(\mathcal {A},\mathcal {B}^{\bot })_{\infty }^{\wedge }).$

(3) Let $\mathcal {C}$ be an n-coherent category [12, Def. 4.1]. An object $M\in \mathcal {C}$ is Gorenstein $\mathcal{F}\mathcal{P}_{n}$-injective if there is an exact sequence

$$\begin{aligned} \eta :\,\cdots \rightarrow I_{2}\rightarrow I_{1}\rightarrow I_{0}\overset{f}{\rightarrow }E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \cdots \end{aligned}$$

with $I_{i},E_{i}\in \textrm{Inj}(\mathcal {C})$ $\forall i\ge 1$, such that $M={\text {Im}}\left( f\right)$ and the complex $\textrm{Hom}_{\mathcal {C}}(J, \eta )$ is acyclic for any $J\in \mathcal{F}\mathcal{P}_{n}^{\bot _{1}}$. Let $\mathcal{G}\mathcal{I}$ denote the class of all the Gorenstein $\mathcal{F}\mathcal{P}_{n}$-injective objects and $\mathcal {W}:={}^{\bot _{1}}\mathcal{G}\mathcal{I}$. In [12, Lem. 5.2], it is proved that $\mathcal{G}\mathcal{I}=(\textrm{Inj}(\mathcal {C}),\mathcal {G}\mathcal {I})_{\infty }^{\wedge }$.

(4) The above example is a particular case of the following observation. Let $\mathcal {X}$ be a class in an abelian category $\mathcal {C}$ and $0\in \omega \subseteq \mathcal {X}$. Then, $\mathcal {X}=(\omega ,\mathcal {X})_{\infty }^{\wedge }$ if, and only if, $\omega$ is a relative generator in $\mathcal {X}$.

We recall the following known Lemma.

Lemma 4.3

[18, Lem. 2.13(a)] Let $\mathcal {C}$ be an abelian category and $\mathcal {X},\mathcal {Y}\subseteq \mathcal {C}$. Then, $\textrm{pd}_{\mathcal {Y}}(\mathcal {X}^{\vee })=\textrm{pd}_{\mathcal {Y}}(\mathcal {X})$.

The next theorem shows the existence of the main approximations that we will be using in the relative class $\,\mathcal {U}_{\mathcal {V}}^{\vee }$. This theorem is a generalization of the dual result of [6, Thm. 1.1] and will play an important role in the development of n-$\mathcal {X}$ tilting theory in [4]. For example, it is fundamental in the proof that the pair $({}^\perp (\mathcal {T}^\perp ),\mathcal {T}^\perp )$ is $\mathcal {X}$-complete if $\mathcal {T}$ is an n-$\mathcal {X}$-tilting class in $\mathcal {C}.$

It is worth also to mention that the statement of Theorem 4.4 is inspired in [10, Thm. 2.8], where the authors present the Auslander–Buchweitz results with the minimum hypotheses needed.

In the following result, the expression $\textrm{coresdim}_\omega ^{\mathcal {V}}(C_Z)=-1$ just means that $C_Z=0$.

Theorem 4.4

For $\mathcal {U}\subseteq \mathcal {V}\subseteq \mathcal {C}$ classes which are closed under extensions, $\omega$ a relative generator in $\mathcal {U}$ and $0\in \mathcal {U},$ the following statements hold true.

• (a) For any $Z\in \mathcal {U}_{\mathcal {V}}^{\vee }$, with $n:=\textrm{coresdim}{}_{\mathcal {U}}^{\mathcal {V}}\left( Z\right)$, there are short exact sequences

  $$\begin{aligned}&0 \rightarrow Z \xrightarrow {g_{Z}} M_{Z} \rightarrow C_{Z} \rightarrow \rightarrow 0 \text{ with } C_{Z}\in (\mathcal {V},\omega ){}^{\vee },\,M_{Z}\in \mathcal {U} \text{ and } \\&0 \rightarrow K_{Z} \rightarrow B_{Z} \xrightarrow {f_{Z}} Z \rightarrow 0 \text{ with } B_{Z}\in \omega _{\mathcal {V}}^{\vee }, K_{Z}\in \mathcal {U}\text{, } \end{aligned}$$

  where $\textrm{coresdim}{}_{\omega }^{\mathcal {V}}\left( C_{Z}\right) =n-1$ and $\textrm{coresdim}{}_{\omega }^{\mathcal {V}}\left( B_{Z}\right) \le n$.

• (b) $B_{Z}\in (\mathcal {V},\omega ){}^{\vee }$ if $Z\in (\mathcal {V},\mathcal {U}){}^{\vee }.$

• (c) Let $\omega \subseteq {}{}^{\bot }\mathcal {U}.$ Then, $\omega ^{\vee }\subseteq {}{}^{\bot }\mathcal {U}$, $f_{Z}$ is a $\omega ^{\vee }$-precover, and $g_{Z}$ is an $\mathcal {U}$-preenvelope.

Proof

(a) We proceed by induction on $n=\textrm{coresdim}{}_{\mathcal {U}}^{\mathcal {V}}\left( Z\right) =n<\infty$.

Let $n=0$. Then, $Z\in \mathcal {U}\cap \mathcal {V}$. Now, since $\omega$ is a relative generator in $\mathcal {U}$, there is an exact sequence $0 \rightarrow U' \rightarrow W \rightarrow Z \rightarrow 0 \text{, }$ with $W\in \omega \subseteq (\mathcal {V}, \omega )^{\vee }$ and $U'\in \mathcal {U}$. We can also consider the exact sequence $0 \rightarrow Z \xrightarrow {1} Z \rightarrow 0 \rightarrow 0\text{, }$ where $0\in (\mathcal {V}, \omega )^{\vee }$. Note that these are the desired exact sequences.

Let $n>0$. Hence, there is an exact sequence $0 \rightarrow Z \rightarrow U_{0} \xrightarrow {f} V \rightarrow 0$, with $U_{0}\in \mathcal {U}$, $V\in \mathcal {V}$, and $\textrm{coresdim}{}_{\mathcal {U}}^{\mathcal {V}}\left( V\right) =n-1$. Now, by inductive hypothesis, there is a short exact sequence

$$\begin{aligned} 0\rightarrow K_{V} \rightarrow B_{V} \xrightarrow {f_{V}} V \rightarrow 0 \end{aligned}$$

with $B_{V}\in \omega _{\mathcal {V}}^{\vee }$, $K_{V}\in \mathcal {U}\subseteq \mathcal {Y}$, and $\textrm{coresdim}{}_{\omega }^{\mathcal {V}}\left( B_{V}\right) \le n-1$. Considering the pull-back of $f_{V}$ and f, we get a short exact sequence

$$\begin{aligned} \eta :\quad 0 \rightarrow Z \xrightarrow {h} E \rightarrow B_{V} \rightarrow 0 \end{aligned}$$

with $E\in \mathcal {U}$ and $B_{V}\in \mathcal {V}$. Note that $\eta$ is the first of the two exact sequences we are looking for.

Now, since $E\in \mathcal {U}$, there is an exact sequence

$$\begin{aligned} 0 \rightarrow L \rightarrow W \xrightarrow {h'} E \rightarrow 0 \end{aligned}$$

with $W\in \omega$ and $L\in \mathcal {X}$. Hence, considering the pull-back of h and $h'$, we get an exact sequence

$$\begin{aligned} \eta ':\quad 0 \rightarrow L \rightarrow Z' \rightarrow Z \rightarrow 0 \end{aligned}$$

with $Z'\in \omega _{\mathcal {V}}^{\vee }$ and $L\in \mathcal {U}$. Note that $\eta '$ is the second of the two exact sequences we are looking for. Moreover, observe that $\textrm{coresdim}{}_{\mathcal {\omega }}^{\mathcal {V}}\left( Z'\right) \le n$.

It remains to prove that $\textrm{coresdim}{}_{\omega }^{\mathcal {V}}\left( B_{V}\right) =n-1$. Since $\omega \subseteq \mathcal {U}$, we have

$$\begin{aligned} \textrm{coresdim}{}_{\mathcal {U}}^{\mathcal {V}}\left( B_{V}\right) \le \textrm{coresdim}{}_{\omega }^{\mathcal {V}}\left( B_{V}\right) \le n-1\text{. } \end{aligned}$$

Moreover, by the exact sequence $\eta$, we have

$$\begin{aligned} n=\textrm{coresdim}{}_{\mathcal {U}}^{\mathcal {V}}\left( Z\right) \le 1+\textrm{coresdim}{}_{\mathcal {U}}^{\mathcal {V}}\left( B_{V}\right) \le n\text{, } \end{aligned}$$

Therefore, $\textrm{coresdim}{}_{\omega }^{\mathcal {V}}\left( B_{V}\right) =n-1$.

(b) From the last pull-back in the proof of (a), we have that $Z'\in \mathcal {V}$ if $Z\in \mathcal {V}$.

(c) Let $\omega \subseteq {}^{\bot }\mathcal {U}.$ Then, by Lemma 4.3, we have $\omega ^{\vee }\subseteq {}^{\bot }\mathcal {U}.$ Let $Z\in \mathcal {U}_{\mathcal {V}}^{\vee }$. Consider the first exact sequence in (a). Observe that $C_{Z}\in {}^{\bot }\mathcal {U}$ and $M_{Z}\in \mathcal {U}\subseteq \left( \omega ^{\vee }\right) ^{\bot }$.

To show that $g_{Z}$ is an $\mathcal {U}$-preenvelope, observe that any morphism $f:Z\rightarrow U$, with $U\in \mathcal {U}$, factors through $g_{Z}$ if, and only if, the exact sequence induced by the push-out of f and $g_{Z}$ splits (which is the case since $C_{Z}\in {}^{\bot }\mathcal {U}$). The fact that $f_Z$ is a $\omega ^\vee$-precover is proved by dual arguments.

$\square$

Let us commence the study of the relations between the relative homological dimensions and the relative (co)resolution dimensions. We begin with the intuitive extension of some known results. It is worth mentioning that the following result is used in [4] to study the properties of an n-$\mathcal {X}$-tilting class $\mathcal {T}.$

Proposition 4.5

Let $\mathcal {C}$ be an abelian category, $\mathcal {X},\mathcal {T}\subseteq \mathcal {C}$ and $\alpha \subseteq \mathcal {T}^{\bot }\cap \mathcal {X}^{\bot }$ be a relative cogenerator in $\mathcal {X}.$ Then, the following inequalities hold true.

• (a) $\textrm{coresdim}{}_{\mathcal {T}^{\bot }\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {X}\right) \le \textrm{pd}_{\mathcal {X}}(\mathcal {T}).$

• (b) $\textrm{pd}_{\mathcal {X}}({}{}^{\bot }\left( \mathcal {T}^{\bot }\right) )\le \textrm{pd}_{\mathcal {X}}({}{}^{\bot }\left( \mathcal {T}{}^{\bot }\cap \mathcal {X}\right) )\le \textrm{pd}_{\mathcal {X}}(\mathcal {T})$.

Proof

(a) We can assume that $n:=\textrm{pd}_{\mathcal {X}}(\mathcal {T})<\infty$. If $n=0$, then $\mathcal {X}\subseteq \mathcal {T}^{\bot }$, and thus, $\textrm{coresdim}{}_{\mathcal {T}^{\bot }\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {X}\right) =0.$

Let $n\ge 1$. Since $\alpha$ is a relative cogenerator in $\mathcal {X}$, for every $A\in \mathcal {X},$ there is an exact sequence $0\rightarrow A\rightarrow W_{0}\rightarrow \cdots \rightarrow W_{n-1}\rightarrow Q\rightarrow 0\text{, }$ with $Q\in \mathcal {T}^{\bot }\cap \mathcal {X}$ and $W_{i}\in \alpha$ $\forall i\in [0,n-1]$ (see Proposition 2.6). Therefore, $\textrm{coresdim}{}_{\mathcal {T}^{\bot }\cap \mathcal {X}}^{\mathcal {X}}\left( A\right) \le n$.

(b) Since $^{\bot }\left( \mathcal {T}^{\bot }\right) \subseteq {}^{\bot }(\mathcal {T}^{\bot }\cap \mathcal {X})$, it is enough to prove that $\textrm{pd}_{\mathcal {X}}(^{\bot }(\mathcal {T}^{\bot }\cap \mathcal {X}))\le \textrm{pd}_{\mathcal {X}}(\mathcal {T})$. Assume $n:=\textrm{pd}_{\mathcal {X}}(\mathcal {T})<\infty$. Then, by (a), there is an exact sequence $0\rightarrow A\rightarrow W_{0}\rightarrow \cdots \rightarrow W_{n-1}\rightarrow W_n\rightarrow 0,$ where $W_i\in \mathcal {T}^{\bot }\cap \mathcal {X}$ $\forall i\in [0,n].$ Let $Y\in {}{}^{\bot }\left( \mathcal {T}^{\bot }\cap \mathcal {X}\right)$ and $A\in \mathcal {X}.$ Since $W_i\in \mathcal {T}^{\bot }\cap \mathcal {X},$ we get that $W_i\in Y^\perp$ $\forall \,i.$ Hence, by the shifting Lemma, $0=\textrm{Ext}^{k}_{\mathcal {C}}(Y,W_n)\cong \textrm{Ext}^{k+n}_{\mathcal {C}}(Y,A)\quad \forall k>0\text{. }$ Therefore, $\textrm{pd}_{\mathcal {X}}(Y)\le n$. $\square$

Lemma 4.6

For an exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ in $\mathcal {C}$ and $\mathcal {X}\subseteq \mathcal {C},$ the following inequalities hold true:

• (a) $\textrm{id}_{\mathcal {X}}(B)\le \max \left\{ \textrm{id}_{\mathcal {X}}(A),\textrm{id}_{\mathcal {X}}(C)\right\} .$

• (b) $\textrm{id}_{\mathcal {X}}(A)\le \max \left\{ \textrm{id}_{\mathcal {X}}(B),\textrm{id}_{\mathcal {X}}(C)+1\right\} .$

• (c) $\textrm{id}_{\mathcal {X}}(C)\le \max \left\{ \textrm{id}_{\mathcal {X}}(B),\textrm{id}_{\mathcal {X}}(A)-1\right\} .$

Proof

The proof is straightforward. $\square$

Lemma 4.7

[6, p.16] $\textrm{pd}_{\mathcal {Y}}(\mathcal {X})=\textrm{id}_{\mathcal {X}}(\mathcal {Y}),$ for any $\mathcal {X},\mathcal {Y}\subseteq \mathcal {C}.$

Theorem 4.8

[18, Thm. 2.1] $\textrm{id}_{\mathcal {X}}(L)\le \textrm{id}_{\mathcal {X}}(\mathcal {Y})+\textrm{coresdim}{}_{\mathcal {Y}}^{,}\left( L\right)$ for all $L\in \mathcal {C}$ and $\mathcal {X},\mathcal {Y}\subseteq \mathcal {C}.$

The following result is a generalization of Theorem 4.8.

Theorem 4.9

Let $\mathcal {C}$ be an abelian category and $\mathcal {X}$,$\mathcal {Y}$,$\mathcal {Z}\subseteq \mathcal {C}.$ Then

$$\begin{aligned} \textrm{id}_{\mathcal {X}}(L)\le \textrm{id}_{\mathcal {X}}(\mathcal {Y})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) \quad \forall L\in \mathcal {C}\text{. } \end{aligned}$$

Furthermore, if $\mathcal {Z}$ is closed under extensions, then

$$\begin{aligned} \textrm{id}_{\mathcal {X}}(L)\le \textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) \quad \forall L\in \mathcal {Z}. \end{aligned}$$

Proof

Note that $\textrm{coresdim}{}_{\mathcal {Y}}^{\,}\left( L\right) \le \textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right)$ $\forall L\in \mathcal {C}$. Hence, by Theorem 4.8, we have that $\textrm{id}_{\mathcal {X}}(L)\le \textrm{id}_{\mathcal {X}}(\mathcal {Y})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right)$ $\forall L\in \mathcal {C}.$

Let $\mathcal {Z}$ be closed under extensions and $L\in \mathcal {Z}.$ Assume that $\textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})=n<\infty$ and $\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) =m<\infty$. We prove, by induction on m,  that $\textrm{id}_{\mathcal {X}}(L)\le n+m$.

If $m=0$, we have $L\cong M\in \mathcal {Y}\cap \mathcal {Z}$. Let $m=1$. Since $\mathcal {Z}$ is closed under extensions, there is an exact sequence $0 \rightarrow L \rightarrow Y_{0} \rightarrow Y_{1} \rightarrow 0$ with $Y_{0},Y_{1}\in \mathcal {Y}\cap \mathcal {Z}.$ Thus, for every $X\in \mathcal {X}$, we have the exact sequence

$$\begin{aligned} \textrm{Ext}^{k-1}_{\mathcal {C}}(X,Y_{1})\rightarrow \textrm{Ext}^{k}_{\mathcal {C}}(X,L) \rightarrow \textrm{Ext}^{k}_{\mathcal {C}}(X,Y_{0})\text{, } \end{aligned}$$

where $\textrm{Ext}^{k-1}_{\mathcal {C}}(X,Y_{1})=0$ and $\textrm{Ext}^{k}_{\mathcal {C}}(X,Y_{0})=0,$ for any $k>n+1$. Therefore, $\textrm{id}_{\mathcal {X}}(L)\le n+1=\textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) .$

Let $m\ge 2$. Then, using that $\mathcal {Z}$ is closed under extensions, we get an exact sequence $0\rightarrow L{\mathop {\rightarrow }\limits ^{f}}Y_{0}\rightarrow \cdots \rightarrow Y_{m}\rightarrow 0,$ where $Y_{i}\in \mathcal {Y}\cap \mathcal {Z}$ $\forall i\in [0,m]$, and $\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( K\right) =m-1$ for $K:={\text {Coker}}(f)$. Hence, by inductive hypothesis

$$\begin{aligned} \textrm{id}_{\mathcal {X}}(K)\le \textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( K\right) =\textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) -1. \end{aligned}$$

Finally, by Lemma 4.6, it follows that:

$$\begin{aligned} \textrm{id}_{\mathcal {X}}(L)&\le \max \left\{ \textrm{id}_{\mathcal {X}}(Y_{0}),\textrm{id}_{\mathcal {X}}(K)+1\right\} \\&\le \max \left\{ \textrm{id}_{\mathcal {X}}(Y_{0}),\textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) \right\} \\&\le \textrm{id}_{\mathcal {X}}(\mathcal {Y}\cap \mathcal {Z})+\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {Z}}\left( L\right) \text{. } \end{aligned}$$

$\square$

In the following lemma, we study the relative dimensions induced by the classes in an $\mathcal {X}$-complete pair $(\mathcal {A},\mathcal {B}).$

Lemma 4.10

For $\mathcal {X}\subseteq \mathcal {C}$ and a right $\mathcal {X}$-complete pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {C},$ the following statements hold true.

• (a) For any $M\in \mathcal {C},$ we have

  $$\begin{aligned} \textrm{id}_{\mathcal {X}}(M)=\max \left\{ \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M),\textrm{id}_{\mathcal {B}\cap \mathcal {X}}(M)\right\} \le \max \left\{ \textrm{id}_{\mathcal {A}}(M),\textrm{id}_{\mathcal {B}}(M)\right\} . \end{aligned}$$

• (b) If $\mathcal {A}\cup \mathcal {B}\subseteq \mathcal {X}$, then $\textrm{id}_{\mathcal {X}}(M)=\max \left\{ \textrm{id}_{\mathcal {A}}(M),\textrm{id}_{\mathcal {B}}(M)\right\} \,\forall M\in \mathcal {C}\text{. }$

• (c) If $(\mathcal {A}, \mathcal {B})$ is $\mathcal {X}$-hereditary, then $\textrm{id}_{\mathcal {X}}(M)=\textrm{id}_{\mathcal {B}\cap \mathcal {X}}(M)\,\forall M\in \mathcal {B}\cap \mathcal {X}\text{. }$

Proof

(a) Since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete, for every $N\in \mathcal {X}$, we have an exact sequence $0 \rightarrow N \rightarrow B \rightarrow A \rightarrow 0$ with $B\in \mathcal {B}\cap \mathcal {X}$ and $A\in \mathcal {A}\cap \mathcal {X}$. Let $M\in \mathcal {C}$. Then, by Lemma 4.6 and Lemma 4.7, it follows that:

$$\begin{aligned} \textrm{id}_{\{N\}}(M)=\textrm{pd}_{\{M\}}(N)&\le \max \left\{ \textrm{pd}_{\{M\}}(B),\textrm{pd}_{\{M\}}(A)-1\right\} \\&\le \max \left\{ \textrm{pd}_{\{M\}}(\mathcal {B}\cap \mathcal {X}),\textrm{pd}_{\{M\}}(\mathcal {A}\cap \mathcal {X})\right\} \\&\le \max \left\{ \textrm{id}_{\mathcal {B}\cap \mathcal {X}}(M),\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)\right\} \\&\le \max \left\{ \textrm{id}_{\mathcal {B}}(M),\textrm{id}_{\mathcal {A}}(M)\right\} \text{. } \end{aligned}$$

Hence, $\textrm{id}_{\mathcal {X}}(M)\le \max \left\{ \textrm{id}_{\mathcal {B}\cap \mathcal {X}}(M),\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)\right\} \le \max \left\{ \textrm{id}_{\mathcal {B}}(M),\textrm{id}_{\mathcal {A}}(M)\right\} \text{. }$ On the other hand, it is clear that $\textrm{id}_{\mathcal {X}}(M)\ge \max \left\{ \textrm{id}_{\mathcal {B\cap \mathcal {X}}}(M),\textrm{id}_{\mathcal {A\cap \mathcal {X}}}(M)\right\} .$

(b) It follows by (a).

(c) It follows from (a), since $\textrm{id}_{\mathcal {A\cap X}}(\mathcal {B}\cap \mathcal {X})=0.$ $\square$

Proposition 4.11

Let $\mathcal {X}\subseteq \mathcal {C}$ and $(\mathcal {A}, \mathcal {B})$ be a right $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pair in $\mathcal {C}$, such that $\left( \mathcal {A}\cap \mathcal {X}\right) ^{\bot }\cap \mathcal {A}\cap \mathcal {X}\subseteq \mathcal {B}\cap \mathcal {X}$ and $\mathcal {B}=\textrm{smd}(\mathcal {B})$. Then, for any $X\in \mathcal {X},$ we have

• (a) $\textrm{coresdim}{}_{\mathcal {B}}^{\,}\left( X\right) \le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B})+{\text {coresdim}}_{\mathcal {B}}\left( X\right) \text{; }$

• (b) $\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right) {=}\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X){=}{\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( X\right) {\le } {\le }{\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) +1;$

• (c) $\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( X\right) =\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right)$ if $\mathcal {X}$ is closed under extensions;

• (d) $\textrm{coresdim}{}_{\mathcal {B}}^{\,}\left( X\right) \le \textrm{id}_{\mathcal {A}}(X)\le \textrm{id}_{\mathcal {A}}(\mathcal {B})+{\text {coresdim}}_{\mathcal {B}}\left( \mathcal {A}\right) +1\text{; }$

• (e) $(\mathcal {A}\cap \mathcal {X})^{\bot }\cap \mathcal {X}\subseteq \mathcal {B}$.

Proof

Let $X\in \mathcal {X}$. We prove first that

$$\begin{aligned} \textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right) \le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\text{. } \end{aligned}$$

(4.A)

We may assume that $n:=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)<\infty .$ Let $n=0$. To show that $\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right) =0,$ it is enough to prove that $X\in \mathcal {B}$. Since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete, there is an exact sequence

$$\begin{aligned} \eta :\,0 \rightarrow X \rightarrow B \rightarrow A \rightarrow 0 \text{, } \text{ with } B\in \mathcal {B}\cap \mathcal {X} \text{ and } A\in \mathcal {A}\cap \mathcal {X}. \end{aligned}$$

Observe that $\eta$ splits, since $n=0$. Hence, X is a direct summand of B, and thus, $X\in \mathcal {B}$. Note that this argument gives a proof of (e).

Let $n\ge 1$. Since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete, there is an exact sequence

$$\begin{aligned} \epsilon :\;0\rightarrow X{\mathop {\rightarrow }\limits ^{g_{0}}}B_{0}{\mathop {\rightarrow }\limits ^{g_{1}}}B_{1}{\mathop {\rightarrow }\limits ^{g_{2}}}\cdots {\mathop {\rightarrow }\limits ^{g_{n-1}}}B_{n-1}\rightarrow A_{n}\rightarrow 0 \end{aligned}$$

with $B_{i}\in \mathcal {B}\cap \mathcal {X}$ and $A_{i+1}:={\text {Coker}}(g_{i})\in \mathcal {A}\cap \mathcal {X}$ $\forall i\in [0,n-1]$. Now, since $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$ and $n=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)$, we have

$$\begin{aligned} \textrm{Ext}^{i}_{\mathcal {C}}(A,A_{n})\cong \textrm{Ext}^{n+i}_{\mathcal {C}}(A,X)=0\,\forall A\in \mathcal {A}\cap \mathcal {X}\,\forall i\ge 1. \end{aligned}$$

Hence, $A_{n}\in \left( \mathcal {A}\cap \mathcal {X}\right) ^{\bot }\cap \mathcal {A}\cap \mathcal {X}\subseteq \mathcal {B}\cap \mathcal {X}$, and thus, from $\epsilon$, we have $\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( X\right) \le n;$ proving (4.A).

Note that (a) follows from (4.A) and Theorem 4.8. On the other hand, using that $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$, (4.A) and Theorem 4.8, we get

$$\begin{aligned} \begin{aligned} \textrm{coresdim}{}_{\mathcal {B}}^{\,}\left( X\right)&\le \textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( X\right) \\&\le \textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right) \\&\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X) \\&\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})+{\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( X\right) \\&={\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( X\right) \\&\le \textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right) \text{. } \end{aligned} \end{aligned}$$

(4.B)

Note that (4.B) proves the first two equalities of (b). Next, let us prove that

$$\begin{aligned} \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\le \textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( \mathcal {A}\cap \mathcal {X}\right) +1\text{, } \forall X\in \mathcal {X}\text{. } \end{aligned}$$

Consider $X\in \mathcal {X}$. Since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete, there is an exact sequence

$$\begin{aligned} 0 \rightarrow X \rightarrow B \rightarrow A \rightarrow 0 \text{, } \text{ with } B\in \mathcal {B}\cap \mathcal {X} \text{ and } A\in \mathcal {A}\cap \mathcal {X}. \end{aligned}$$

Then, by Lemma 4.6 and Theorem 4.8, we have

$$\begin{aligned} \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)&\le \max \left\{ \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(B),\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(A)+1\right\} \\&=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(A)+1\\&\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})+1\\&\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})+{\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) +1\\&={\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) +1\text{. } \end{aligned}$$

Therefore, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\le {\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) +1$ $\forall X\in \mathcal {X}$.

To prove (c), we assume that $\mathcal {X}$ is closed under extensions. Then, by Theorem 4.9, we have

$$\begin{aligned} \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})+\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( X\right) =\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( X\right) \,\forall X\in \mathcal {X}\text{, } \end{aligned}$$

and by (4.B), it follows (c).

Finally, since ${\text {coresdim}}_{\mathcal {B}}\left( X\right) \le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\le \textrm{id}_{\mathcal {A}}(X)$ $\forall X\in \mathcal {X}$ by (4.B), the proof of (d) is similar to the proof of (b). Indeed, since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete, for any $X\in \mathcal {X}$, there is an exact sequence

$$\begin{aligned} 0 \rightarrow X \rightarrow B \rightarrow A \rightarrow 0\text{, } \text{ with } B\in \mathcal {B}\cap \mathcal {X} \text{ and } A\in \mathcal {A}\cap \mathcal {X}. \end{aligned}$$

Then, by Lemma 4.6 and Theorem 4.8, we have

$$\begin{aligned} \textrm{id}_{\mathcal {A}}(X)&\le \max \left\{ \textrm{id}_{\mathcal {A}}(B),\textrm{id}_{\mathcal {A}}(A)+1\right\} \\&\le \max \left\{ \textrm{id}_{\mathcal {A}}(\mathcal {B}),\textrm{id}_{\mathcal {A}}(\mathcal {A})+1\right\} \\&\le \max \left\{ \textrm{id}_{\mathcal {A}}(\mathcal {B}),\textrm{id}_{\mathcal {A}}(\mathcal {B})+{\text {coresdim}}_{\mathcal {B}}\left( \mathcal {A}\right) +1\right\} \\&=\textrm{id}_{\mathcal {A}}(\mathcal {B})+{\text {coresdim}}_{\mathcal {B}}\left( \mathcal {A}\right) +1\text{. } \end{aligned}$$

$\square$

Corollary 4.12

For a closed under extensions class $\mathcal {X}\subseteq \mathcal {C}$ and a right $\mathcal {X}$-complete pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {C}$ which is $\mathcal {X}$-hereditary and $\mathcal {B}=\textrm{smd}(\mathcal {B}),$ the following statements hold true.

• (a) $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( M\right)$ $\forall M\in (\mathcal {X},\mathcal {B})^{\vee }.$

• (b) Let $\left( \mathcal {A}\cap \mathcal {X}\right) ^{\bot }\cap \mathcal {A}\cap \mathcal {X}\subseteq \mathcal {B}\cap \mathcal {X}.$ Then, for any $X\in \mathcal {X},$ we have

  $$\begin{aligned} \textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( X\right)&=\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( X\right) \\&=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(X)\\&=\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( X\right) \\&\le \textrm{coresdim}{}_{\mathcal {B\cap \mathcal {X}}}^{ }\left( \mathcal {A}\cap \mathcal {X}\right) +1. \end{aligned}$$

Proof

(a) Let $M\in \mathcal {B}_{\mathcal {X}}^{\vee }\cap \mathcal {X}$. We proceed by induction on $n:=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( M\right)$.

If $n=0$, then $M\cong N\in \mathcal {B}\cap \mathcal {X}$. Thus, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)=0$, and so, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$.

We assert that $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)\ge 1$ if $n>0.$ Indeed, since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete and $M\in \mathcal {X}$, there is an exact sequence

$$\begin{aligned} 0 \rightarrow M\rightarrow B\rightarrow A\rightarrow 0\text{, } \text{ with } B\in \mathcal {B}\cap \mathcal {X} \text{ and } A\in \mathcal {A}\cap \mathcal {X}. \end{aligned}$$

Note that this sequence does not split since $M\notin \mathcal {B}\cap \mathcal {X}$ ($\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( M\right) =n>0$). Hence, $\textrm{Ext}^{1}_{\mathcal {C}}(A,M)\ne 0$, and thus, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)\ge 1;$ proving our assertion.

Let $n=1$. Since $\mathcal {X}$ is closed under extensions, there is an exact sequence

$$\begin{aligned} 0 \rightarrow M \rightarrow Y_{0} \rightarrow Y_{1} \rightarrow 0 \text{, } \text{ with } Y_{0},Y_{1}\in \mathcal {B}\cap \mathcal {X}\text{. } \end{aligned}$$

By Lemma 4.6, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)\le \max \left\{ \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(Y_{0}),\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(Y_{1})+1\right\} =1\text{. }$ Therefore, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)=1$ since $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)\ge 1$.

Let $n>1$. By inductive hypothesis, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(N)=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( N\right)$ for any $N\in \mathcal {X}$ with $\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( N\right) \le n-1$. Since $\mathcal {X}$ is closed under extensions, we have an exact sequence

$$\begin{aligned} 0\rightarrow M\rightarrow Y_{0}\overset{f_{1}}{\rightarrow }Y_{1}\rightarrow \cdots \overset{f_{n}}{\rightarrow }Y_{n}\rightarrow 0\text{, } \end{aligned}$$

with $Y_{n}\in \mathcal {B}\cap \mathcal {X}$, $Y_{i}\in {\mathcal {B}}\cap \mathcal {X}$, ${\text {Im}}\left( f_{i}\right) \in \mathcal {X}$ $\forall i\in [0,n-1]$. Moreover, for $K:={\text {Im}}\left( f_{1}\right)$, we have $\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( K\right) =n-1$. Consider the exact sequence

$$\begin{aligned} 0 \rightarrow M \rightarrow Y_{0} \rightarrow K \rightarrow 0 \text{. } \end{aligned}$$

Note that $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(K)=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( K\right) =n-1$ by inductive hypothesis. Hence, by Lemma 4.6, we have

$$\begin{aligned} \textrm{id}_{\mathcal {A\cap \mathcal {X}}}(M)&\le \max \left\{ \textrm{id}_{\mathcal {A\cap \mathcal {X}}}(Y_{0}),\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(K)+1\right\} =n\text{, }\quad \text{ and } \\ n-1&=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(K)\le \max \left\{ \textrm{id}_{\mathcal {A\cap \mathcal {X}}}(Y_{0}),\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)-1\right\} =\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)-1, \end{aligned}$$

since $\textrm{id}_{\mathcal {A\cap \mathcal {X}}}(M)\ge 1$. Therefore, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)=n.$

(b) It follows from Proposition 4.11(c,b). $\square$

Before going any further in the study of $\mathcal {X}$-complete pairs, we recall from [10] the notion of thick class. We will be interested in reviewing the principal properties of this kind of classes with regard to their generators and cogenerators.

Definition 4.13

[10, Def. 2.2] Let $\mathcal {C}$ be an abelian category and $\mathcal {X}\subseteq \mathcal {C}.$

• (a) $\mathcal {X}$ is right thick if it is closed under extensions, direct summands, and mono-cokernels.

• (b) $\mathcal {X}$ is left thick if it is closed under extensions, direct summands, and epi-kernels.

• (c) $\mathcal {X}$ is thick if it is right and left thick.

Lemma 4.14

Let $\omega \subseteq \mathcal {X}\subseteq \mathcal {C}$ with $\mathcal {X}$ closed under epi-kernels. Then, $\omega ^{\vee }\subseteq \mathcal {X}$.

Proof

The proof is straightforward and we left it to the reader. $\square$

The next Lemma is a generalization of [10, Prop. 2.7]. The only difference between them is a subtle precision on the kind of the resolutions that are used.

Lemma 4.15

For the classes $\omega ,\mathcal {X}\subseteq \mathcal {C}$ such that $\omega$ is $\mathcal {X}$-injective, the following statements hold true.

• (a) $\omega ^{\wedge }$ is $\mathcal {X}$-injective.

• (b) If $\omega =\textrm{smd}(\omega )$ is a relative cogenerator in $\mathcal {X},$ then

  $$\begin{aligned} \mathcal {X}\cap \mathcal {X}^{\bot }=\mathcal {X}\cap \omega ^{\wedge }=(\omega ,\mathcal {X})^{\wedge }=\omega \text{. } \end{aligned}$$

Proof

The item (a) follows from the dual of Lemma 4.3. Let us show (b). Indeed, from (a), it is clear that $(\omega ,\mathcal {X})^{\wedge }\subseteq \mathcal {X}\cap \omega ^{\wedge }\subseteq \mathcal {X}\cap \mathcal {X}^{\bot }$.

Let $X\in \mathcal {X}\cap \mathcal {X}^{\bot }$. Since $\omega$ is a relative cogenerator in $\mathcal {X}$, there is an exact sequence $\eta :\;0 \rightarrow X \rightarrow W \rightarrow X' \rightarrow 0,$ with $W\in \omega$ and $X'\in \mathcal {X}.$ Moreover, the exact sequence $\eta$ splits, since $X\in \mathcal {X}^{\bot }$ and, thus, $X\in \omega$. Therefore

$$\begin{aligned} (\omega ,\mathcal {X})^{\wedge }\mathcal {\subseteq X}\cap \omega ^{\wedge }\subseteq \mathcal {X}\cap \mathcal {X}^{\bot }\subseteq \omega \text{. } \end{aligned}$$

Finally $\omega \subseteq (\omega ,\mathcal {X})^{\wedge }$, since $\omega \subseteq \mathcal {X}$ and $0\in \omega .$ $\square$

The following lemma is a compilation of results from the Auslander–Buchweitz theory; see [6, Prop. 4.2, Lem. 4.3 and Prop. 4.7].

Lemma 4.16

For the classes $\omega ,\mathcal {X}\subseteq \mathcal {C}$, such that $\omega =\textrm{smd}(\omega )$ is an $\mathcal {X}$-injective relative cogenerator in $\mathcal {X},$ the following statements hold true.

• (a) $\mathcal {X}\cap \omega ^{\vee }=\left\{ X\in \mathcal {X}\,|\,\textrm{id}_{\mathcal {X}}(X)<\infty \right\} .$

• (b) $\textrm{id}_{\mathcal {X}}(M)={\text {coresdim}}_{\omega }\left( M\right)$ $\forall M\in \mathcal {X}\cap \omega ^{\vee }.$

• (c) If $\mathcal {X}$ is left thick, then $\omega ^{\vee }$ is left thick and

  $$\begin{aligned} \omega ^{\vee }=\left\{ X\in \mathcal {X}\,|\,\textrm{id}_{\mathcal {X}}(X)<\infty \right\} \text{. } \end{aligned}$$

The following result is a generalization of the previous lemma and will play an important role in the proof of Theorem 4.20.

Lemma 4.17

For the classes $(\mathcal {W}, \nu )\subseteq \mathcal {C}^{2}$ and $\mathcal {X}\subseteq \mathcal {C}$ such that $\nu =\textrm{smd}(\nu )$ is a $\mathcal {W}\cap \mathcal {X}$-injective relative cogenerator in $\mathcal {W}\cap \mathcal {X},$ the following statements hold true.

• (a) $\mathcal {W}\cap \mathcal {X}\cap \nu {}_{\mathcal {X}}^{\vee }=\left\{ W\in \mathcal {W}\cap \mathcal {X}\,|\,\textrm{id}_{\mathcal {W}\cap \mathcal {X}}(W)<\infty \right\} .$

• (b) $\textrm{id}_{\mathcal {W}\cap \mathcal {X}}(M)=\textrm{coresdim}{}_{\nu }^{\mathcal {X}}\left( M\right)$ $\forall M\in \mathcal {W}\cap \mathcal {X}\cap \nu _{\mathcal {X}}^{\vee }.$

• (c) If $\mathcal {W}\cap \mathcal {X}$ is left thick, then $\nu _{\mathcal {X}}^{\vee }$ is left thick and

  $$\begin{aligned} \nu _{\mathcal {X}}^{\vee }=\left\{ W\in \mathcal {W}\cap \mathcal {X}\,|\,\textrm{id}_{\mathcal {W}\cap \mathcal {X}}(W)<\infty \right\} \text{. } \end{aligned}$$

Proof

Let $M\in \mathcal {W}\cap \mathcal {X}\cap \nu _{\mathcal {X}}^{\vee }$. By Theorem 4.9, we have

$$\begin{aligned} \textrm{id}_{\mathcal {W}\cap \mathcal {X}}(M)\le \textrm{id}_{\mathcal {W}\cap \mathcal {X}}(\nu )+\textrm{coresdim}{}_{\nu }^{\mathcal {X}}\left( M\right) =\textrm{coresdim}{}_{\nu }^{\mathcal {X}}\left( M\right) <\infty \text{. } \end{aligned}$$

For every $M\in \mathcal {W}\cap \mathcal {X}$ with $\textrm{id}_{\mathcal {W}\cap \mathcal {X}}(M)=n<\infty$, we have $\textrm{coresdim}{}_{\nu }^{\mathcal {X}}\left( M\right) \le n$. Indeed, since $\nu$ is a relative cogenerator in $\mathcal {W}\cap \mathcal {X}$, there is an exact sequence

$$\begin{aligned} \eta :\,0 \rightarrow M \rightarrow N_{0} \rightarrow Z \rightarrow 0 \text{, } \text{ with } N_{0}\in \nu \text{ and } Z\in \mathcal {W}\cap \mathcal {X}. \end{aligned}$$

Now, if $n=0$, then $\eta$ splits and, thus, $\textrm{coresdim}{}_{\nu }^{\mathcal {X}}\left( M\right) =0$. Let $n\ge 1$. We can build an exact sequence

$$\begin{aligned} 0\rightarrow M\rightarrow N_{0}\overset{f_{0}}{\rightarrow }N_{1}\rightarrow \cdots \rightarrow N_{n-2}\overset{f_{n-2}}{\rightarrow }N_{n-1}\rightarrow Z\rightarrow 0\text{, } \end{aligned}$$

with $N_{i}\in \nu \;\forall i\in [0,n-1]$, ${\text {Im}}\left( f_{i}\right) \in \mathcal {W}\cap \mathcal {X}\;\forall i\in [0,n-2]$, and $Z\in \mathcal {W}\cap \mathcal {X}$. Hence, by the Shifting Lemma, it follows that:

$$\begin{aligned} \textrm{Ext}^{k}_{\mathcal {C}}(W,Z)=\textrm{Ext}^{n+k}_{\mathcal {C}}(W,M)=0\quad \forall \, W\in \mathcal {W}\cap \mathcal {X},\;\forall k>0\text{. } \end{aligned}$$

Therefore, by Lemma 4.15(b), $Z\in \mathcal {W}\cap \mathcal {X}\cap (\mathcal {W}\cap \mathcal {X})^{\bot }=\nu$, and thus, $\textrm{coresdim}{}_{\nu }^{\mathcal {X}}\left( M\right) \le n,$ proving (a) and (b).

Let $\mathcal {W}\cap \mathcal {X}$ be left thick. Then, by Lemma 4.14, we have $\nu ^{\vee }\subseteq \mathcal {W}\cap \mathcal {X}$. Therefore, the equality of (c) follows from (a), since $\nu _{\mathcal {X}}^{\vee }\subseteq \nu ^{\vee }\subseteq \mathcal {W}\cap \mathcal {X}$. Finally, the left thickness of $\nu _X^\vee$ follows from Lemma 4.6. $\square$

In what follows, we will be interested in the study of the closure properties of the classes $(\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$, $(\mathcal {X},\mathcal {Y})^{\vee }$ and $(\mathcal {X},\mathcal {Y})_{n}^{\vee }$. In particular, we will get sufficient conditions for this classes to be thick.

Proposition 4.18

For an abelian category $\mathcal {C}$ and $\mathcal {X}$,$\mathcal {Y}\subseteq \mathcal {C},$ the following statements hold true.

• (a) Let $\mathcal {Y}=\mathcal {Y}^{\oplus _{<\infty }},$ $\mathcal {X}$ be closed under extensions and $(\mathcal {X},\mathcal {Y})_{\infty }^{\vee }\subseteq {}{}^{\bot _{1}}\mathcal {Y}$. Then, for a given exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \text{, }$ with $A,C\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$, we have that $\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( B\right) \le \max \left\{ \textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( A\right) ,\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( C\right) \right\} \text{. }$ Furthermore, $(\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$ and $(\mathcal {X},\mathcal {Y})^{\vee }$ are closed under extensions.

• (b) Let $\mathcal {X}=\textrm{smd}(\mathcal {X})$ and $(\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$ be both closed under extensions. Then, $(\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$ is closed under direct summands.

Proof

(a) Let $\eta _{0}:\;0 \rightarrow A \xrightarrow {u} B \xrightarrow {v} C \rightarrow 0$ be an exact sequence with $A,C\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }.$ Then, $B\in \mathcal {X}$, since $\mathcal {X}$ is closed under extensions. On the other hand, by definition, there are exact sequences

$$\begin{aligned} \eta _{A}^{0}:\;0 \rightarrow A \xrightarrow {a} Y_{A} \rightarrow A_{1} \rightarrow 0\, \text{ and } \,\eta _{C}^{0}:\;0 \rightarrow C \xrightarrow {c} Y_{C} \rightarrow C_{1}\rightarrow 0\text{, } \end{aligned}$$

with $Y_{A},Y_{C}\in \mathcal {Y}$ and $A_{1},C_{1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$. Since $C\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }\subseteq {}{}^{\bot _{1}}\mathcal {Y}\subseteq {}{}^{\bot _{1}}Y_{A}\text{, }$ we have the exact sequence

$$\begin{aligned} 0\rightarrow \textrm{Hom}_{\mathcal {C}}(C, Y_{A})\rightarrow \textrm{Hom}_{\mathcal {C}}(B, Y_{A})\rightarrow \textrm{Hom}_{\mathcal {C}}(A, Y_{A})\rightarrow 0\text{. } \end{aligned}$$

Therefore, there is a morphism $\alpha :B\rightarrow Y_{A}$, such that $\alpha u=a$. Consider the morphism $b:=\left( {\begin{matrix}\alpha \\ cv \end{matrix}}\right) :B\rightarrow Y_{A}\oplus Y_{C}\text{. }$ Since

$$\begin{aligned} \left( {\begin{matrix}1\\ 0 \end{matrix}}\right) a=\left( {\begin{matrix}a\\ 0 \end{matrix}}\right) =\left( {\begin{matrix}\alpha u\\ 0 \end{matrix}}\right) =\left( {\begin{matrix}\alpha \\ cv \end{matrix}}\right) u\quad \text{ and } \quad \left( {\begin{matrix}0&1\end{matrix}}\right) \left( {\begin{matrix}\alpha \\ cv \end{matrix}}\right) =cv\text{, } \end{aligned}$$

by the Snake Lemma, we get the exact sequences

$$\begin{aligned} \eta _{B}^{0}:\;0 \rightarrow B \xrightarrow {b} Y_{0} \rightarrow B_{1} \rightarrow 0\; \text{ and } \;\eta _{1}:\;0 \rightarrow A_{1} \rightarrow B_{1} \rightarrow C_{1} \rightarrow 0\text{, } \end{aligned}$$

where $A_{1}$,$C_{1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$, $Y_{0}:=Y_{A}\oplus Y_{C}\in \mathcal {Y}$ and $B_{1}\in \mathcal {X}$. To repeat the argument recursively, assume we have exact sequences

$$\begin{aligned} \eta _{B}^{k-1}:\;0 \rightarrow B_{k-1} \xrightarrow {b_{k-1}} Y_{k-1} \rightarrow B_{k}\rightarrow 0 \; \text{ and } \;\eta _{k}:\;0 \rightarrow A_{k} \xrightarrow {u_{k}} B_{k} \xrightarrow {v_{k}} C_{k} \rightarrow 0\text{, } \end{aligned}$$

where $A_{k},$ $C_{k}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$, $X_{k}\in \mathcal {Y}$ and $B_{k}\in \mathcal {X}$ $\forall \, k\le n$. Observe that there are exact sequences

$$\begin{aligned} \eta _{A}^{n}:\;0 \rightarrow A_{n} \xrightarrow {a_{n}} Y_{A,n} \rightarrow A_{n+1} \rightarrow 0 \, \text{ and } \,\eta _{C}^{n}:\;0\rightarrow C_{n} \xrightarrow {c_{n}} Y_{C,n} \rightarrow C_{n+1} \rightarrow 0 \text{, } \end{aligned}$$

with $Y_{A,n},Y_{C,n}\in \mathcal {Y}$ and $A_{n+1},C_{n+1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$. Since $C_{n}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }\subseteq {}{}^{\bot _{1}}\mathcal {Y}\text{, }$ we have the exact sequence

$$\begin{aligned} 0\rightarrow \textrm{Hom}_{\mathcal {C}}(C_{n}, Y_{A,n})\rightarrow \textrm{Hom}_{\mathcal {C}}(B_{n}, Y_{A,n})\rightarrow \textrm{Hom}_{\mathcal {C}}(A_{n}, Y_{A,n})\rightarrow 0\text{. } \end{aligned}$$

Therefore, there is a morphism $\alpha _{n}:B_{n}\rightarrow Y_{A,n}$, such that $\alpha _{n}u_{n}=a_{n}$. Consider the morphism $b_{n}:=\left( {\begin{matrix}\alpha _{n}\\ c_{n}v_{n} \end{matrix}}\right) :B_{n}\rightarrow Y_{A,n}\oplus Y_{C,n}\text{. }$ Since

$$\begin{aligned} \left( {\begin{matrix}1\\ 0 \end{matrix}}\right) a_{n}=\left( {\begin{matrix}a_{n}\\ 0 \end{matrix}}\right) =\left( {\begin{matrix}\alpha _{n}u_{n}\\ 0 \end{matrix}}\right) =\left( {\begin{matrix}\alpha _{n}\\ c_{n}v_{n} \end{matrix}}\right) u_{n}\quad \text{ and } \quad \left( {\begin{matrix}0&1\end{matrix}}\right) \left( {\begin{matrix}\alpha _{n}\\ c_{n}v_{n} \end{matrix}}\right) =c_{n}v_{n}\text{, } \end{aligned}$$

by the Snake Lemma, we get the exact sequences

$$\begin{aligned} \eta _{B}^{n}:\;0 \rightarrow B_{n} \xrightarrow {b_{n}} Y_{n} \rightarrow B_{n+1} \rightarrow 0 \; \text{ and } \;\eta _{n+1}:\;0 \rightarrow A_{n+1}\rightarrow B_{n+1} \rightarrow C_{n+1} \rightarrow 0\text{, } \end{aligned}$$

where $A_{n+1}$,$C_{n+1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$, $Y_{n}:=Y_{A,n}\oplus Y_{C,n}\in \mathcal {Y}$ and $B_{n+1}\in \mathcal {X}$. Observe that the family of short exact sequences $\left\{ \eta _{B}^{i}\right\} _{i=0}^{\infty }$ induces a long exact sequence from where we get that $B\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$. Furthermore, if $A,C\in (\mathcal {X},\mathcal {Y})^{\vee }$, then the families of exact sequences $\left\{ \eta _{A}^{k}\right\} _{k=0}^{\infty }$ and $\left\{ \eta _{B}^{k}\right\} _{k=0}^{\infty }$ can be chosen in a way that they form a $(\mathcal {X},\mathcal {Y})$-coresolution of minimal length. Hence, for

$$\begin{aligned} m:=\max \left\{ \textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( A\right) ,\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( C\right) \right\} \text{, } \end{aligned}$$

we have $A_{k}=0=C_{k}\,\forall k>m$. Thus, by considering the family of exact sequences $\left\{ \eta _{k}\right\} _{k=1}^{\infty }$, we have $B_{k}=0\,\forall k>m$. Therefore, $\textrm{coresdim}{}_{\mathcal {Y}}^{\mathcal {X}}\left( B\right) \le m$.

(b) Consider a split exact sequence

$$\begin{aligned} 0 \rightarrow W \rightarrow V \xrightarrow {f} U \rightarrow 0 \text{, } \text{ with } V\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee } \text{. } \end{aligned}$$

Then, $U,W\in \mathcal {X}$, since $\mathcal {X}=\textrm{smd}(\mathcal {X}).$ Let us show that $W\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }.$ Indeed, since $V\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee },$ there is an exact sequence

$$\begin{aligned} 0 \rightarrow V \xrightarrow {g} Y_{0} \rightarrow V_{1} \rightarrow 0 \text{, } \text{ with } Y_{0}\in \mathcal {Y} \,and\, V_{1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee } \text{. } \end{aligned}$$

Now, by considering the push-out of f with g, we get the exact sequences

$$\begin{aligned} \eta :\;&0 \rightarrow U \rightarrow W_{1} \rightarrow V_{1} \rightarrow 0 \text{, }\\ \mu _{0}:\;&0 \rightarrow W \rightarrow Y_{0} \rightarrow W_{1} \rightarrow 0 \text{. } \end{aligned}$$

Since $U,V_{1}\in \mathcal {X}$, from $\eta$, we get that $W_{1}\in \mathcal {X}$. Moreover, by making the coproduct of $\eta$ with the short exact sequence

$$\begin{aligned} 0 \rightarrow W \xrightarrow {1} W \rightarrow 0 \rightarrow 0 \text{, } \end{aligned}$$

we get the short exact sequence

$$\begin{aligned} 0 \rightarrow V \rightarrow W \oplus W_{1} \rightarrow V_{1} \rightarrow 0 \text{. } \end{aligned}$$

Observe that $V,V_{1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$. Hence, $W\oplus W_{1}\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$, since $(\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$ is closed under extensions. Then, by repeating the above argument, we get a family of exact sequences $\left\{ \mu _{i}:\;0 \rightarrow W_{i} \rightarrow Y_{i} \rightarrow W_{i+1} \rightarrow 0 \right\} _{i=0}^{\infty },$ where $W_{0}:=W$, $Y_{i}\in \mathcal {Y}$ and $W_{i}\in \mathcal {X}\,\forall \, i\ge 0$. Therefore, $W\in (\mathcal {X},\mathcal {Y})_{\infty }^{\vee }$. $\square$

The following theorem is inspired by [11, Thm. 3.29], which in turn is a generalization of [7, Prop. 5.1]. The closure properties of the relative classes, involved in Theorem 4.19, Theorem 4.20 and Corollary 4.21, play an important role in the study and development of n-$\mathcal {X}$-tilting theory in [4].

Theorem 4.19

Let $(\mathcal {X},\mathcal {Y})\subseteq \mathcal {C}^2$ be classes, such that $\mathcal {Y}=\mathcal {Y}^{\oplus _{<\infty }},$ $\mathcal {X}=\textrm{smd}(\mathcal {X})$ is closed under extensions and $\textrm{Ext}^{1}_{\mathcal {C}}(\mathcal {X}, \mathcal {X}\cap \mathcal {Y})=0.$ Then, the following statements hold true.

• (a) $(\mathcal {X}, \mathcal {Y})_{\infty }^{\vee }=(\mathcal {X}, \mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }$ and it is closed under extensions and direct summands. Moreover, $(\mathcal {X},\mathcal {Y})^{\vee }=(\mathcal {X},\mathcal {X}\cap \mathcal {Y})^{\vee }$ and it is closed under extensions.

• (b) $(\mathcal {X}, \mathcal {Y})_{\infty }^{\vee }$ is left thick if $\mathcal {X}$ is left thick.

Proof

(a) The equality $(\mathcal {X}, \mathcal {Y})_{\infty }^{\vee }=(\mathcal {X}, \mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }$ follows from the fact that $\mathcal {X}$ is closed under extensions. Since $\textrm{Ext}^{1}_{\mathcal {C}}(\mathcal {X}, \mathcal {X}\cap \mathcal {Y})=0$, we have that $(\mathcal {X},\mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }\subseteq \mathcal {X}\subseteq {}^{\bot _{1}}(\mathcal {X}\cap \mathcal {Y})\text{. }$ Therefore, we get (a) by applying Proposition 4.18 to the pair $(\mathcal {X},\mathcal {X}\cap \mathcal {Y}).$

(b) Let $\mathcal {X}$ be closed under epi-kernels. Then, by (a), it is enough to show that $(\mathcal {X}, \mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }$ is closed under epi-kernels. Consider an exact sequence $0 \rightarrow A \xrightarrow {a} B \rightarrow C \rightarrow 0,$ with $B,C\in (\mathcal {X}, \mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }.$ In particular, there is an exact sequence

$$\begin{aligned} 0 \rightarrow B \xrightarrow {b} W_{0} \rightarrow C_{0}\rightarrow 0\text{, } \end{aligned}$$

with $W_{0}\in \mathcal {X}\cap \mathcal {Y}$ and $C_{0}\in (\mathcal {X},\mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }$. By the composition $ba:A\rightarrow W_0$ and the Snake Lemma, we get the exact sequences

$$\begin{aligned} \eta :\,0 \rightarrow A \rightarrow W_{0} \rightarrow C' \rightarrow 0&\text{, }\\ \eta ':\,0 \rightarrow C \rightarrow C' \rightarrow C_{0} \rightarrow 0&\text{. } \end{aligned}$$

Since $C,C_{0}\in (\mathcal {X},\mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }$, it follows from (a) that $C'\in (\mathcal {X},\mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee },$ and since $\mathcal {X}$ is left thick, we have $A\in \mathcal {X}.$ Therefore, $\eta$ shows that $A\in (\mathcal {X},\mathcal {X}\cap \mathcal {Y})_{\infty }^{\vee }$.

$\square$

Theorem 4.20

Let $(\mathcal {Z}, \nu )\subseteq \mathcal {C}^{2}$ be classes, such that $\mathcal {Z}=\textrm{smd}(\mathcal {Z})$ is closed under extensions, ${\text {add}}(\nu )=\nu$ and $\nu$ is $\mathcal {Z}$-injective. Then, the following statements hold true.

• (a) The classes $(\mathcal {Z}, \nu )_{\infty }^{\vee }$ and $(\mathcal {Z}, \nu )^{\vee }$ are closed under extensions and direct summands. Moreover, we have the equalities

  • (a1) $(\mathcal {Z}, \nu )_{\infty }^{\vee }=(\mathcal {Z}, \mathcal {Z}\cap \nu )_{\infty }^{\vee },$

  • (a2) $(\mathcal {Z}, \nu )^{\vee }=(\mathcal {Z}, \mathcal {Z}\cap \nu )^{\vee }=\left\{ M\in (\mathcal {Z},\nu )_{\infty }^{\vee }\,|\,\textrm{id}_{(\mathcal {Z},\nu )_{\infty }^{\vee }}(M)<\infty \right\} .$

• (b) $\textrm{id}_{(\mathcal {Z},\nu )_{\infty }^{\vee }}(M)=\textrm{coresdim}{}_{\nu }^{\mathcal {Z}}\left( M\right) \;\forall \, M\in (\mathcal {Z},\nu )^{\vee }$.

• (c) The classes $(\mathcal {Z},\nu )_{\infty }^{\vee }$ and $(\mathcal {Z},\nu )^{\vee }$ are left thick if $\mathcal {Z}$ is left thick.

Proof

(a) & (b) By Theorem 4.19, we get: the first two equalities in (a), $(\mathcal {Z},\nu )_{\infty }^{\vee }$ is closed under extensions and direct summands and $(\mathcal {Z},\nu )^{\vee }$ is closed under extensions.

Let $\mathcal {W}=(\mathcal {Z},\nu )_{\infty }^{\vee }.$ Since $\nu$ is $\mathcal {Z}$-injective, it follows that $\nu$ is a $\mathcal {W}$-injective relative cogenerator in $\mathcal {W}$. Hence, by Lemma 4.17 (a)

$$\begin{aligned} (\mathcal {Z}, \nu )^{\vee }=\mathcal {W}\cap (\mathcal {Z}, \nu )^{\vee }=\mathcal {W}\cap \mathcal {Z}\cap \nu _{\mathcal {Z}}^{\vee }=\left\{ M\in \mathcal {W}\cap \mathcal {Z}\,|\,\textrm{id}_{\mathcal {W}\cap \mathcal {Z}}(M)<\infty \right\} \text{. } \end{aligned}$$

Note that $\mathcal {W}\cap \mathcal {Z}=\mathcal {W}$. Consequently

$$\begin{aligned} (*)\quad {\small (\mathcal {Z},\nu )^{\vee }=\left\{ M\in (\mathcal {Z},\nu )_{\infty }^{\vee }\,|\,\textrm{id}_{(\mathcal {Z},\nu )_{\infty }^{\vee }}(M)<\infty \right\} =\left\{ M\in \mathcal {W}\,|\,\textrm{id}_{\mathcal {W}}(M)<\infty \right\} } \text{. } \end{aligned}$$

Furthermore, we have (b) from Lemma 4.17(b).

Let us show that $(\mathcal {Z},\nu )^{\vee }$ is closed under direct summands. Indeed, let $M\in (\mathcal {Z},\nu )^{\vee }$ and $M=M_{1}\oplus M_{2}$. Since $(\mathcal {Z},\nu )^{\vee }\subseteq (\mathcal {Z},\nu )_{\infty }^{\vee }$ and $(\mathcal {Z},\nu )_{\infty }^{\vee }$ is closed under direct summands, we have $M_{1},M_{2}\in (\mathcal {Z},\nu )_{\infty }^{\vee }$. Now, using $(*)$ and that

$$\begin{aligned} \max \left\{ \textrm{id}_{\mathcal {W}}(M_{1}),\textrm{id}_{\mathcal {W}}(M_{2})\right\} =\textrm{id}_{\mathcal {W}}(M)<\infty , \end{aligned}$$

it follows that $M_{1},M_{2}\in (\mathcal {Z},\nu )^{\vee }$.

(c) Let $\mathcal {Z}$ be left thick. By Theorem 4.19 (b), we have that $(\mathcal {Z},\nu )_{\infty }^{\vee }$ is left thick. Now, from (a) and the proof of Theorem 4.19 (b), it follows that $(\mathcal {Z},\nu )^{\vee }$ is left thick.

$\square$

Corollary 4.21

Let $\mathcal {X}\subseteq \mathcal {C}$ be left thick and $\mathcal {T}={\text {add}}(\mathcal {T})\subseteq \mathcal {C}$ be such that $\mathcal {T}\subseteq \mathcal {T}^{\bot }\cap \mathcal {X}.$ Then, the following statements hold true.

• (a) The class $\mathcal {Q}:=({}^{\bot }\mathcal {T}\cap \mathcal {X},\mathcal {T})_{\infty }^{\vee }$ is left thick.

• (b) $\mathcal {T}_{\mathcal {X}}^{\vee }=\mathcal {T}^{\vee }=\{M\in \mathcal {Q}\;|\;\textrm{id}_{\mathcal {Q}}(M)<\infty \}$ and it is left thick.

Proof

Note that $\mathcal {Q}\subseteq {}^{\bot }\mathcal {T}\cap \mathcal {X},$ $\mathcal {Q}\cap \mathcal {X}=\mathcal {Q}$ and $\mathcal {T}\subseteq {}^{\bot }\mathcal {T}\cap \mathcal {Q}.$ Using that $\textrm{Ext}^{i}_{\mathcal {C}}(^{\bot }\mathcal {T},\mathcal {T})=0$ $\forall i>0$, it follows that $\mathcal {T}$ is $^{\bot }\mathcal {T}\cap \mathcal {X}$-injective and, thus, $\mathcal {T}$ is $\mathcal {Q}$-injective, since $\mathcal {Q}\subseteq {}^{\bot }\mathcal {T}\cap \mathcal {X}.$ Now, by applying Theorem 4.20 to the pair $({}^{\bot }\mathcal {T}\cap \mathcal {X},\mathcal {T}),$ we get that $\mathcal {Q}$ is left thick; proving (a). Finally, by applying Lemma 4.16 and Lemma 4.17 to the pair $(\mathcal {Q},\mathcal {T})$ and the class $\mathcal {X},$ it can be seen that (b) holds true. $\square$

We can now return to the study of the $\mathcal {X}$-complete pairs in an abelian category $\mathcal {C}.$ We will be focusing on deepening our understanding of the relations between the different induced relative dimensions. Furthermore, we will see that, under certain hypotheses, for an $\mathcal {X}$-complete pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {C},$ the class $\mathcal {A}\cap \mathcal {B}\cap \mathcal {X}$ is a relative generator in $\mathcal {A}\cap \mathcal {X}$ and a relative cogenerator in $\mathcal {B}\cap \mathcal {X}.$ Let us start by recalling the following result proved by M. Auslander and R. O. Buchweitz in [6].

Proposition 4.22

[6, Prop. 2.1] Let $(\mathcal {X}, \omega )\subseteq \mathcal {C}^{2}$ be classes which are closed under direct summands, $\mathcal {X}$ be closed under extensions, and let $\omega$ be an $\mathcal {X}$-injective relative cogenerator in $\mathcal {X}.$ Then

$$\begin{aligned} \textrm{pd}_{\omega ^{\wedge }}(C)=\textrm{pd}_{\omega }(C)={\text {resdim}}_{\mathcal {X}}\left( C\right) \quad \forall \, C\in \mathcal {X}^{\wedge }. \end{aligned}$$

In the case of an $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pair, the above result can be strengthened, see Proposition 4.23. As we will see in [4], for an n-$\mathcal {X}$-tilting subcategory of $\mathcal {C},$ the pair $({}^\perp (\mathcal {T}^\perp ), \mathcal {T}^\perp )$ is always $\mathcal {X}$-complete and $\mathcal {X}$-hereditary. Therefore, Proposition 4.23 and Theorem 4.24 will play an important role in the development of the n-$\mathcal {X}$-tilting theory in [4].

Proposition 4.23

For a class $\mathcal {X}\subseteq \mathcal {C},$ an $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {C}$ such that $\mathcal {A},$ $\mathcal {X}$ and $\mathcal {B}$ are closed under extensions and direct summands, and $\omega :=\mathcal {A}\cap \mathcal {B}\cap \mathcal {X},$ the following statements hold true.

• (a) The class $\omega$ is an $\mathcal {A}\cap \mathcal {X}$-injective relative cogenerator in $\mathcal {A}\cap \mathcal {X}.$

• (b) The class $\omega$ is an $\mathcal {B}\cap \mathcal {X}$-projective relative generator in $\mathcal {B}\cap \mathcal {X}.$

• (c) For any $M\in \left( \mathcal {A},\mathcal {X}\right) ^{\wedge },$ we have the equalities

  $$\begin{aligned} \textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=\textrm{pd}_{\omega }(M)=\textrm{pd}_{\omega ^{\wedge }}(M)=\textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( M\right) ={\text {resdim}}_{\mathcal {X}\cap \mathcal {A}}\left( M\right) . \end{aligned}$$

• (d) $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right) \;$ $\forall \, M\in (\omega ,\mathcal {X})^{\wedge }.$

• (e) For any $M\in \left( \mathcal {X},\mathcal {B}\right) ^{\vee },$ we have the equalities

  $$\begin{aligned} \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)=\textrm{id}_{\omega }(M)=\textrm{id}_{\omega ^{\vee }}(M)=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( M\right) ={\text {coresdim}}_{\mathcal {X}\cap \mathcal {B}}\left( M\right) . \end{aligned}$$

• (f) $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(M)=\textrm{coresdim}{}_{\omega }^{\mathcal {X}}\left( M\right) \;$ $\forall \, M\in (\mathcal {X},\omega )^{\vee }.$

Proof

Note that (b) is the dual of (a), (e) is the dual of (c), and (f) is the dual of (d).

(a) We have that $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$, since $(\mathcal {A}, \mathcal {B})$ is $\mathcal {X}$-hereditary. In particular, $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\omega )\le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})=0$, and thus, $\omega$ is $\mathcal {A}\cap \mathcal {X}$-injective.

Let us show that $\omega$ is a relative cogenerator in $\mathcal {A}\cap \mathcal {X}.$ Indeed, since $(\mathcal {A}, \mathcal {B})$ is right $\mathcal {X}$-complete, for every $X\in \mathcal {A}\cap \mathcal {X},$ there is an exact sequence

$$\begin{aligned} 0 \rightarrow X \rightarrow W \rightarrow X' \rightarrow 0 \text{, } \text{ with } W\in \mathcal {B}\cap \mathcal {X} \text{ and } X'\in \mathcal {A}\cap \mathcal {X}. \end{aligned}$$

Furthermore, $W\in \mathcal {A}\cap \mathcal {X}$, since $\mathcal {A}\cap \mathcal {X}$ is closed under extensions. Hence, W belongs to $\mathcal {A}\cap \mathcal {B}\cap \mathcal {X}=\omega ,$ proving (a).

(c) Observe, first, that $\left( \mathcal {A},\mathcal {X}\right) ^{\wedge }\subseteq \left( \mathcal {A}\cap \mathcal {X}\right) ^{\wedge }$ since $\mathcal {X}$ is closed under extensions.

Let $M\in \left( \mathcal {A},\mathcal {X}\right) ^{\wedge }.$ Then, by (a) and Proposition 4.22, $\textrm{pd}_{\omega ^{\wedge }}(M)=\textrm{pd}_{\omega }(M)={\text {resdim}}_{\mathcal {A}\cap \mathcal {X}}\left( M\right) \text{. }$ By the dual of Corollary 4.12 (a), $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=\textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( M\right) \text{. }$ Moreover, ${\text {resdim}}_{\mathcal {A}\cap \mathcal {X}}\left( M\right) \le \textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( M\right)$, because $\mathcal {X}$ is closed under extensions. Since we have that

$$\begin{aligned} \textrm{pd}_{\omega }(M)=\textrm{pd}_{\omega ^{\wedge }}(M)={\text {resdim}}_{\mathcal {A}\cap \mathcal {X}}\left( M\right) \le \textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( M\right) =\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)\text{, } \end{aligned}$$

it is enough to show that $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)\le \textrm{pd}_{\omega }(M).$ To prove that, we can assume that $\textrm{pd}_{\omega }(M)=m<\infty$. Then

$$\begin{aligned} \textrm{pd}_{\omega }(M)\le \textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=\textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( M\right) <\infty \text{, } \end{aligned}$$

and there is some $t\ge 0$, such that $\textrm{pd}_{\mathcal {B\cap \mathcal {X}}}(M)=m+t$. Let $B\in \mathcal {B}\cap \mathcal {X}$. By (b), we know that $\omega$ is a $\mathcal {B}\cap \mathcal {X}$-projective relative generator in $\mathcal {B}\cap \mathcal {X}$. Hence, there is an exact sequence $0\rightarrow B_{t}\rightarrow A_{t-1}\rightarrow \cdots \rightarrow A_{0}\rightarrow B\rightarrow 0\text{, }$ with $B_{t}\in \mathcal {B}\cap \mathcal {X}$ and $A_{i}\in \omega \,\forall i\in [0,t-1].$ Since $A_{i}\in \omega \subseteq M^{\bot _{>m}}\;\forall \, i\in [0,t-1]\text{, }$ by the Shifting Lemma, we have $\textrm{Ext}^{k}_{\mathcal {C}}(M,B)\cong \textrm{Ext}^{k+t}_{\mathcal {C}}(M,B_{t})\,\forall k>m\text{. }$ Now, using that $\textrm{pd}_{\mathcal {B\cap \mathcal {X}}}(M)=m+t$, it follows that:

$$\begin{aligned} \textrm{Ext}^{k}_{\mathcal {C}}(M,B)\cong \textrm{Ext}^{k+t}_{\mathcal {C}}(M,B_{t})=0\quad \forall \, k>m\text{. } \end{aligned}$$

Therefore, $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)\le m=\textrm{pd}_{\omega }(M)$.

(d) Let $M\in (\omega ,\mathcal {X})^{\wedge }$. By (b), we have that $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(\omega )=0$. Then, by the dual of Theorem 4.9, we get that

$$\begin{aligned} \textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)\le \textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(\omega )+\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right) =\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right) \text{. } \end{aligned}$$

We shall prove, by induction on $n=\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right)$, that $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right)$.

If $n=0$, then $M\in \omega$, and thus, $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=0.$

Let $n>0$. By inductive hypothesis, $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(N)=\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( N\right) ,$ for every $N\in \mathcal {X}$ with $\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( N\right) <n$. Since $n=\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right)$, we have an exact sequence

$$\begin{aligned} \eta :\quad 0 \rightarrow K \rightarrow W_{0} \rightarrow M \rightarrow 0 \text{, } \text{ with } W_{0}\in \omega ,\,K\in (\omega ,\mathcal {X})^{\wedge }\text{, } \end{aligned}$$

and $\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( K\right) =n-1.$ Thus, $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(K)=n-1$. Besides, by Lemma 4.6

$$\begin{aligned} n-1=\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(K)\le \max \{\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)-1, \textrm{pd}_{\mathcal {B} \cap \mathcal {X}}(W_{0})\}\text{. } \end{aligned}$$

(1)

We assert that $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)>0.$ Suppose that $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=0$. Since $\omega \subseteq \mathcal {A}$, we have $(\omega ,\mathcal {X})^{\wedge }\subseteq (\mathcal {A},\mathcal {X})^{\wedge }$. Hence, by (c), $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=\textrm{pd}_{\omega ^{\wedge }}(M)$. Thus, $\eta$ splits since $K\in \omega ^{\wedge }.$ Consequently, $M\in \omega$, contradicting that $\textrm{resdim}{}_{\omega }^{\mathcal {X}}\left( M\right) >0;$ and the assertion follows. Then, by (1), we get that $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)=n$, since $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(M)>0$ and $\textrm{pd}_{\mathcal {B}\cap \mathcal {X}}(W_{0})=0.$ $\square$

Theorem 4.24

For a class $\mathcal {X}\subseteq \mathcal {C},$ an $\mathcal {X}$-complete and $\mathcal {X}$-hereditary pair $(\mathcal {A}, \mathcal {B})$ in $\mathcal {C}$ such that $\mathcal {A},$ $\mathcal {X}$ and $\mathcal {B}$ are closed under extensions and direct summands, and $\omega :=\mathcal {A}\cap \mathcal {B}\cap \mathcal {X},$ the following statements hold true.

• (a) $\omega =\left( \mathcal {A}\cap \mathcal {X}\right) ^{\bot }\cap \mathcal {A}\cap \mathcal {X}=\mathcal {A}\cap \mathcal {X}\cap \omega ^{\wedge }=(\omega ,\mathcal {A}\cap \mathcal {X})^{\wedge }.$ Furthermore,

  • (a1) we have that

    $$\begin{aligned} \textrm{pd}_{\mathcal {X}}(\mathcal {A}\cap \mathcal {X})&=\textrm{pd}_{\mathcal {A\cap \mathcal {X}}}(\mathcal {A}\cap \mathcal {X})&={\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) \\&=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( \mathcal {X}\right)&=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) \\&={\text {coresdim}}_{\omega }\left( \mathcal {A}\cap \mathcal {X}\right)&={\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right) \\&=\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {A}\cap \mathcal {X}\right)&=\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {X}\right) \text{; } \end{aligned}$$

  • (a2) $\textrm{pd}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})<\infty$ if, and only if, $\mathcal {A}\cap \mathcal {X}\subseteq \omega ^{\vee }$ and $\textrm{pd}_{\mathcal {X}}(\omega )<\infty .$ Moreover, for $\textrm{pd}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})<\infty ,$ we have that

    $$\begin{aligned} \mathcal {X}\subseteq (\mathcal {X},\mathcal {B})^{\vee }\subseteq \left( \mathcal {B}\cap \mathcal {X}\right) ^{\vee } \text{ and } \textrm{pd}_{\mathcal {X}}(\mathcal {A}\cap \mathcal {X})=\textrm{pd}_{\mathcal {X}}(\omega ). \end{aligned}$$

• (b) $\omega ={}^\perp \left( \mathcal {B}\cap \mathcal {X}\right) \cap \mathcal {B}\cap \mathcal {X}=\mathcal {B}\cap \mathcal {X}\cap \omega ^{\vee }=(\mathcal {B}\cap \mathcal {X},\omega )^{\vee }.$ Furthermore,

  • (b1) we have that

    $$\begin{aligned} \textrm{id}_{\mathcal {X}}(\mathcal {B}\cap \mathcal {X})&=\textrm{id}_{\mathcal {\mathcal {B}\cap \mathcal {X}}}(\mathcal {B}\cap \mathcal {X})&={\text {resdim}}_{\mathcal {A}\cap \mathcal {X}}\left( \mathcal {B}\cap \mathcal {X}\right) \\&=\textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( \mathcal {X}\right)&=\textrm{resdim}{}_{\mathcal {A}}^{\mathcal {X}}\left( \mathcal {B}\cap \mathcal {X}\right) \\&={\text {resdim}}_{\omega }\left( \mathcal {B}\cap \mathcal {X}\right)&={\text {resdim}}_{\mathcal {A}\cap \mathcal {X}}\left( \mathcal {B}\cap \mathcal {X}\right) \\&=\textrm{resdim}{}_{\mathcal {A}\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {B}\cap \mathcal {X}\right)&=\textrm{resdim}{}_{\mathcal {A}\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {X}\right) \text{; } \end{aligned}$$

  • (b2) $\textrm{id}_{\mathcal {B}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})<\infty$ if, and only if, $\mathcal {B}\cap \mathcal {X}\subseteq \omega ^{\wedge }$ and $\textrm{id}_{\mathcal {X}}(\omega )<\infty .$ Moreover, for $\textrm{id}_{\mathcal {B}\cap \mathcal {X}}(\mathcal {B}\cap \mathcal {X})<\infty ,$ we have that

    $$\begin{aligned} \mathcal {X}\subseteq (\mathcal {A},\mathcal {X})^{\wedge }\subseteq \left( \mathcal {A}\cap \mathcal {X}\right) ^{\wedge } \text{ and } \textrm{id}_{\mathcal {X}}(\mathcal {B}\cap \mathcal {X})=\textrm{id}_{\mathcal {X}}(\omega ). \end{aligned}$$

Proof

Note first that (b) is the dual of (a). Thus, we need to prove (a).

By Proposition 4.23 (a), $\omega$ is an $\mathcal {A}\cap \mathcal {X}$-injective relative cogenerator in $\mathcal {A}\cap \mathcal {X}$. Then, by Lemma 4.15 (b), $\omega$ satisfies the desired equalities of (a). Let us show the statements of (a1) and (a2).

(a1) By Corollary 4.12, we have

$$\begin{aligned} \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {X})&=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( \mathcal {X}\right) =\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( \mathcal {X}\right) =\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {X}\right) \text{ and } \\ \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}')&=\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( \mathcal {A}'\right) =\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( \mathcal {A}'\right) =\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\mathcal {X}}\left( \mathcal {A}'\right) , \end{aligned}$$

where $\mathcal {A}':=\mathcal {A}\cap \mathcal {X}$. On the other hand, by Lemma 4.10 (c) and Lemma 4.7

$$\begin{aligned} \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {X})=\textrm{pd}_{\mathcal {X}}(\mathcal {A}\cap \mathcal {X})=\textrm{pd}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X}). \end{aligned}$$

Now, since $\omega \subseteq \mathcal {B}\cap \mathcal {X}$, we have

$$\begin{aligned} {\text {coresdim}}_{\mathcal {B}\cap \mathcal {X}}\left( {\mathcal {A}\cap \mathcal {X}}\right) \le {\text {coresdim}}_{\omega }\left( {\mathcal {A}\cap \mathcal {X}}\right) \text{. } \end{aligned}$$

We claim that ${\text {coresdim}}_{\omega }\left( {\mathcal {A}\cap \mathcal {X}}\right) \le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})$. To show it, we can assume $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})<\infty$. By Proposition 4.23 (a), we can apply Lemma 4.16 to the pair $(\mathcal {A}\cap \mathcal {X},\omega )$, and since $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})<\infty$, it follows that:

$$\begin{aligned} \mathcal {A}\cap \mathcal {X}=\left\{ Z\in \mathcal {A}\cap \mathcal {X}\,|\,\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(Z)<\infty \right\} =\mathcal {A}\cap \mathcal {X}\cap \omega ^{\vee }\subseteq \omega ^{\vee } \end{aligned}$$

and $\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})=\textrm{coresdim}{}_{\omega }^{\,}\left( \mathcal {A}\cap \mathcal {X}\right) \text{. }$ Hence

$$\begin{aligned} \textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( \mathcal {A}'\right) \le \textrm{coresdim}{}_{\omega }^{\,}\left( \mathcal {A}'\right) \le \textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}')=\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( \mathcal {A}'\right) \end{aligned}$$

and thus, $\textrm{coresdim}{}_{\mathcal {B}\cap \mathcal {X}}^{\,}\left( \mathcal {A}\cap \mathcal {X}\right) =\textrm{coresdim}{}_{\omega }^{\,}\left( \mathcal {A}\cap \mathcal {X}\right) ;$ proving (a1).

(a2) $(\Rightarrow )$ Let $\textrm{pd}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X})<\infty .$ Then, by (a1), $\textrm{coresdim}{}_{\mathcal {B}}^{\mathcal {X}}\left( \mathcal {X}\right) <\infty$, and thus, $\mathcal {X}\subseteq (\mathcal {X},\mathcal {B}){}^{\vee }\subseteq (\mathcal {B}\cap \mathcal {X})^{\vee }$, since $\mathcal {X}$ is closed under extensions. Moreover, in the proof of (a1), we showed that $\mathcal {A}\cap \mathcal {X}\subseteq \omega ^{\vee }$. Then, by (a1) and Proposition 4.23 (e), we have $\textrm{pd}_{\mathcal {X}}(\omega )=\textrm{id}_{\omega }(\mathcal {X})=\textrm{id}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {X})=\textrm{pd}_{\mathcal {X}}(\mathcal {A}\cap \mathcal {X})\text{. }$

$(\Leftarrow )$ Let $\mathcal {A}\cap \mathcal {X}\subseteq \omega ^{\vee }$ and $\textrm{pd}_{\mathcal {X}}(\omega )=n<\infty$. Since $\omega \subseteq \mathcal {A}\cap \mathcal {X}\subseteq \omega ^{\vee }$, by (a1) and Lemma 4.3, we have $\textrm{pd}_{\mathcal {X}}(\omega )=\textrm{pd}_{\mathcal {X}}(\omega ^{\vee })\ge \textrm{pd}_{\mathcal {X}}(\mathcal {A}\cap \mathcal {X})=\textrm{pd}_{\mathcal {A}\cap \mathcal {X}}(\mathcal {A}\cap \mathcal {X}).$ $\square$

5 The class ${\text {Fac}}^{\mathcal {X}}_{n}(M)$

In the theory of infinitely generated tilting modules of finite projective dimension, Bazzoni [9] and Wei [25] presented the class ${\text {Gen}}_{n}(M)$ as a tool in the characterization of tilting modules. The goal of this section is to present a generalization of such class, and to review some basic properties, which will be used in [4] to characterize when a class $\mathcal {T}\subseteq \mathcal {C}$ is n-$\mathcal {X}$-tilting.

Definition 5.1

Let $\mathcal {C}$ be an abelian category, $n\ge 1$ and $\mathcal {X},\mathcal {T}\subseteq \mathcal {C}$. We denote by ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})$ the class of all the objects $C\in \mathcal {C}$ admitting an exact sequence

$$\begin{aligned} 0\rightarrow K\rightarrow T_{n}\xrightarrow {f_{n}}T_{n-1}\rightarrow \cdots \rightarrow T_2\xrightarrow {f_{2}}T_{1}\xrightarrow {f_{1}}C\rightarrow 0\text{, } \end{aligned}$$

with $\textrm{Ker}(f_{i})\in \mathcal {X}$ and $T_{i}\in \mathcal {T}\cap \mathcal {X}$ $\forall \, i\in [1,n]$.

We also define ${\text {Gen}}_{n}^{\mathcal {X}}(\mathcal {T}):={\text {Fac}}^{\mathcal {X}}_{n}(\mathcal {T}^{\oplus })$ and ${\text {gen}}_{n}^{\mathcal {X}}(\mathcal {T}):={\text {Fac}}^{\mathcal {X}}_{n}(\mathcal {T}^{\oplus _{<\infty }})$. For an object $T\in \mathcal {C}$, we define ${\text {Gen}}_{n}^{\mathcal {X}}(T):={\text {Gen}}_{n}^{\mathcal {X}}({\text {Add}}(T))$ and ${\text {gen}}_{n}^{\mathcal {X}}(T):={\text {gen}}_{n}^{\mathcal {X}}({\text {add}}(T))$. If $\mathcal {X}=\mathcal {C}$, we set ${\text {Fac}}_{n}(\mathcal {T}):={\text {Fac}}^{\mathcal {C}}(\mathcal {T})$, ${\text {Gen}}_{n}(\mathcal {T}):={\text {Gen}}_{n}^{\mathcal {C}}(\mathcal {T})$ and ${\text {gen}}_{n}(\mathcal {T}):={\text {gen}}_{n}^{\mathcal {C}}(\mathcal {T})$.

The following result is a generalization of [25, Prop. 3.7].

Proposition 5.2

Let $\mathcal {C}$ be an abelian category, $\mathcal {T}\subseteq \mathcal {C},$ $\mathcal {X}=\textrm{smd}(\mathcal {X})\subseteq \mathcal {C}$ be closed under extensions, ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ be closed under extensions and let ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}={\text {Fac}}_{n+1}^{\mathcal {X}}(\mathcal {T}) \cap \mathcal {X}.$ Then

$$\begin{aligned} {\text {Fac}}_{k}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T}))\cap \mathcal {X}= {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X} \forall k\ge 1 \end{aligned}$$

and ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ is closed by n-quotients in $\mathcal {X}$.

Proof

Since $\mathcal {T}\cap \mathcal {X}\subseteq {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$, it follows that:

$$\begin{aligned} \text{ Fac}_{k}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T}))\cap \mathcal {X}\supseteq {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}\text{. } \end{aligned}$$

Hence, we need to show that $\text{ Fac}_{k}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})) \cap \mathcal {X}\subseteq {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$. We proceed by induction on k.

Let $k=1$ and $X\in \text{ Fac}_{k}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T}))\cap \mathcal {X}.$ Then, there is an exact sequence

$$\begin{aligned} 0 \rightarrow K \xrightarrow {f} M' \rightarrow X \rightarrow 0 \text{ with } X,K\in \mathcal {X} \text{ and } M'\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}. \end{aligned}$$

Moreover, using that ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}\subseteq {\text {Fac}}_{1}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X},$ there is an exact sequence

$$\begin{aligned} 0 \rightarrow K_{1} \rightarrow M_{1} \xrightarrow {g} M' \rightarrow 0 \text{ with } M_{1}\in \mathcal {T}\cap \mathcal {X} \text{ and } K_{1}\in \mathcal {X}\text{. } \end{aligned}$$

By taking the pull-back of f and g and since $\mathcal {X}$ is closed under extensions, we get that $X\in {\text {Fac}}^{\mathcal {X}}_{1}(\mathcal {T})\cap \mathcal {X}$. Therefore, $\text{ Fac}_{1}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T}))\cap \mathcal {X}\subseteq {\text {Fac}}_{1}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}.$

Let $k>1$ and $M\in \text{ Fac}_{k+1}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T}))\cap \mathcal {X}.$ Then, there is an exact sequence

$$\begin{aligned} 0\rightarrow K\rightarrow C_{k+1}\overset{f_{k+1}}{\rightarrow }C_{k}\rightarrow ...\rightarrow C_{1}\overset{f_{1}}{\rightarrow }M\rightarrow 0, \end{aligned}$$

with $M\in \mathcal {X}$, $C_{i}\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ and $\textrm{Ker}(f_{i})\in \mathcal {X}$ $\forall \, i\in [1,k+1]$. Observe that

$$\begin{aligned} M_{1}:=\textrm{Ker}(f_{1})\in \text{ Fac}_{k}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T}))\cap \mathcal {X}\text{. } \end{aligned}$$

By inductive hypothesis, $M_{1}\in {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X},$ and thus, there is an exact sequence

$$\begin{aligned} 0\rightarrow M_{1}\xrightarrow {i}C_{1}\xrightarrow {f_1}M\rightarrow 0,\hbox { with }M_{1}\in {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}\hbox { and } C_{1}\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}. \end{aligned}$$

On the other hand, since ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}={\text {Fac}}_{n+1}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ and $C_{1}\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$, there is an exact sequence

$$\begin{aligned} 0\rightarrow C'\rightarrow T_{1}\overset{p}{\rightarrow }C_{1}\rightarrow 0, \end{aligned}$$

with $T_{1}\in \mathcal {T}\cap \mathcal {X}$ and $C'\in {\text {Fac}}^{\mathcal {X}}_{n}(\mathcal {T})\cap \mathcal {X}$. Now, by taking the pull-back of i and p, we get an exact sequence

$$\begin{aligned} 0 \rightarrow C' \rightarrow Y \xrightarrow {p'} M_{1}\rightarrow 0\text{; } \end{aligned}$$

and since $M_{1}\in {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})$, we have an exact sequence

$$\begin{aligned} 0 \rightarrow M'_{1} \rightarrow T'_{1} \xrightarrow {p''} M_{1} \rightarrow 0, \end{aligned}$$

with $T'_{1}\in \mathcal {T}\cap \mathcal {X}$ and $M'_{1}\in {\text {Fac}}_{k-1}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$. By taking the pull-back of $p'$ and $p''$,

we get an exact sequence

$$\begin{aligned} 0 \rightarrow C' X \rightarrow T'_{1} \rightarrow 0, \end{aligned}$$

with $C'\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ and

$$\begin{aligned} T'_{1}\in \mathcal {T}\cap \mathcal {X}\subseteq {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}\text{, } \end{aligned}$$

and thus, $X\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$. On the other hand, from the second pull-back, we also get an exact sequence

$$\begin{aligned} 0 \rightarrow M'_{1} \rightarrow X \rightarrow Y \rightarrow 0 \text{, } \end{aligned}$$

with $M'_{1}\in {\text {Fac}}_{k-1}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}\subseteq \text{ Fac } _{k-1}^{\mathcal {X}}\left( {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\right) \cap \mathcal {X}$ and $X\in {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$.

Hence, $Y\in \text{ Fac } _{k}^{\mathcal {X}}\left( {\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\right) \cap \mathcal {X}\subseteq {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ by inductive hypothesis. Finally, from the first pull-back, we get the exact sequence

$$\begin{aligned} 0 \rightarrow Y \rightarrow T_{1} \rightarrow M \rightarrow 0\text{, } \end{aligned}$$

where $Y\in {\text {Fac}}_{k}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ and $T_{1}\in \mathcal {T}\cap \mathcal {X}$. Therefore, $M\in {\text {Fac}}_{k+1}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X};$ proving the result.

Finally, the fact that ${\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$ is closed by n-quotients follows from the equality $\text{ Fac}_{n}^{\mathcal {X}}({\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})) \cap \mathcal {X}={\text {Fac}}_{n}^{\mathcal {X}}(\mathcal {T})\cap \mathcal {X}$. $\square$