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HIGHER IDEAL APPROXIMATION THEORY

JAVAD ASADOLLAHI AND SOMAYEH SADEGHI

Abstract. Let C be an n-cluster tilting subcategory of an exact category (A , E ), where n ≥ 1is an integer. It is proved by Jasso that if n > 1, then C although is no longer exact, buthas a nice structure known as n-exact structure. In this new structure conflations are calledadmissible n-exact sequences and are E -acyclic complexes with n+ 2 terms in C . Since theirintroduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely totheir links and applications to many research areas, many of them unexpected. On the otherhand, ideal approximation theory, that is a gentle generalization of the classical approximationtheory and deals with morphisms and ideals instead of objects and subcategories, is an activearea that has been the subject of several researches. Our aim in this paper is to introducethe so-called ‘ideal approximation theory’ into ‘higher homological algebra’. To this end,we introduce some important notions in approximation theory into the theory of n-exactcategories and prove some results. In particular, the higher version of the notions such asideal cotorsion pairs, phantom ideals, Salce’s Lemma and Wakamatsu’s Lemma for idealswill be introduced and studied. Our results motivate the definitions and show that n-exactcategories are the appropriate context for the study of ‘higher ideal approximation theory’.

Contents

1. Introduction 12. Higher homological algebra 33. Higher ideal cotorsion pairs 94. Higher phantom ideals 125. Salce’s Lemma 166. Special precovering ideals 197. Wakamatsu’s Lemma 21References 23

1. Introduction

One of the cornerstones of the modern representation theory is the Auslander’s theorem [Au]proving that there is a bijective correspondence between Morita equivalence classes of Artinalgebras Λ of finite representation type and Morita equivalence classes of algebras Γ satisfyinghomological conditions that global dimension of Γ is less than or equal to two and its dominantdimension [T] is greater or equal than two.

In a successful attempt to build up a higher version of Auslander’s correspondence and alsogeneralizing the classical theory of almost split sequences of Auslander-Reiten, Iyama [I1–I3]

2010 Mathematics Subject Classification. 18E05, 18G25, 18G15, 18E99, 16E30.Key words and phrases. Ideal approximation theory, n-exact categories, n-cluster tilting subcategories, higher

phantom morphisms, complete cotorsion pairs.1

http://arxiv.org/abs/2010.13203v1

2 JAVAD ASADOLLAHI AND SOMAYEH SADEGHI

introduced the notion of n-cluster tilting subcategories, where n is an integer greater or equalthan 1. Soon it is realized that these subcategories play a crucial role in the theory and so clustertilting subcategories became the subject of several researches.

In particular, study of the structure of such subcategories leads Jasso [Ja] to a higher versionof the classical homological algebra and as a consequence new notions such as n-abelian andn-exact categories were born. These notions provide appropriate higher version of the classicalabelian and exact categories, in the sense that 1-abelian and 1-exact categories are the usualabelian and exact categories. Instead of the usual kernels and cokernels, resp. inflations anddeflations, in these categories we have the notions of n-kernels and n-cokernels and the role ofshort exact sequences are played by exact complexes with n+ 2 terms.

Since their introduction, various authors studied further several properties of n-abelian andn-exact categories, also trying to introduce notions of classical homological algebra into thishigher version, which is now known as higher homological algebra. For some attempts in thisdirection see e.g. [Jor], [JK], [AHS] and [Kv].

Considering the fact that approximation theory is one of the efficient tools for studying com-plicated objects in a category, the aim of the present paper is to contribute to this project byintroducing some notions of ideal approximation theory into higher homological algebra.

The starting point of approximation theory is the discovery of the existence of injective en-velopes by Baer in 1940. Approximation theory, that is approximation of complicated objectsof a category by simpler objects in a specific subcategory, is essentially based on the notionsof preenvelopes and precovers. Recall that a class F of R-modules is precovering if for everyR-module M , there exists a morphism ϕ : F −→ M with F ∈ F such that the induced mor-phism HomR(F

′, F ) −→ HomR(F′,M) is surjective, for all F ′ ∈ F . Dually the notion of (pre-)

enveloping classes is defined. An important problem in this context is to investigate whethera class of modules is (pre-) enveloping or/and (pre)covering. For instance, flat cover conjec-ture, posed by Enochs [E] states that the class of of flat modules is a precovering (and hence acovering) class. This conjecture proved affirmatively after about 30 years [BEE].

Thanks to the researches of Auslander and his colleagues for artin algebras, in particular, thoseof Auslander, Reiten and Smalø, [AS1,AS2,AR2], approximation theory also plays a central rolein representation theory of algebras under the name of left approximations (pre-envelopings)and right approximations (precoverings). For a good account on approximation theory see themonograph [GT].

In classical approximation theory, approximation is done by objects from a subcategory. But anice generalization of the classical approximation theory, known as ideal approximation theory isstudied systematically in [FGHT] that give morphisms and ideals of categories equal importanceas objects and subcategories. In this theory, the role of the objects and subcategories in classicalapproximation theory is replaced by morphisms and ideals of the category.

An ideal of a category is an additive subfunctor of the Hom functor, which is closed undercompositions by morphisms from left and right. For instance, the phantom ideal and phantomcover in module category are studied extensively by Herzog in [H1]. Moreover, continuing the ideaof [FGHT], Fu and Herzog in [FH] studied ideal versions of some pillars of classical approximationtheory, such as cotorsion pairs, Salce’s Lemma and Wakamatsu’s Lemma. Very recently, Breazand Modoi [BM] studied ideal cotorsion pairs in extension closed subcategories of triangulatedcategories. For more research in this direction see [EAO] and [O].

Following these ideas, the general goal of this paper is to introduce ideal approximation theoryinto the higher homological algebra. Our results show that the correct context in which to carrythese arguments out is that of an n-cluster tilting subcategory of an exact category. By [Ja, §4]

HIGHER IDEAL APPROXIMATION THEORY 3

we know that these subcategories are n-exact, i.e. with ‘admissible’ sequences with n+ 2 termsas conflations. Using this structure, a ‘higher ideal approximation theory’ is developed in thispaper. We state and prove some foundational results in this subject to motivate the theory.

The paper is structured as follows. In section 3, we introduce and study a higher versionof the notion of ideal cotorsion pairs [FGHT, Definition 12]. For a good account for cotorsionpairs in abelian categories see [BR, §V.3]. In section 4, phantom morphisms in n-cluster tiltingsubcategories of exact categories will be studied. It is a higher version of the notion of phantommorphisms studied by Herzog [H1,H2]. As an example, we study n-pure phantom morphisms.We recall that phantom morphisms were introduced by McGibbon [Mc] for use in topology tostudy maps between CW-complexes. Neeman [N] introduced this concept into the context oftriangulated categories. The theory also was developed in the stable category of a finite groupring in a series of works of Benson and Gnacadja [BG1,BG2,G].

Higher version of Salce’s Lemma for n-ideal cotorsion pairs is treated in section 5. Salce’sLemma [S] is one of the main theorems in the classical approximation theory. It relates thenotions of (special) precoverings, (special) preenvelopings and cotorsion pairs. By introducingan interesting exact structure on the morphism category of an exact category, called ME-exactstructure, an ideal version of Salce’s Lemma is proved in [FH, Theorem 6.3].

The main purpose of section 6 is to study connections between special precovering idealsand n-phantom morphisms in an n-cluster tilting subcategory. In particular, we show that everyspecial precovering ideal, under some conditions, can be represented as an ideal of n-F -phantommorphisms, for some bifunctor F of Extn.

In the last section of the paper, we state and prove an ideal version of the Wakamatsu’sLemma [W] in higher homological algebra. Note that an ideal version of Wakamatsu’s Lemmain an exact category is proved in [FH, Theorem 10.3]. Moreover a version of Wakamatsu’sLemma for (n+ 2)-angulated categories is formulated and proved in [Jor, §3].

2. Higher homological algebra

In this section we collect some basic facts and backgrounds on higher homological algebrawe need throughout the paper. We are mainly work in an exact category (A , E ), where Ais an additive category and E is a class of composable pairs (also called kernel-cokernel pairs)of morphisms in A which is closed under isomorphisms and satisfies axioms of Definition 2.1of [B]. The composable pair (i, p) in E is denoted by A′

A։A′′, while i : A′ → A is called anE -admissible monic and p : A→ A′′ is called an E -admissible epic. When the exact structure Eis clear from the context we just say admissible instead of E -admissible. Admissible pairs, admis-sible monics and admissible epics also called conflations, inflations and deflations, respectively,see [Q] and [K]. For definitions and properties of exact categories we refer to [B].

2.1. n-exact categories. Let n ≥ 1 be a fixed integer. The notion of n-exact categories isdefined by Jasso in [Ja, §4] as a natural generalization of exact categories. Its idea appearednaturally in a series of papers by Iyama, studying a higher version of the Auslander’s corre-spondence [I1–I3] and then axiomatized by Jasso [Ja]. Let us recall the definitions and basicproperties.

Let C be an additive category. Let f0 : X0 −→ X1 be a morphism in C . An n-cokernel off0 is a sequence

X1 f1

−→ X2 −→ · · · −→ Xn fn

−→ Xn+1


of morphisms in C such that for every X ∈ C the induced sequence

0 −→ C (Xn+1, X)fn

∗−→ · · ·f1∗−→ C (X1, X)

f0∗−→ C (X0, X)

of abelian groups is exact. Here and throughout we write C (−,−) instead of HomC (−,−).We denote the n-cokernel of f0 by (f1, f2, · · · , fn). The notion of n-kernel of a morphismfn : Xn −→ Xn+1 is defined similarly, or rather dually.

A sequence X0 f0

−→ X1 −→ · · · −→ Xn fn

−→ Xn+1 of objects and morphisms in C , is calledn-exact [Ja, Definitions 2.2, 2.4] if (f0, f1, · · · , fn−1) is an n-kernel of fn and (f1, f2, · · · , fn) isan n-cokernel of f0. An n-exact sequence like the above one, usually will be denoted by

X0 f0

X1 f1

−→ X2 −→ · · · −→ Xnfn

։ Xn+1.

Consider the complex

X : X0 f0

−→ X1 −→ · · · −→ Xn−1 fn−1

−→ Xn

and morphism g0 : X0 −→ Y 0 in C . An n-pushout diagram of X along g0 is a morphism

(2.1)

X0 X1 · · · Xn

Y 0 Y 1 · · · Y n

g0 g1

of complexes such that in the mapping cone C = cone(g)

X0 d−1

C−→ X1 ⊕ Y 0 d0C−→ · · ·

dn−2

C−→ Xn ⊕ Y n−1 dn−1

C−→ Y n

the sequence (d0C, d1

C, · · · , dn−1

C) is an n-cokernel of d−1

C. The maps in the mapping cone are

defined as usual. For more details and properties of the n-pushout, and also n-pullback diagrams,see Subsection 2.3 of [Ja].

A morphism f : X −→ Y of n-exact sequences is called a weak isomorphism if fk and fk+1

are isomorphisms for some k ∈ 0, 1, · · · , n+ 1, where we set n+ 2 := 0.An n-exact structure on C is a class X of n-exact sequences

η : X0 X1 · · · Xn Xn+1f0 fn

in C , called X -admissible n-exact sequences, that is closed under weak isomorphisms and satis-fies the following axioms. f0, resp. fn, is then called an X -admissible monomorphism, resp. anX -admissible epimorphism. We just write admissible instead of X -admissible, when the classX is clear from the context.

(E0) The sequence 0 0→ · · · → 0 ։ 0 is an admissible n-exact sequence.(E1) The class of admissible monomorphisms is closed under composition.(E2) The n-pushout of an admissible n-exact sequence (d0X , d

1X , ..., d

nX) along morphism f :

X0 → Y 0 exists and d0Y is an admissible monomorphism.

(2.2)

X0 X1 · · · Xn−1 Xn Xn+1

Y 0 Y 1 · · · Y n−1 Y n

d0X

f

d1X

dn−1

Xdn

X

d0Y

dn−1

Y

(E1op) The class of admissible epimorphisms is closed under composition.


(E2op) The n-pullback of an admissible n-exact sequence (d0X , d1X , ..., d

nX) along morphism g :

Y n+1 → Xn+1 exists and dnY is an admissible epimorphism.

(2.3)

Y 1 Y 2 · · · Y n Y n+1

X0 X1 X2 · · · Xn Xn+1

d1Y

d2Y

dn

Y

g

d0X

d1X

d2X

dn

X

An n-exact category is a pair (C ,X ) where C is an additive category and X is an n-exactstructure on C .

Let (C ,X ) be an n-exact category. By [Ja, Proposition 4.8], the n-pushout diagram 2.2 canbe completed to the commutative diagram

(2.4)

X0 X1 · · · Xn Xn+1

Y 0 Y 1 · · · Y n Xn+1

d0X

f

d1X

dn

X

d0Y

dn−1

Ydn

Y

where the lower row also is an admissible n-exact sequence. Similar result holds true for then-pullback diagrams 2.3. For more details and properties of n-exact categories see [Ja, Section4].

The most known examples of n-exact categories are n-cluster tilting subcategories of exactcategories [Ja, Theorem 4.14], introduced by Iyama in a series of papers, see e.g. [I1, I2]. Let usrecall their definitions and some basic properties we need later in the paper.

2.2. Let (A , E ) be an exact category. A morphism A −→ A′ in A is called proper if it admitsa factorization A ։ K A′, such that A ։ K is an E -admissible epic and K A′ is anE -admissible monic. A sequence

· · · Ai−1 Ai Ai+1 · · ·

Ki−1 Ki

of proper morphisms is called an E -acyclic complex if for all i ∈ Z the sequenceKi−1 Ai

։ Ki

is an E -admissible exact sequence. An E -acyclic complex such as

δ : A′ A1 → A2 → · · · → Ai

։ A

is called an E -acyclic complex of length i with end-terms A and A′.

2.3. Let (A , E ) be an exact category. Let

δ : A′ B1 → B2 → · · · → Bi

։ A

be an E -acyclic complex of length i with end-terms A and A′. A morphism between δ and δ′,denoted by δ −→ δ′, is a commutative diagram

δ : A′ A1 A2 · · · Ai A

δ′ : A′ B1 B2 · · · Bi A


We say that δ and δ′ are Yoneda equivalent if there is a sequence of morphisms

δ0 = δ −→ δ1 ←− δ2 −→ · · · −→ δt−1 ←− δt = δ′

of E -acyclic complexes of length i with end-terms A and A′. It is known that the collection ofYoneda equivalence classes of all E -acyclic complexes of length i with the same end-terms form a(big) abelian group under Baer sum. For a detailed explanation, see Subsection 6.2 of [FS]. For

objects A and A′ in A , this group is denoted by ExtiE (A,A′). Note that in general ExtiE (A,A

′)is not a set [FS, Remark 6.2 and Remark 6.20]. But if A is a small category or contains enough

E -projective or enough E -injective objects, then ExtiE (A,A′) will be a set [FS, Remark 6.44].

Definition 2.4. ( [Ja, Definition 4.13]) Let (A , E ) be a small exact category. A subcategory Cof A is called an n-cluster tilting subcategory if it satisfies the following conditions.

(i) For every object A ∈ A , there exists an admissible monomorphism A C, which isalso a left C -approximation of A.

(ii) For every object A ∈ A , there exists an admissible epimorphism C′։ A, which is also

a right C -approximation of A.(iii) There exists equalities C⊥n = C = ⊥nC , where

C⊥n = A ∈ A : ExtiE (C,A) = 0 for all C ∈ C and all 1 ≤ i ≤ n− 1,

⊥nC = A ∈ A : ExtiE (A,C) = 0 for all C ∈ C and all 1 ≤ i ≤ n− 1.

Note that if n = 1, C = A is the unique 1-cluster tilting subcategory of A . Followingtheorem provides a source of examples of n-exact categories.

Theorem 2.5. ( [Ja, Theorem 4.14]) Let C be an n-cluster tilting subcategory of the exactcategory (A , E ). Set

X = C : C0 C1 → · · · → Cn

։ Cn+1 | C is E -acyclic and Ci ∈ C , ∀i ∈ 0, 1, . . . , n+ 1.

Then (C ,X ) is an n-exact category.

Remark 2.6. By the above theorem any n-cluster tilting subcategory of an exact category isn-exact. In particular, n-pullbacks and n-pushouts exist in C . Let us explain the constructionof n-pullback diagrams. n-pushout diagrams are constructed similarly, see e.g. [JK, Proposition2.18]. Consider the following n-pullback diagram

X0 Y 1 Y 2 · · · Y n Y n+1

η : X0 X1 X2 · · · Xn Xn+1

d0Y

d1Y

d2Y

dn

Y

g

d0X

d1X

d2X

dn

X

obtained from n-exact sequence η along g. Each square

Y i Y i+1

X i X i+1

gi+1

di

X

for i ∈ 1, 2, . . . , n, is constructed by taking usual pullback diagram of the morphism (gi+1, diX)in the ambient exact category (A , E ) and then taking a right C -approximation of the outcome,


that exists by definition of n-cluster tilting subcategories. In other words, as it can be seen fromthe following diagram

Y i Y i+1

Zi

X i X i+1

gi

di

Y

γi

gi+1

αi

βi

di

X

(Zi, αi, βi) is a pullback of (gi+1, diX) and γi : Y i −→ Zi is a right C -approximation of Zi.

Remark 2.7. Let (A , E ) be an exact category, C be an n-cluster tilting subcategory of A andX denote the n-exact structure of C . For each C,C′ ∈ C , set

YextnC (C,C′) := [δ] | δ : C′ C1 → C2 → · · · → Cn

։ C is an E -acyclic complex in C ,

where [δ] denotes the Yoneda equivalence class of δ similar to what is defined in 2.3. One canfollow the same argument as in Subsection 6.2 of [FS] to deduce that YextnC (C,C′), based onn-pullbacks and n-pushouts in C , could be equipped with a Baer sum making it into a group.In fact, YextnC (−,−) becomes a bifunctor on C , see also [HS, Theorem IV.9.1] for a proof in anabelian category.

Moreover, for a morphism f : X ′ −→ Y ′ in C , similar argument as in [FS, pp. 197-198],implies that the induced morphism

f∗ := YextnC (X, f) : YextnC (X,X ′) −→ YextnC (X,Y ′)

for every X ∈ C , can be computed in terms of n-pushout diagrams. That is for every element

η : X ′ X1 → X2 → · · · → Xn

։ X

of YextnC (X,X ′), f∗(η) is the n-exact sequence obtained by extending n-pushout of η along f ,that is

η : X ′ X1 · · · Xn−1 Xn X

η′ : Y ′ Y 1 · · · Y n−1 Y n X

f

For a similar discussion in the case where A = mod-Λ is an abelian category see [Fe, Remark3.8(b) and Lemma 3.13]. Similarly the morphism

f∗ := YextnC (f,X) : YextnC (Y ′, X) −→ YextnC (X ′, X)

for every X ∈ C is defined using n-pullback diagrams.

Remark 2.8. Let R be a commutative local ring and A be an abelian R-category with enoughprojective objects. Let B be a resolving subcategory of A , that is, B contains projective objectsof A and is closed under extensions and kernel of surjections. Moreover assume that the stablecategory B is a dualising R-variety [AR1]. Since B is a full extension closed subcategory ofabelian category A , it is an exact category. Let E denote the exact structure of B. Now letC be an n-cluster tilting subcategory of B. So C is an n-exact category. Let X denote then-exact structure of C . By Section 2.5.1 of [I2] higher-dimensional Auslander-Reiten theory


exists for subcategory C of B. Hence, by [I1, A.1. Proposition], if X0, Xn+1 ∈ C , every elementin ExtnE (X

n+1, X0) is Yoneda equivalent to an admissible n-exact sequence

η : X0 X1 → · · · → Xn

։ Xn+1

in C , i.e. all middle terms lie in C . Therefore, in this case, the bifunctor YextnC is the same asthe bifunctor ExtnE restricted to C . See [Fe, Remark 2.9] for the special case A = B = mod-Λ,where Λ is a finite dimensional artin algebra.

Notation 2.9. In view of the above remark, under some mild conditions, bifunctor YextnC (−,−)

is nothing but the usual bifunctor ExtiE (−,−), recalled in 2.3, restricted to C . Because of thisfact, from now on, YextnC (−,−) will be denoted by ExtnX (−,−).

Example 2.10. As an example of the above discussion, set A = mod-Λ and let B = Gprj-Λ bethe full subcategory of A consisting of all Gorenstein-projective Λ-modules. By [AHS, Theorem3.16], if C is an n-cluster tilting subcategory of mod-Λ with enough injectives, then C ∩Gprj-Λis an n-cluster tilting subcategory with enough injectives of Gprj-Λ, see also [Kv, Theorem 7.3].Hence in this case, ExtnC∩Gprj-Λ is the same as the functor ExtnGprj-Λ on C ∩Gprj-Λ.

Let (A , E ) be an exact category and f : X → A and g : B → Y be morphisms in A . Thenit is known [FS, Remark 6.7] that

ExtnE (f, g) = ExtnE (X, g)ExtnE (f,B) = ExtnE (f, Y )ExtnE (A, g).

Here we show that the same result holds true for ExtnX , where (C ,X ) is an n-cluster tiltingsubcategory of (A , E ).

Proposition 2.11. Let C be an n-cluster tilting subcategory of the exact category (A , E ). Letf : X → A and g : A′ → Y be morphisms in C . Then

ExtnX (f, g) = ExtnX (X, g)ExtnX (f,A′) = ExtnX (f, Y )ExtnX (A, g),

that is, the following diagram of abelian groups is commutative

ExtnX (A,A′)Extn

X(f,A′)

//

ExtnX

(A,g)

ExtnX (X,A′)

ExtnX

(X,g)

ExtnX (A, Y )Extn

X(f,Y )

// ExtnX (X,Y ).

Proof. Suppose that η : A′ X1 → · · · → Xn

։ A ∈ ExtnX (A,A′) is given. By takingn-pullback of η along f and n-pushout of it along g, we get the following diagram.

A′ U1 · · · Un X

A′ X1 · · · Xn A

Y V 1 · · · V n A

f

g


Now by taking n-pushout of A′ → U1 → · · · → Un along g and applying Proposition 4.9 of [Ja],we get the following diagram

A′ U1 · · · Un X

γ : Y W 1 · · · Wn X

η : A′ X1 · · · Xn A

Y V 1 · · · V n A

g

f

f

g

Note that by (dual of) the statement (iv)⇒ (i) of Proposition 4.8 of [Ja], diagram

Y W 1 · · · Wn X

Y V 1 · · · V n A

f

is an n-pullback diagram. Hence the admissible n-exact sequence γ can be described as

ExtnX (X, g)ExtnX (f,A′)(η) = γ = ExtnX (f, Y )ExtnX (A, g)(η).

We end this section by the definition of n-proper classes that will be used later.

2.12. n-proper classes. Let (C ,X ) be an n-cluster tilting subcategory of an exact category(A , E ). A class F of E -acyclic complexes of length n in C is called an n-proper class if itcontains all split (contractible) E -acyclic complexes, is closed under isomorphisms and finitedirect sums and is closed under n-pullbacks and n-pushouts along any other morphisms in A .

It is easy to see that any n-proper class of E -acyclic complexes of length n gives rise to anadditive subfunctor ExtnF of ExtnX . On the other hand, any additive subfunctor of ExtnX inducesan n-proper class of E -acyclic complexes of length n. For a similar discussion in the classicalcase n = 1, see [DRSS, 1.2] for exact categories and [ASo] for abelian categories.

3. Higher ideal cotorsion pairs

In this section we introduce and study a higher version of the notion of ideal cotorsion pairs[FGHT, Definition 12]. Our setting for this purpose, and throughout the paper, is n-clustertilting subcategories of exact categories. Let us begin by recalling the notion of an ideal of anadditive category.

3.1. Ideal approximation theory. Let A be an additive category. A two sided ideal I ofA is a subfunctor

I (−,−) : A op ×A −→ A b

of the bifunctor A (−,−) that associates to every pair A and A′ of objects in A a subgroupI (A,A′) ⊆ A (A,A′) such that

(i) If f ∈ I (A,A′) and g ∈ A (A′, C), then gf ∈ I (A,C),(ii) If f ∈ I (A,A′) and g ∈ A (D,A), then fg ∈ I (D,A′).


Let I be an ideal of A and A ∈ A be an object of A . An I -precover of A is a morphism

Cϕ−→ A in I such that any other morphism C′ ϕ′

−→ A in I factors through ϕ, i.e. thereexists a morphism ψ : C′ −→ C such that ϕψ = ϕ′. I is called a precovering ideal if everyobject A ∈ A admits an I -precover. The notions of I -preenvelope and preenveloping idealsare defined dually. See [H1, Section 3] for the special case in module category and [FGHT] forthe general definitions and properties in an exact category.

We say that an object A of A is in I if the identity morphism 1A is in I (A,A).

Notation 3.2. Let M be a collection of morphisms in C . We define the right and left orthog-onals of M , respectively, as follows

M⊥ := g : ExtnX (m, g) = 0 for all m ∈M ,

⊥M := f : ExtnX (f,m) = 0 for all m ∈M .

Note that maybe the better notations for the above orthogonals are M⊥n and ⊥nM . Sincen is fixed throughout, we drop it, both for the ease of notation and also to avoid confusion withthe orthogonal notations used in the definition of n-cluster tilting subcategories.

Following proposition is a higher version of [FGHT, Proposition 9]. In this theorem and alsothroughout the section, for the ease of notation, we shall use Extn instead of ExtnX .

Proposition 3.3. Let M be a collection of morphisms in C . Then both M⊥ and ⊥M are idealsof C .

Proof. We just prove that M⊥ is an ideal, the proof of the other one follows similarly. Letg1, g2 : A′ → Y be morphisms in M⊥. We should show that not only g1 + g2 ∈ M⊥ but alsofor all morphisms h : Y → Y ′ and k : Y ′′ → A′ in C , hg1, g1k ∈ M⊥. These all follows usingthe equalities of Proposition 2.11. To see this, let f : X → A be a morphism in M and considerthe following equalities

Extn(f, g1 + g2) = Extn(X, g1 + g2)Extn(f,A′)

= [Extn(X, g1) + Extn(X, g2)]Extn(f,A′)

= Extn(X, g1)Extn(f,A′) + Extn(X, g2)Ext

n(f,A′)

= Extn(f, g1) + Extn(f, g2) = 0.

to deduce that g1 + g2 ∈M⊥. To prove hg1 ∈M⊥, by another use of Proposition 2.11 we havethe following equalities

Extn(f, hg1) = Extn(X,hg1)Extn(f,A′)

= Extn(X,h)Extn(X, g1)Extn(f,A′)

= Extn(X,h)Extn(f, g1) = 0,

which, in turn, implies that hg1 ∈M⊥. The statement g1k ∈M⊥ follows similarly. Hence theproof is complete.

The above proposition, in particular, implies that I ⊥ and ⊥I are ideals of C , whenever Iis so.


Definition 3.4. Let I and J be ideals of C . A pair (I ,J ) is called an n-orthogonal pair ofideals if for every f ∈ I and every g ∈J , the pair (f, g) is an n-orthogonal pair of morphisms,that is, the morphism

Extn(f, g) : Extn(A,B) −→ Extn(X,Y )

of abelian groups is zero.

For instance, the pair (1A, g) is an n-orthogonal pair of morphisms if and only if Extn(A, g) =0. We infer this using the following equalities

Extn(1A, g) = Extn(A, g)Extn(1A, B) = Extn(A, g),

proved in Proposition 2.11. Similar argument implies that the pair (1A, 1B) is n-orthogonal pairof morphisms if and only if Extn(A,B) = 0.

Definition 3.5. The n-orthogonal pair (I ,J ) of ideals in C is called an n-ideal cotorsion pairif I = ⊥J and J = I ⊥.

Definition 3.6. (see [FGHT, page 762]) Let I be an ideal of C and A ∈ C be an arbitraryobject. A morphism i : Xn → A in I is called a special I -precover of A if it obtained asthe rightmost morphism in an n-pushout of an admissible n-exact sequence η along a morphismj : Y−→A′ ∈ I ⊥. It is depicted by the following diagram

η : Y Y 1 · · · Y n A

η′ : A′ X1 · · · Xn A.

j

i

The ideal I is called a special precovering ideal if every object A ∈ C has a special I -precover.The notions of special I -preenvelopes and special preenveloping ideals are defined dually.

Remark 3.7. Let i : Xn → A be a special I -precover of A. Then it, necessarily, is an I -precover of A. To see this, let i′ : X ′ → A be a morphism in I . We show that it factors throughi. By the above diagram, η′ = Extn(A, j)(η). By assumption j ∈ I ⊥, so the n-pullback of η′

along i′ is

Extn(i′, A′)(η′) = Extn(i′, A′)Extn(A, j)(η) = Extn(i′, j)(η) = 0,

Hence we have the desired factorization

X ′

Xn A.

i′

i

Definition 3.8. An n-ideal cotorsion pair (I ,J ) is called complete if every object in C admitsa special I -precover and a special J -preenvelope.

In the following theorem we present a source of examples of n-ideal cotorsion pairs.

Theorem 3.9. Let I be a special precovering ideal of C . Then the n-orthogonal pair of ideals(I ,I ⊥) is an n-ideal cotorsion pair.

Proof. By definition, we just need to show that I = ⊥(I ⊥). The inclusion I ⊆ ⊥(I ⊥)is trivial. So assume that i′ : X ′ → A is in ⊥(I ⊥). We show that i′ ∈ I . Since I is a


special precovering ideal, there is a special I -precover i : Xn → A of A, which is the rightmostmorphism in an admissible n-exact sequence

η′ : A′ X1 → · · · → Xn i

։ A

that is obtained from an n-pushout diagram of an admissible n-exact sequence η along amorphism j : Y −→ A′ in I ⊥. That is η′ = Extn(A, j)(η), for some admissible n-exactsequence η. Now by taking the n-pullback of η′ along i′ and using the assumption thati′ : X ′ −→ A ∈ ⊥(I ⊥), we have

Extn(i′, A′)(η′) = Extn(i′, A′)Extn(A, j)(η) = Extn(i′, j)(η) = 0.

Hence we get the factorization

X ′

Xn A.

g i′

i

which implies that i′ = ig ∈ I . The proof is hence complete.

4. Higher phantom ideals

In this section, we introduce and study the notion of phantom morphisms in n-cluster tiltingsubcategories of exact categories. It will provide a higher version of the notion of phantommorphisms introduced and studied by Herzog [H1,H2].

Setup 4.1. Throughout the section, C is an n-cluster tilting subcategory of an exact category(A , E ) with n-exact structure X . Moreover, F denotes an additive subfunctor of ExtnX . As itis explained in 2.12, every subfunctor F of ExtnX induces an n-proper subclass of X , that alsowill be denoted by F . The admissible n-exact sequences in F are called F -admissible n-exactsequences. Throughout we denote ExtnX by Extn, for simplicity.

Let us begin with the following definition.

Definition 4.2. With the above notations, a morphism ϕ in C is called an n-F -phantommorphism if the n-pullback of every X -admissible n-exact sequence along ϕ is an F -admissiblen-exact sequence. In other words, ϕ : X −→ A in C is an n-F -phantom morphism if for everyobject A′ in C , the morphism

Extn(ϕ,A′) : Extn(A,A′) −→ Extn(X,A′)

of abelian groups takes values in the subgroup F (X,A′). We denote the collection of all n-F -phantom morphisms by Φ(F ). Note that it is easy to see that Φ(F ) forms an ideal ofC .

To see an example of phantom morphisms we need to recall following definitions/notationsfrom [FGHT, page 759].

Definition 4.3. A morphism f : X −→ A in C is called F -projective if for every object Bin C , F (f,B) = 0. In other words, f : X → A in C is F -projective if the n-pullback of anyF -admissible n-exact sequence along f is contractible. An object A in C is called F -projectiveif the identity morphism is an F -projective morphism. The ideal of F -projective morphisms isdenoted by F -proj. The notions of F -injective morphisms and F -injective objects are defineddually. The ideal of F -injective morphisms is denoted by F -inj.


Example 4.4. (n-pure phantom morphisms) In this example, we assume that (A , E ) is anexact category with arbitrary direct sums. An object C of A is called a compact object [EN,Definition 2.3] if any morphism from C to a nonempty coproduct

⊕i∈I Ai factors through a

sub-coproduct⊕

j∈J Aj , where J ⊆ I is a finite subset. A is called compactly generated if for

every A ∈ A there is an admissible epimorphism h :⊕

i∈I Ci −→ A such that Ci is compact foreach i ∈ I.

Let (A , E ) be a compactly generated exact category and (C ,X ) be an n-cluster tiltingsubcategory of A containing all compact objects. An admissible n-exact sequence

A′ X1 → · · · → Xn

։ A

in C is called pure n-exact if for every compact object C, the induced sequence

0 −→ C (C,A′) −→ C (C,X1) −→ C (C,X2) −→ · · · −→ C (C,Xn) −→ C (C,A) −→ 0

of abelian groups is acyclic [EN, Definition 2.4].We claim that the collection of all pure n-exact sequences forms an n-proper class of X . To

see this, note that it is obviously closed under isomorphisms and also contains all contractiblen-exact sequences. It also is closed under n-pushout by any other morphism. We show that itis closed under n-pullback along any other morphism. Let

η : A′ X1 → · · · → Xn

։ A

be a pure n-exact sequence and consider its n-pullback along a morphism A′′ −→ A

δ : A′ Y 1 Y 2 · · · Y n A′′

η : A′ X1 X2 · · · Xn A

g

dn

Y

h

dn

X

Now let u : C −→ A′′ be a morphism from a compact object C to A′′. Since η is pure n-exact,hu factors through dnX . Now the construction of the n-pullback diagram explained in Remark2.6, implies easily that u factors also through dnY . Hence the claim follows.

So, by 2.12, it gives rise to a subfunctor of Extn, that we will denote by n-Pext. By ann-pure phantom morphism we mean an n-Pext-phantom morphism. The collection of all n-purephantom morphisms will be denoted by Φ(n-Pext). The n-Pext-injective objects will be calledn-pure injective objects. The pair of ideals (Φ(n-Pext), n-Pext-inj) is an n-orthogonal pair. Tosee this, let f be an n-pure phantom and g be an n-pure injective morphism. Then n-pullbackof admissible n-exact sequence η along f is a pure admissible n-exact sequence and n-pushoutof this sequence along g is contractible. Hence Extn(f, g) = 0.

Notation 4.5. Let I be an ideal of C . The collection of all admissible n-exact sequences thatare obtained from n-pullbacks of admissible n-exact sequences along morphisms in I is denotedby PB(I ).

Proposition 4.6. Let I be an ideal of C . Then PB(I ) is an n-proper subclass of X .

Proof. By 2.12, we have to show that PB(I ) contains contractible n-exact sequences, is closedunder isomorphisms, closed under direct sums, n-pullbacks and n-pushouts. Obviously it isclosed under isomorphisms. Let η : A′ → X1 → · · · → Xn → A be a contractible n-exactsequence. Then n-pullback of η along morphism 0 : A → A is just η, so η ∈ PB(I )(A,A′).Now let η and η′ be in PB(I ) such that arise as n-pullbacks of admissible n-exact sequences γand γ′ along morphisms i and i′, respectively. Then direct sum of η and η′ is the n-pullback of


γ⊕γ′ along morphism

[i 00 i′

], and so PB(I ) is closed under direct sums. To see that PB(I ) is

closed under n-pullbacks, let η : A′ → X1 → · · · → Xn → A be an admissible n-exact sequencethat is obtained from the n-pullback of γ along morphism i : A → A′′. The n-pullback of ηalong morphism f : X → A is an n-pullback of γ along if which is in I . Finally we show thatPB(I ) is closed under n-pushouts. Let η : A′ → X1 → · · · → Xn → A be the n-pullback ofγ : A′ → Y 1 → · · · → Y n → A′′ along i : A → A′′. By Proposition 2.11, the n-pushout of ηalong g : A′ → Z is

ExtnX (A, g)(η) = ExtnX (A, g)ExtnX (i, B)(γ) = ExtnX (i, Z)ExtnX (A′′, g)(γ),

which is the n-pullback of admissible n-exact sequence ExtnX (A′′, g)(γ) along i.

Remark 4.7. The above proposition, in view of 2.12, implies that PB(I ) induces an additivesubfunctor of ExtnX , we denote it by the same notation PB(I ).

Lemma 4.8. Let I be an ideal of C . Then I ⊥ = F -inj, where F = PB(I ).

Proof. This follows from the fact that a morphism j is PB(I )-injective if and only if for everyi ∈ I , Extn(i, j) = 0 and this holds true if and only if j ∈ I ⊥.

Proposition 4.9. Let I be a special precovering ideal. Then I is the ideal of PB(I )-phantommorphisms, that is I = Φ(PB(I )).

Proof. It is clear that every morphism in I is an n-PB(I )-phantom morphism. Now supposethat i is an n-PB(I )-phantom morphism, so every n-pullback of an admissible n-exact sequencealong i is a PB(I )-admissible n-exact sequence. If j ∈ I ⊥, then by Lemma 4.8, j is PB(I )-injective, so Extn(i, j) = 0 and i ∈ ⊥(I ⊥). Since by Theorem 3.9, I = ⊥(I ⊥), we concludethat i ∈ I .

Corollary 4.10. Let I be a special precovering ideal of C . Set F = PB(I ). Then (Φ(F ),F -inj)is an n-ideal cotorsion pair.

Proof. By Proposition 4.9, I = Φ(F ). Also by Lemma 4.8, I ⊥ = F -inj. Now the resultfollows by Theorem 3.9.

Following lemma is a special case of the Obscure axiom [Ja, Proposition 4.11].

Lemma 4.11. Let i0 : A′ −→ Y 1 be an F -admissible monomorphism that factors through anX -admissible monomorphism i : A′ −→ X1

A′ X1

Y 1

i

i0 g

Then i is an F -admissible monomorphism.

Proof. Since i is an X -admissible monomorphism, there exists an X -admissible n-exact se-quence

A′i X1 −→ X2 −→ · · · −→ Xn

։ A.

Similarly, for i0 we have F -admissible n-exact sequence

A′i0 Y 1 −→ Y 2 −→ · · · −→ Y n

։ A.


Using the morphism g : X1 → Y 1 and factorization property of the weak cokernels we canconstruct the commutative diagram

η : A′ X1 X2 · · · Xn A

δ : A′ Y 1 Y 2 · · · Y n A′′

i

g h

i0

By the dual of (iv) ⇒ (i) of Proposition 4.8 of [Ja], this diagram is an n-pullback diagram.Now since δ is an F -admissible n-exact sequence and F is an n-proper class, η also should beF -admissible n-exact sequence and so i is an F -admissible monomorphism.

We say that an additive subfunctor F ⊆ Extn has enough injective morphisms if for everyobject A′ ∈ C there exists an F -admissible n-exact sequence

η : A′ e→ X1 → · · · → Xn → A

where e : A′ → X1 is an F -injective morphism. The subfunctor F ⊆ Extn has enough specialinjective morphism if for every object A′ ∈ C , there exists an F -admissible sequence η asabove that obtains from the n-pullback of an admissible n-exact sequence along an F -phantommorphism.

Proposition 4.12. Let F ⊆ Extn be an additive subfunctor admitting enough injective mor-phisms. Then Φ(F ) = ⊥(F -inj).

Proof. First note that the pair of ideals (Φ(F ),F -inj) is an n-orthogonal pair. To see this,let f be an n-F -phantom and g be an F -injective morphism. Then n-pullback of admissiblen-exact sequence η along f is an F -admissible n-exact sequence and n-pushout of this sequencealong g is trivial. Hence Extn(f, g) = 0. This, in particular, implies that Φ(F ) ⊆ ⊥(F -inj).Now we show that the converse inclusion ⊥(F -inj) ⊆ Φ(F ) also holds true. Assume thatf : X → A ∈ ⊥(F -inj). We show that f is an n-F -phantom morphism, that is, the n-pullbackof any admissible n-exact sequence

η : A′ X1 → · · · → Xn

։ A

along f is an F -admissible sequence.Since F has enough injective morphisms, there exists F -injective F -admissible monomor-

phism e : A′ → Y . By Proposition 4.9 of [Ja], we get the following diagram

η′ : A′ U1 · · · Un X

γ : Y W 1 · · · Wn X

η : A′ X1 · · · Xn A

Y V 1 · · · V n A

e

i

f

f

e

The admissible sequence γ is contractible, since Extn(f, e) = 0. So the F -admissible monomor-phism e factors as e = gi for some g : U1 → Y . By Lemma 4.11, the morphism i is anF -admissible monomorphism. Thus η′, n-pullback of admissible n-exact sequence η along f , isan F -admissible sequence.


Corollary 4.13. Let I be an ideal of C such that n-ideal cotorsion pair (I ,I ⊥) is complete.Set F = PB(I ). Then F ⊆ Extn is an additive subfunctor admitting enough special injectivemorphisms.

Proof. By 2.12 we know that F is an additive subfunctor of Extn. We show that it has enoughspecial injective morphisms. By Lemma 4.8 we know that I ⊥ = F -inj. Since the n-idealcotorsion pair (I ,I ⊥) is complete, F -inj is special preenveloping ideal. So every A′ ∈ C has

a special F -inj-preenvelope A′ j→ X1 such that it obtained as the leftmost morphism in an

n-pullback of an admissible n-exact sequence η along a morphism i : A−→Y ∈ ⊥(F -inj). It isdepicted by the following diagram.

η′ : A′ X1 · · · Xn A

η : A′ Z1 · · · Zn Y

j

i

Also since the n-ideal cotorsion pair (I ,I ⊥) is complete, I is special precovering. So byProposition 4.9 we get I = Φ(F ). On the other hand I = ⊥(F -inj) implies that i ∈ Φ(F ).Hence we get admissible n-exact sequence η′ with leftmost morphism j ∈ F -inj, obtained fromthe n-pullback of an admissible n-exact sequence η along an F -phantom morphism. ThereforeF has enough special injective morphisms.

5. Salce’s Lemma

Salce’s Lemma [S] is one of the main theorems in the classical approximation theory. Itrelates the notions of (special) precoverings, (special) preenvelopings and cotorsion pairs. Byintroducing an interesting exact structure on the morphism category of an exact category, calledME-exact structure, an ideal version of Salce’s Lemma is proved in [FH, Theorem 6.3]. Our aimin this section is to provide a higher ideal version of this result.

Setup 5.1. Throughout the section, n ≥ 1 is a fixed integer and C is an n-cluster tiltingsubcategory of the exact category (A , E ) with the n-exact structure X . Since (C ,X ) is fixed,for simplicity, we just write Extn instead of ExtnX .

The notions of injective and projective objects in an n-exact category are defined by using theexactness of the Hom functor. For example, an object E ∈ A is called X -injective if, for everyadmissible monomorphism f : X0 → X1 in X , the induced morphism C (X1, E) −→ C (X0, E)is an epimorphism.

Definition 5.2. ( [Ja, Definition 5.3]) We say that an n-exact category (C ,X ) has enough X -injectives if for every object X ∈ C there exists injective objects E1, · · · , En and an admissiblen-exact sequence

X E1 → · · · → En։ X ′.

By abuse of notation, we denote X ′ by Ω−nX and will call it the nth X -cosyzygy of X . Thenotion of having enough X -projectives is defined dually. The nth X -syzygy of X is denotedby ΩnX .


Definition 5.3. Let (C ,X ) be an n-cluster tilting subcategory of A and

X0 X1 X2 · · · Xn Xn+1

Y 0 Y 1 Y 2 · · · Y n Y n+1

f0 f1 f2 fn fn+1

be a morphism of admissible n-exact sequences in X . Let I be an ideal of C . We say that Iis closed under n-extensions by X -injective objects, if whenever f0 ∈ I and Xn+1 is an X -injective object, then we can deduce that all the middle morphisms f i, for i ∈ 1, 2, . . . , n, arein I . Dually one can define the notion of an ideal closed under n-coextensions by X -projectiveobjects.

Note that if n = 1, and hence C = A , the ideal M⊥, where M is a collection of morphismsin A , is always closed under (1-)extensions by injective objects [FGHT, Proposition 9].

Construction 5.4. Consider the following commutative diagram

Un+1

V n+1

η : X0 X1 X2 · · · Xn Xn+1

δ : Y 0 Y 1 Y 2 · · · Y n Y n+1

ϕ

ψ

f0

in which rows are admissible n-exact sequences. By taking n-pullback diagrams of η along ϕand of δ along ψ and then using the properties of n-pullbacks in n-exact categories, the abovediagram can be completed as follows

X0 U1 U2 · · · Un Un+1

Y 0 V 1 V 2 · · · V n V n+1

X0 X1 X2 · · · Xn Xn+1

Y 0 Y 1 Y 2 · · · Y n Y n+1

f0ϕ

ψ

f0

where all squares are commutative and rows are admissible n-exact sequences. To constructmorphisms U i −→ V i, one should apply the construction of n-pullbacks, explained in Remark2.6.

Now we are in a position to state and prove the main result of this section. For a proof in thecase n = 1, see [FH, Theorem 6.3].

Theorem 5.5. (Salce’s Lemma) Let (C ,X ) be an n-cluster tilting subcategory of an exact cate-gory (A , E ). Let (I ,J ) be an n-ideal cotorsion pair such that I is closed under n-coextensionsby X -projective objects and J is closed under n-extensions by X -injective objects. If C hasenough X -injective objects, then I is a special precovering ideal if and only if J is a specialpreenveloping ideal.


Proof. We assume that I is a special precovering ideal and prove that J is special preenvelop-ing. The converse statement can be proved dually. Let A be an object of C . Since C has enoughX -injective objects, there is an admissible n-exact sequence η : A E1 → · · · → En

։ X suchthat Ei, for 1 ≤ i ≤ n, is X -injective. Since I is special precovering, there exists an n-pushoutdiagram

Y Z1 · · · Zn−1 Zn X

C′ X1 · · · Xn−1 Xn X

j′ h

i

where i : Xn → X in I is a special I -precover of X and j′ ∈ I ⊥ = J . Now considern-pullback of η along i : Xn → X to get the following diagram

A C1 C2 · · · Cn Xn

A E1 E2 · · · En X

j

i

To complete the proof it is enough to show that j : A → C1 belongs to J . To this end, taken-pullback of η along ih, to get the following commutative diagram

Y

C′

Z1

X1

.

..

...

A Y 1 Y 2 · · · Y n Zn

A C1 C2 · · · Cn Xn

A E1 E2 · · · En X

A E1 E2 · · · En X

j′

hj

i

Now we start from the rightmost column of the above diagram and use Construction 5.4step by step to get the following diagram, in which all rows and columns are admissible n-exact


sequences.

Y Y · · · Y Y

C′ C′ · · · C′ C′

U11

U12

· · · U1n Z1

V 11 V 1

2 · · · V 1n X1

..

....

..

....

.

.....

.

.....

A Y 1 Y 2 · · · Y n Zn

A C1 C2 · · · Cn Xn

A E1 E2 · · · En X

A E1 E2 · · · En X

j′ j′ j′ j′

j1 hj

i

Since J is closed under n-extensions by injective objects, we infer that j1 is in J . Hence jfactors through j1 and so j ∈J , as we desired.

6. Special precovering ideals

The main purpose of this section is to study connections between special precovering idealsand n-phantom morphisms in an n-cluster tilting subcategory. In particular, we show that everyspecial precovering ideal, under some conditions, can be represented as an ideal of n-F -phantommorphisms, for some bifunctor F of Extn. Throughout the section (C ,X ) is an n-cluster tiltingsubcategory of an exact category (A , E ).

Let us begin with the following useful lemma.

Lemma 6.1. Let A ∈ C and γ : K Y 1 → · · · → Y np։ A be an admissible n-exact sequence

such that p is a projective morphism. Then a morphism ϕ : X → A is an n-F -phantommorphism if and only if the n-pullback of γ along ϕ is an F -admissible n-exact sequence.

Proof. The ‘only if’ part is obvious, by definition. For the proof of the ‘if’ part, Let

η : A′ X1 → · · · → Xn

։ A

be an arbitrary admissible n-exact sequence. We show that n-pullback of η along ϕ is an F -admissible n-exact sequence. Since p is a projective morphism, there is a morphism Y n → Xn

that makes the rightmost square of the following diagram commutative. Now by the factorization


property of weak kernels we can complete the diagram and in fact get a morphisms of admissiblen-exact sequences

γ : K · · · Y n−1 Y n A

η : A′ · · · Xn−1 Xn A

g

p

So, by Proposition 4.8 of [Ja], η is the n-pushout of γ along the morphism g. Hence η =Extn(A, g)(γ) and is a part of the following diagram

γ′ : K Z1 · · · Zn X

η′ : A′ W 1 · · · Wn X

γ : K Y 1 · · · Y n A

η : A′ X1 · · · Xn A

g

i

ϕ

ϕ

g

By assumption, γ′, the n-pullback of γ along ϕ, is an F -admissible n-exact sequence. Now byProposition 2.11 we have

η′ = Extn(ϕ,A′)(η)

= Extn(ϕ,A′)Extn(A, g)(γ)

= Extn(X, g)Extn(ϕ,K)(γ)

= Extn(X, g)(γ′).

So η′ is the n-pushout of F -admissible n-exact sequence γ′ and hence is an F -admissible n-exactsequence.

Theorem 6.2. Let (C ,X ) be an n-cluster tilting subcategory with enough projective morphismsand F ⊆ Extn be a subfunctor with enough injective morphisms. Let A ∈ C and consider the

n-pushout of admissible n-exact sequence γ : K Y 1 → · · · → Y np։ A, where p is projective

admissible epimorphism, along an F -injective F -admissible monomorphism e : K → A′

γ : K Y 1 · · · Y n A

A′ X1 · · · Xn A

e

p

ϕ

Then ϕ : Xn → A is a special n-F -phantom precover of A.

Proof. Since e ∈ Φ(F )⊥, to show that ϕ is a special n-F -phantom precover of A, it is enoughto show that ϕ is an n-F -phantom morphism. This we do. To this end, by Lemma 6.1, it isenough to show that n-pullback of γ along ϕ : Xn → A is an F -admissible n-exact sequence.


The diagram

γ′ : K Z1 · · · Zn Xn

γ : K Y 1 · · · Y n A

A′ X1 · · · Xn A

i

ϕ

e

p

ϕ

give us, by composition, the following morphism of admissible n-exact sequences

γ′ : K Z1 · · · Zn Xn

A′ X1 · · · Xn A

i

eg 1X

ϕ

ϕ

Since there exists morphism 1X : X −→ X such that ϕ = ϕ1X , this composition is null-homotopic, see e.g. [Fe, Lemma 3.6]. So there exists morphism g : Z1 −→ A′ such that e =gi. Now since e is an F -admissible monomorphism, by Lemma 4.11, i is an F -admissiblemonomorphism and hence γ′ is an F -admissible n-exact sequence.

In particular, we have proved the following theorem that is the main result of this section.

Theorem 6.3. Let (C ,X ) be an n-cluster tilting subcategory of A with enough projectivemorphisms. Then the ideal I of C is special precovering if there exists an additive subfunctorF ⊆ Extn with enough injective morphisms and I = Φ(F ).

Last result of this section, shows that every special precovering ideal I , under some condi-tions, can be represented as an ideal of n-PB(I )-phantom morphisms.

Theorem 6.4. Let (C ,X ) be an n-cluster tilting subcategory of A with enough X -injectiveobjects. Let I be an ideal of C such that I is closed under n-coextensions by projectives and I ⊥

is closed under n-extensions by injectives. If I is a special precovering ideal then there exists anadditive subfunctor F ⊆ Extn with enough special injective morphisms such that I = Φ(F ).

Proof. Since I is a special precovering ideal, by Theorem 3.9, (I ,I ⊥) is an n-ideal cotorsionpair. Also since I is closed under n-coextensions by projectives and I ⊥ is closed under n-extensions by injectives, Salce’s Lemma 5.5, implies that the n-ideal cotorsion pair (I ,I ⊥) iscomplete. Now put F = PB(I ) and use Corollary 4.13.

7. Wakamatsu’s Lemma

In this section we state and prove an ideal version of the Wakamatsu’s Lemma [W] in higherhomological algebra. An ideal version of Wakamatsu’s Lemma in an exact category is provedin [FH, Theorem 10.3]. For a version in (n+ 2)-angulated categories and a version in n-abeliancategories see [Jor, §3] and [AMS, Theorem 4.2], respectively.

Throughout the section (C ,X ) is an n-cluster tilting subcategory of an exact category (A , E ).We need the following version of [IJ, Definition-Proposition 2.15] in the proof of the main

result of this section. The proof follows using similar argument so we skip it. As before, forsimplicity of notation, we write Exti instead of ExtiX .


Proposition 7.1. Let (C ,X ) be an n-cluster tilting subcategory of the exact category (A , E ).The following conditions are equivalent.

(a) Exti(C ,C ) = 0, for all i /∈ nZ.(b) C is closed under n-syzygies, that is, Ωn(C ) ⊆ C .(c) C is closed under n-cosyzygies, that is, Ω−n(C ) ⊂ C .(d) For each X ∈ C and for each admissible n-exact sequence

LM1 −→M2 −→ · · · −→Mn։ N

in X , there exists an exact sequence

0 −→ C (X,L) −→ C (X,M1) −→ · · · −→ C (X,Mn) −→ C (X,N)

−→ Extn(X,L) −→ Extn(X,M1) −→ · · · −→ Extn(X,Mn) −→ Extn(X,N)

−→ Ext2n(X,L) −→ Ext2n(X,M1) −→ · · · −→ Ext2n(X,Mn) −→ Ext2n(X,N)

−→· · ·

of abelian groups.(e) For each X ∈ C and for each admissible n-exact sequence

LM1 −→M2 −→ · · · −→Mn։ N

in X , there exists an exact sequence

0 −→ C (N,X) −→ C (Mn, X) −→ · · · −→ C (M1, X) −→ C (L,X)

−→ Extn(N,X) −→ Extn(Mn, X) −→ · · · −→ Extn(M1, X) −→ Extn(L,X)

−→ Ext2n(N,X) −→ Ext2n(Mn, X) −→ · · · −→ Ext2n(M1, X) −→ Ext2n(L,X)

−→· · ·

of abelian groups.

Recall that an object A of C is said to be in I if the identity morphism 1A is in I (A,A).

Definition 7.2. Let I be an ideal of C . We say that I is left closed under n-extensions byobjects of I , if for every morphism of X -admissible n-exact sequences in C such as

X0 X1 X2 · · · Xn Xn+1

Y 0 Y 1 Y 2 · · · Y n Xn+1

f0 f1 f2 fn

X1 is an object of I if f0 ∈ I and Xn+1 is an object of I .

Finally let us recall the definition of an I -cover.

Definition 7.3. Let I be an ideal of C and A be an object of C . An I -precover i : C −→ Aof A is called an I -cover if every endomorphism f : C −→ C with the property that if = i isnecessarily an automorphism.

Now we have the necessary ingredients to state and prove the main result of this section. Fora version in (n+ 2)-angulated categories see Lemma 3.1 of [Jor].


Theorem 7.4. (Wakamatsu’s Lemma) Let (C ,X ) be an n-cluster tilting subcategory of anexact category (A , E ) with enough X -injective objects. Let I be an ideal of C which is leftclosed under n-extensions by objects in I . Let A be an object of C and i : I −→ A be theI -cover of A. Then for every X ∈ I , there exists the exact sequence

0 −→ Extn(X,Kn) −→ Extn(X,Kn−1) −→ · · · −→ Extn(X,K1) −→ 0,

of abelian groups, where Kn −→ Kn−1 −→ · · · −→ K1 is an n-kernel of i.

Proof. Consider n-exact sequence

Kn Kn−1 −→ · · · −→ K1 −→ I ։ A

in X . Since C has enough X -injective objects, it is closed under n-cosyzygies and hence byequivalent conditions of Proposition 7.1, we have the following long exact sequence

· · · C (X, I) C (X,A)

Extn(X,Kn) Extn(X,Kn−1) · · · Extn(X, I) Extn(X,A) · · ·

i∗

i

of abelian groups. To prove the theorem, it is enough to show that i∗ is surjective and i isinjective. This we do. Since X ∈ Ob(I ) and i is an I -cover, it follows by definition that i∗

is surjective. So it remains to prove that i is injective. This follows from [Fe, Lemma 5.1]. Wereproduce the proof here. Let

η : I X1 −→ X2 −→ · · · −→ Xn։ X

be an element of Extn(X, I) that maps to zero in Extn(X,A), that is, the n-pushout of η along i,say η′, is a contractible n-exact sequence. We show that η itself should be a contractible n-exactsequence. To see this consider the following n-pushout diagram

η : I X1 X2 · · · Xn X

η′ : A Y 1 Y 2 · · · Y n X

i i

d0X

d1X

f1

d2X

f2

dn

X

fn

d0Y

d1Y

d2Y

dn

Y

To show that η is contractible, by [Ja, Proposition 2.6], it is enough to show that d0X is asplit monomorphism. Since η′ is contractible, there exists morphism sn+1 : X → Y n such

that dnY sn+1 = 1X . This, in view of Lemma 3.6 of [Fe] implies that i : η −→ η′ should be

null-homotopic. So, in particular, there exists a morphism s1 : X1 −→ A such that s1d0X = i.Now since I is left closed under n-extensions by objects of I and i ∈ I and X is an object

in I , X1 ∈ Ob(I ). So s1 : X1 −→ A is a morphism in I and hence factors through i. Letα : X1 −→ I be such that iα = s1. By composing with d0X we get iαd0X = s0d0X = i. Since iis an I -cover, αd0X : I −→ I is an isomorphism. So d0X is a split monomorphism. Hence η iscontractible, as it was desired.

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Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan,P.O.Box: 81746-73441, Isfahan, Iran

Email address: [email protected], [email protected]

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