Generalized tilting theory in functor categories

Abstract This paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories 
$\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$
 with 
$\mathcal{C}$
 an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory 
$\mathcal{T}$
 of 
$\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$
 is constructed. Some applications of these two results include the equivalence of Grothendieck groups 
$K_0(\mathcal{C})$
 and 
$K_0(\mathcal{T})$
 , the existences of a new abelian model structure on the category of complexes 
$\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$
 , and a t-structure on the derived category 
$\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$
 .


Introduction
Tilting theory arises as a universal method for constructing equivalences between categories. Since its advent, it has been an essential tool in the study of many areas of mathematics, including algebraic group theory, commutative and noncommutative algebraic geometry, and algebraic topology.
Tilting theory can trace its history back to the article by Bernstein et al. [8], where they used reflection functors to construct recursively all the indecomposable modules from simple modules over a representation-finite hereditary algebra. The major milestone in the development of tilting theory was the article by Brenner and Butler [9]. They introduced the notion of a tilting module over a finite dimensional algebra and established the so-called Brenner-Butler theorem by a tilting module. In this article, the behavior of the associated quadratic forms was investigated as well. Dropping a more restrictive notion of tilting module defined by Brenner and Butler, Happel and Ringel [15] successfully simplified the definition of original tilting modules. A few years later, Miyashita [26] generalized the concept of tilting modules allowing modules of any finite projective dimension and over any ring, for which a generalization of the Brenner-Butler theorem was still valid. Indeed, the authors, like Brenner and Butler [9], Happel and Ringel [15], and Miyashita [26], considered finitely generated tilting modules, obtaining portions of the Brenner-Butler theorem. Colpi and Trlifaj [10] generalized the notion of tilting module to not necessarily finitely generated modules. Later on, Angeleri-Hügel and Coelho [1] did the same with the concept of Miyashita. On the other hand, functor categories, introduced by Auslander [2], are used as a potent tool for solving some important problems in representation theory. Martsinkovsky and Russell have studied the injective stabilization of additive functors (see [23][24][25]). Recently, Martínez-Villa and Ortiz-Morales [20][21][22] initialed the study of tilting theory in arbitrary functor categories with applications to the functor category Mod(A) for A a category of modules over a finite dimensional algebra. The first one [20] in a series of three is to deal with the concept of tilting subcategory T of Mod(C), which is the category of contravariant functors from a skeletally small additive category C, to the category of abelian groups. They showed that the Brenner and Butler theorem holds for T . In the second and third papers [21,22], replacing a tilting subcategory with a generalized tilting subcategory T of Mod(C), they continued Xi Tang the project of extending tilting theory to the same functor category with particular focusing on the equivalence of the derived categories of bounded complexes D b (Mod(C)) and D b (Mod(T )).
In the same spirit as in the above-mentioned results of Martínez-Villa and Ortiz-Morales, in this paper, we aim at extending some well-known results, relating generalized tilting modules in module category, to a generalized tilting subcategory of Mod(C). We now give a brief outline of the contents of this paper.
In Section 2, we collect preliminary notions and results on functor categories that will be useful throughout the paper and we fix notation. We also give an example of a generalized tilting subcategory of Mod(C) (see Example 2.4).
In Section 3, we are interested in studying a generalized version of the Brenner-Butler theorem in functor category. More precisely, we show in Theorem 3.4 that for a generalized tilting subcategory T of Mod(C) one gets an equivalence between the categories KE ∞ e (T ) and KT ∞ e (T ). As an application of this main theorem, we state in Theorem 3.8 that if C is an abelian category with enough injectives and T is an n-tilting subcategory of mod(C) with pseudokernels, then the Grothendieck groups K 0 (C) and K 0 (T ) are isomorphic.
In Section 4, we prove in Theorem 4.3 that for a generalized tilting subcategory T of Mod(C), ( ⊥∞ (T ⊥∞ ), T ⊥∞ ) is a hereditary and complete cotorsion pair. Furthermore, this induces an abelian model structure on C(Mod(C)), where the trivial objects are the exact complexes, the cofibrant objects are dg-⊥∞ ⇐T ⊥∞ ⇒ complexes, and the class of fibrant objects is given by the complexes whose terms are in T ⊥∞ (see Corollary 4.6).
In Section 5, we use the model structure on C(Mod(C)) to describe the t-structure on the derived category D(Mod(C)), induced by a generalized tilting subcategory T of Mod(C) (see Theorem 5.5).

Preliminaries
Throughout this paper, C will be an arbitrary skeletally small additive category, and Mod(C) will denote the category of additive contravariant functors from C to the category of abelian groups. It follows from [28, Theorem 1.2 and Proposition 1.9] or [20, Section 1.2] that Mod(C) is a Grothendieck category with enough projective objects. In addition, Mod(C) also has enough injective objects by [19,p.384,Theorem B.3]. If M, N ∈ Mod(C), we denote the set Mod(C)(M, N) of natural transformations M → N by Hom C (M, N). Following [3], a functor F is called representable if it is isomorphic to C( , C) for some C ∈ C. A functor F is finitely generated if there is an epimorphism C( , C) → F → 0 with C ∈ C. A functor F is finitely presented, if there exists a sequence of natural transformations C( , C 1 ) → C( , C 0 ) → F → 0 with C 0 , C 1 ∈ C such that for any C ∈ C the sequence of abelian groups C(C, C 1 ) → C(C, C 0 ) → F(C) → 0 is exact. We denote by mod(C) the full subcategory of Mod(C) consisting of finitely presented functors. An object P in Mod(C) is projective (finitely generated projective) if and only if P is a summand of i∈I C( , C i ) for a (finite) family {C i } i∈I of objects in C (see [20, Paragraph 3 of Section 1.2]). We recall from [3, p.188] that an annuli variety is a skeletally small additive category in which idempotents split.
Let A be an abelian category and F ∈ mod(A), then there is an exact sequence A( , X) Proof. (1) and (2)  (3) It suffices to show that (v, φ) is an adjoint pair. Let M ∈ mod(T ) and N ∈ Mod(C), we need to find an isomorphism which is natural in both M and N. Suppose that there is an exact sequence T ( , By the construction of defect functor, we get an exact sequence T 1 . It follows from the Yoneda Lemma that (T ( , T 1 ), φ(N)) ∼ = (T 1 , N) and (T ( , T 0 ), φ(N)) ∼ = (T 0 , N). Then we have the following commutative diagram with exact rows So θ is an isomorphism and it is easy to check that θ is natural in both M and N.
Following [20] and [21], given categories C and T as in Lemma 2.1, since Mod(C) and Mod(T ) have enough projective and injective objects, we can define the nth right derived functors of the functors Hom C (M, ) and Hom C ( , N), which will be denoted by Ext n C (M, ) and Ext n C ( , N), respectively. In the same way, the functor φ : Mod(C) → Mod(T ) has an nth right derived functor, denoted by Ext n C ( , −) T , and they are defined as Ext n C ( , −) T (M) = Ext n C ( , M) T . Analogously, the functor − ⊗ T : Mod(T ) → Mod(C) has an nth left derived functor, denoted by Tor T n ( , T ). Let T be a subcategory of Mod(C). Add(T ) (resp. add(T )) will denote the class of functors isomorphic to summands of (finite) direct sums of objects in T and Gen n (T ) will denote the full subcategory consisting of M ∈ Mod(C) for which there exists an exact sequence of the form T n → · · · → T 2 → T 1 → M → 0 with T i ∈ Add(T ). For any i 1, we write and Im f i ∈ T ⊥∞ . It is easy to see that objects in T X are in Gen n (T ) for each n.
Next we recall the concept of cotorsion pairs in abelian categories, due to Holm and Jørgensen [16, Section 6]. ( (1) There exists a fixed integer n such that every object T in T has a projective resolution with each P i finitely generated. (

2) For each pair of objects T and T in T and any positive integer
There is a fixed integer m such that each representable functor C( , C) has an exact sequence For a general subcategory T of Mod(C), we use pdim T to denote the the supremum of the set of projective dimensions of all T in T . If T is generalized tilting with pdim T n satisfying condition (3 ), and the integer m in condition (3 ) equals n, then we say T is n-tilting. It should be pointed that a tilting subcategory T defined in [20,Definition 8] is exactly 1-tilting when T is closed under taking direct summands.
Finally, we end this section by showing that there exists a natural example of a generalized tilting subcategory T of Mod(C).

Example 2.4.
Let be an artin R-algebra and C = add . Assume that Mod has a classical n-tilting module T. Then we have an n-tilting subcategory T of Mod( add ).
Proof. Since T is a classical n-tilting module, it follows from [6] that T satisfies the following conditions: (1) There exists a projective resolution 0 → P n → · · · → P 1 → P 0 → T → 0 with each P i finitely generated, There is an exact sequence 0 → → T 0 → T 1 → · · · → T n → 0 with T i ∈ add(T). We set T = {C( , T ) | T ∈ add T}. It is easy to verify that T is an n-tilting subcategory of Mod(C).

Equivalences induced by a generalized tilting subcategory
Our purpose in this section is to study category equivalences induced by a generalized tilting subcategory T of Mod(C). First, we observe the following key result, which is vital in proving the main theorem of this section. Proposition 3.1. Assume that T is generalized tilting with pdim T n for some integer n. Then the following statements are equivalent for any M in Mod(C). where Then it follows from Diagram (3.1) that α is epic. Since α is also monic, by Yoneda's lemma, we have . Moreover, this exact sequence remains exact after applying the functor φ to it. Observe that Ext i 1 C (T, T (X) ) = 0 for any T, T ∈ T and any set X by [20,Proposition 4]. So M 1 ∈ T ⊥∞ . Now repeating the process to M 1 , we obtain that M ∈ T X .
(3) ⇒ (1) The case for n = 0 is trivial. Now suppose that n > 0, then by assumption there is an exact N) for any T in T and any i 1 by dimension shift. But the latter equals 0, since pdim T n. Therefore, M ∈ T ⊥∞ .
The following two results, dual to each other, will be used throughout.

Lemma 3.2. Assume that T is generalized tilting and M is an object in
with T i = T∈T T (X i ) and each Im f i ∈ T ⊥∞ . Applying the functor φ to (3.2) yields the following exact sequence Analogously, dualizing the proof of the above lemma, we have the following Proof. Consider the following projective resolution of N Note that T is generalized tilting, we have that P i ⊗ T ∈ T ⊥∞ for any i 0. Moreover, we have the following commutative diagram In order to present the main theorem in this section, we need to introduce the following notions.

Theorem 3.4. Assume that T is generalized tilting and e is a non-negative integer. Then there are two category equivalences
Proof. We can apply Lemmas 3.2 and 3.3 to conclude that the equivalence holds for e = 0. Now assume that e 1 and M ∈ KE ∞ e (T ). Consider an injective resolution of M Because every term except X in the exact sequence belongs to ∞ T , the ith-homology can be computed by it. Therefore, we obtain that Tor T i (X, T ) = 0 for any 0 i < ∞ and i = e, Tor T e (X, T ) ∼ = M. Conversely, suppose that N ∈ KT ∞ e (T ). Consider a projective resolution of N Since Tor T i (Im g e , T ) ∼ = Tor T i+e (N, T ) = 0 for any i 1. It follows from Lemma 3.3 that Im g e ∼ = φ(Im g e ⊗ T ) and Im g e ⊗ T ∈ T ⊥∞ . Applying the functor − ⊗ T to (3.5), we get an exact sequence (N, T ). Because every term except Y in the exact sequence belongs to T ⊥∞ , the ith-cohomology can be computed by it. Therefore, we obtain that Ext i C ( , Y) T = 0 for any Given a 1-tilting subcategory T , Martínez-Villa and Ortiz-Morales in [20,Theorem 3] proved that φ and − ⊗ T induce an equivalence between KE 1 0 (T ) and KT 1 0 (T ). We generalize this result to n-tilting subcategory T as follows. According to [21], we say C has pseudokernels if given a map f : Since Mod(C) is an abelian category, C has pseudokernels if and only if Ker(, f ) is finitely generated for each f : C 1 → C 0 in C. Next we turn to investigating the invariance of Grothendieck groups under generalized tilting. To this end, we need the following. It was proved in [4] that mod(C) is abelian if and only if C has pseudokernels. We will use this result to show the following proposition.

Xi Tang
Proposition 3.7. Let C be an annuli variety and T a generalized tilting subcategory of mod(C). Assume C and T have pseudokernels. Then the following statements hold.
(1) Ext i C ( , M) T ∈ mod(T ) for any M ∈ mod(C) and any i 0. (2) Tor T i (N, T ) ∈ mod(C) for any N ∈ mod(T ) and any i 0.
Proof. (1) Since T is generalized tilting, we may assume that pdim T n for some integer n. Let M ∈ mod(C), then there is an exact sequence 0 → K 1 → C( , C 0 ) → M → 0 with K 1 ∈ mod(C). Applying the functor φ to this exact sequence gives rise to the following exact sequence (2) Let N ∈ mod(T ) and a projective resolution of N. Set L i = Im f i and split the resolution in short exact sequences: Thus it follows from the long homology sequence that there are exact sequences: Since T 1 ∈ mod(C), L 1 ⊗ T is finitely generated. Thus, Im(L 1 ⊗ T → T 0 ) is finitely generated and so N ⊗ T is finitely presented. Similarly, as L 1 ∈ mod(C), L 1 ⊗ T is finitely presented. Note that mod(C) is abelian. We get that Tor We now come to the first application in this section, our method in the following has its origin in [26, Theorem 1.19].
Theorem 3.8. Let C be an abelian category with enough injectives and T an n-tilting subcategory of mod(C) with pseudokernels. Then the Grothendieck groups K 0 (C) and K 0 (T ) are isomorphic.
Proof. We define two group homomorphisms It is easily seen by Proposition 3.7 that F and G are well defined. For any M ∈ mod(C), since mod(C) has enough injectives by [29, Section 6], we have an injective resolution

Given M ∈ Mod(C), since M has an injective envelope by [19, Theorem B.3], we have a minimal injective resolution
Then cTr T M := Coker φ(f 0 ) is called the cotranspose of M with respect to T . The notion is analogous to the cotranspose of a module with respect to a semidualizing bimodule defined in [32,Defintion 3.1]. Using the tool of cotransposes, Tang and Huang in [32,Proposition 3.2] established the so-called dual Auslander sequence. We will apply Theorem 3.4 to conclude that the dual Auslander sequence still holds in functor categories, but the approach used here is different. Corollary 3.9. Assume that T is generalized tilting. Then for any M ∈ Mod(C), there is an exact sequence Proof. Let 0 → M → I 0 → I 1 → · · · be a minimal injective resolution of M. Applying the functor φ to it yields an exact sequence (3.6) where K = Im(φ(I 0 ) → φ(I 1 )). By Theorem 3.4, now applying the functor ⊗T to Diagram (3.6) gives rise to the following diagram Therefore the left most column in the above diagram is as desired.

Cotorsion pair and model category structure
Our goal in this section is to construct a cotorsion pair induced by a generalized tilting subcategory T of Mod(C), allowing us to provide a model category structure on the category of complexes C (Mod(C)). For the definition of a model structure, we refer to the book by Hoevy [18].

Xi Tang
The following lemma is straightforward, but we include a proof as we have not been able to find a suitable reference for it.

Lemma 4.1. Suppose that T is a subcategory of Mod(C). Then the following statements hold.
( Proof. We show the first statement of the lemma. The second statement follows from a dual argument.

1) If T is closed under cokernels of monomorphisms and contains all injective objects in Mod(C),
(1) It is enough to show that ⊥ 1 T ⊆ ⊥∞ T . Let X ∈ ⊥ 1 T and M ∈ T , then we have an exact sequence 0 → M → I 0 → I 1 → · · · → I i → · · · with I i injective. Set K i = Ker(I i → I i+1 ). By assumption, both I i and Now we consider the relation between Gen n (T ) and Gen n ( T X ).

Lemma 4.2. Assume that T is a subcategory of Mod(C). If there is an exact sequence
Proof. The proof is modeled on [33, Lemma 3.5(1)]. We shall prove the statement by induction on n. When n = 1, we have an exact sequence 0 → L → K 1 → M → 0 with K 1 ∈ T X . Thus, we have another exact sequence 0 → U 1 → T 1 → K 1 → 0 with T 1 ∈ Add(T ) and U 1 ∈ T X . Consider the following pull-back diagram (4.1) Then the middle row and left column in Diagram (4.1) are the desired exact sequences. Now assume that the conclusion is true for n − 1. We will show that the conclusion holds for n. Set L = Coker(L → K n ). Then, by the induction assumption, there is an exact sequence 0 → U n−1 → V n−1 → L → 0 for some U n−1 ∈ T X , and some V n−1 such that there is an exact sequence 0 → V n−1 → T n−1 → · · · → T 1 → M → 0 with T i ∈ Add(T ). Then we can construct the following pullback diagram Since U n−1 , K n ∈ T X and T X is closed under extensions by [11,Lemma 8.2.1], we get that X ∈ T X . Thus, there is an exact sequence 0 → U n → T n → X → 0 with T n ∈ Add(T ) and U n ∈ T X . Consider the following pull-back diagram Our main aim in this section is to show that the following holds.  Proof. (1) Since T ⊥∞ is closed under cokernels of monomorphisms and contains all injective objects, it follows from Lemma 4.1(1) that ⊥∞ (T ⊥∞ ) = ⊥ 1 (T ⊥∞ ). On the other hand, since ⊥∞ (T ⊥∞ ) is closed under kernels of epimorphisms and contains all projective objects, we have ( ⊥∞ (T ⊥∞ )) ⊥ 1 = ( ⊥∞ (T ⊥∞ )) ⊥∞ Lemma 4.1 (2). Thus, ( ⊥∞ (T ⊥∞ )) ⊥ 1 = T ⊥∞ . Hence, ( ⊥∞ (T ⊥∞ ), T ⊥∞ ) forms a hereditary cotorsion pair. For each T ∈ T , by assumption, there is a projective resolution 0 → P n (T) → · · · → P 1 (T) Obviously, X is a set and X ⊥ 1 = T ⊥∞ . It implies that ( ⊥∞ (T ⊥∞ ), T ⊥∞ ) is generated by a set X . We conclude by [16] that ( ⊥∞ (T ⊥∞ ), T ⊥∞ ) is a complete cotorsion pair.
(2) Let M ∈ Mod(C), there is an exact sequence 0 → M → I 0 → I 1 → · · · → I n−1 → K → 0 with I i injective. Then I i ∈ T X by Proposition 3.1. Thus, K ∈ Gen n (T ) by Lemma 4.2. It follows from Proposition 3.1 again that K ∈ T ⊥∞ . Given X ∈ ⊥∞ (T ⊥∞ ), we know that Ext i+n C (X, M) ∼ = Ext i C (X, K) = 0 for any i 1. Therefore, the result holds.
Let A be an abelian category. For complexes X and Y, we define the homomorphism complex Hom(X, Y) ∈ C(A) to be the complex to be the group of (equivalence classes) of short We denote by K(A) the homotopic category, that is, the category consisting of complexes such that the morphism set between X, Y ∈ C(A) is given by Hom K(A) (X, Y) = Hom C(A) (X, Y)/ ∼. Furthermore, there is a corresponding derived category D(A), which is also triangulated.
In order to obtain an abelian model structure, we have to introduce the following classes in C(Mod(C)). (1) X is called an A complex if it is exact and Z n (X) ∈ A for all n.
(2) X is called a B complex if it is exact and Z n (X) ∈ B for all n.
(3) X is called a dg-A complex if X n ∈ A for each n, and Hom(X, B) is exact whenever B is a B complex. (4) X is called a dg-B complex if X n ∈ B for each n, and Hom(A, X) is exact whenever A is an A complex.
We denote the class of A complexes byÃ and the class of dg-A complexes by dgÃ. Similarly, the class of B complexes is denoted byB and the class of dg-B complexes is denoted by dgB.
Inspired by [5,Theorem 2.5], we present the following theorem. The homotopy category of this model category is D (Mod(C)).
Proof. Since Mod(C) is a Grothendieck category, it is a bicomplete abelian category by [30,Chapter V] and [20,Section 1.2]. Because ( ⊥∞ (T ⊥∞ ), T ⊥∞ ) is a hereditary and complete cotorsion pair by Theorem 4.3, it follows from [34,Theorem 2.4] that there are two induced complete cotorsion pairs (Ã, dg-B) and (dg-A,B). So we claim that X ∈ dg-A if and only if Ext 1 C(Mod(C)) (X, B) = 0 for any B ∈B and Y ∈ dg-B if and only if Ext 1 C(Mod(C)) (A, Y) = 0 for any A ∈Ã. Next, we know from [13,Corollary 3.8] that there is an abelian model structure on C(Mod(C)). Furthermore, weak equivalences, cofibrantions, and fibrations are described exactly as in the statements. Observe that exact dg-A complexes are exactly A complexes by [34,Theorem 2.5].  [14,Introduction].
According to [17,18], suppose that an abelian category A has a model structure, X is trivial if 0 → X is a weak equivalence, X is cofibrant if 0 → X is a cofibration and X is fibrant if X → 0 is a fibration.

In particular, this applies to the complexes C bounded below and with terms in A.
Proof. We know by Corollary 4.6(3) that F is a fibrant. Then it follows from [14] that Hom K(Mod(C)) (C, F) ∼ = Hom D(Mod(C)) (C, F). In particular, if C is a bounded below complex with terms in A, then C ∈ dg-A by [12,Lemma 3.4(1)]. It implies that C is a cofibrant by Corollary 4.6(2). So the equivalence also applies to C.

A t-structure induced by a generalized tilting subcategory
In this section, we mainly show that there exists a t-structure on the derived category D(Mod(C)), relating a generalized tilting subcategory T of Mod(C). For the sake of completeness, let us recall the definition of a t-structure.
Notation 5.2. Suppose that T is a generalized tilting subcategory of Mod(C). For any k ∈ Z, we denote by D k T and D k T the full subcategories of D(Mod(C)) given by D k T = {X ∈ D( Mod(C)) | Hom D( Mod(C)) ( i T, X) = 0 for any i < k and T ∈ T }, Let A be an abelian category and F be a class of objects in A. Then a morphism ϕ : If every object of A has an F-precover, F is said to be precovering [11]. Given k ∈ Z, we say that a complex X ∈ C [k,∞] (F) if X i = 0 for i < k and X i ∈ F for i k.

Lemma 5.3. Suppose that A is an abelian category and F is a class of objects in A.
If F is precovering, then for every complex X in D(A) and every k ∈ Z, there is a chain map f : F → X with F ∈ C [k,∞] (F) such that Hom K(A) ( i F, f ) is an isomorphism for any F ∈ F and any i k.
Proof. Given a complex X := · · · → X n+1 d X n+1 −→ X n d X n −→ X n−1 → · · · with X n ∈ A. We will inductively construct a chain map f : X → F such that F i ∈ F for i 0. Consider an F-precover of Ker d X k , F k ϕ k −→ Ker d X k , and let f k be the composition of ϕ k with the inclusion Ker d X k → X k . By induction construct f i+1 : F i+1 → X i+1 as follows. Having defined f i : F i → X i and d F i : F i → F i−1 . Let λ i : K i → F i be the kernel of d F i and let g i = f i λ i . Consider the pullback P i+1 of the maps g i and d X i+1 . Let ϕ i+1 : F i+1 → P i+1 be an F-precover of P i+1 and let f i+1 be the obvious composition. All the maps used in the inductive step are depicted in the following diagram where d F i+1 = λ i α i+1 ϕ i+1 . It is easy to see that F is a complex with all terms in F and f is a chain map between F and X. Now we claim that Hom K(A) ( i F , X) = 0 for any F ∈ F and any i k. If i k, given a map h : i F → X in C(A), our task is to prove that (a) h factors through f : F → X, and (b) If h is null-homotopic, so is any such factorization t : i F → F.
Hence, Hom D(Mod(C)) (X , Y ) = 0. Now we will prove (3) of Definition 5.1. Let X ∈ D(Mod(C)), in view of Corollary 4.6, we may assume that X has all the terms in T ⊥∞ . By Lemma 5.3, there is a chain map f : F → X with F ∈ C [0,∞] ( Add(T )) and Hom K(Mod(C)) ( i T, f ) is an isomorphism for any T ∈ T and any i 0. By Corollary 4.7, the same is true for Hom D(Mod(C)) ( i T, f ). Furthermore, it is straightforward to check that F ∈ D 0 T . Let Cone(f ) be the mapping cone of f , that is, we have a triangle