RELATIVE THEORY IN SUBCATEGORIES. Introduction

Transcription

1 RELATIVE THEORY IN SUBCATEGORIES SOUD KHALIFA MOHAMMED Abstract. We generalize the relative (co)tilting theory of Auslander- Solberg [9, 1] in the category mod Λ of finitely generated left modules over an artin algebra Λ to certain subcategories of mod Λ. We then use the theory (relative (co)tilting theory in subcategories) to generalize one of the main result in [19]. 2 Mathematics Subject Classification. 12G1, 16G7 (primary); 16E1 (secondary). Key words and phrases. Subfunctors, approximation dimension, stratifying systems. Introduction Let Λ be an artin algebra, and let mod Λ denote the category of finitely generated left Λ-modules. Auslander and Solberg [9, 1] developed a relative (co)tilting theory in mod Λ which is a generalization of standard (co)tilting theory [8] [12] [15] [25]. In this paper we develop a relative (co)tilting theory in extension closed functorially finite subcategories of mod Λ. Let T be an ordinary tilting module over Λ. Then the module DT, where D is the usual duality between left and right modules, is a cotilting module over the endomorphism ring Γ End Λ pt q op. If T is a relative tilting module, in the sense of [9, 1], then the Γ-module DT is a direct summand of the cotilting module T Hom Λ pt, Iq over Γ, where add I are the relative injective modules for the relative theory. Here we define relative (co)tilting modules relative to a subcategory C of mod Λ. The module Hom Λ pt, Iq, where I is as above, is not a cotilting module in general. However, we will show that when the C-approximation dimension of mod Λ is finite (see below for the definition), then Hom Λ pt, Iq is a cotilting module. In addition, DT does not need to be a direct summand of T, but it has a finite resolution in add T. Another main result is that for a relative tilting and cotliting module in C, there exists an equivalence between the full subcategory add { TC of C consisting of all modules having a finite resolution in add T and the full subcategory add T consisting of all Γ-modules with finite coresolution in add T. This is used to generalize Theorem.1 in [19]. Let T an ordinary tilting Λ-module. Then the classical tilting functor Hom Λ pt, q induces an equivalence between T K, the category of all Λ-modules Y such that Ext i ΛpT, Y q for all i, and its image Hom Λ pt, T K q in mod Γ, where the category Hom Λ pt, T K q is identified with 1

2 2 SOUD KHALIFA MOHAMMED K DT, the category of all Γ-modules X such that Ext i ΓpX, DT q for all i. Similar results were established by Auslander-Solberg [1] for a relative tilting module T in mod Λ. We want to establish similar result for a relative tilting module in subcategories of mod Λ. To do this we need to develop a relative theory in subcategories. Let C 1 be an additive category which is closed under kernels and cokernels, and suppose C is a functorially finite subcategory of C 1. Iyama [16] introduced an invariant of C 1 given by C, namely the right and left C-resolution dimensions of C 1. A special example of this invariant occurs when C 1 is mod Λ. In this case we refer to the right and left C-resolution dimensions as the right and left C-approximation dimensions. Let us call the maximum of the two invariants (the right and left C-approximation dimensions) the C-approximation dimension of mod Λ. Suppose C is closed under extensions, and assume that the C-approximation dimension of mod Λ is zero. Then it will be shown that C is naturally equivalent to a module category over an artin algebra. This means that a relative theory in C can be developed in the sense of [9, 1]. Let us refer to this theory as the relative theory in dimension. We develop a relative theory in dimension n for certain subfunctors F of the bifunctor Ext 1 Λp, q, where n is the C-approximation dimension of mod Λ. Let C be a functorially finite subcategory of mod Λ which is closed under extensions, and let X be a generator subcategory of C in the sense of [2] (i.e. X contains the Ext-projectives in C). In Section 2 we investigate the subfunctors F F X in C. Denote by C X (C X ) the right (left) C-approximation of X. Then we show that P C pf q, the category of F - projectives in C, and I C pf q, the category of F -injectives in C, are related by the formulas P C pf q C TrD I CpF q YPpCq and I C pf q C DTr IC pf q YIpCq, where PpCq and IpCq denote the categories of Ext-projectives and Extinjectives in C respectively. In Section 3 we state some results connected to approximation dimension. Among the results, we show that the subcategories C of mod Λ with C-approximation dimension zero are equivalent to categories mod Λ{I, where I is an ideal of Λ. In Section 4 we investigate relative (co)tilting modules in extension closed functorially finite subcategories C of mod Λ. Consider a subfunctor F in C with enough projectives and injectives in C. Also suppose that T is an F -tilting module in C with pd F T r. In this setting we will generalize the classical tilting equivalence. Suppose that the C- approximation dimension of mod Λ is a nonnegative integer n. Then, if there is an F -tilting module in C, we will show that I C pf q is of finite type. We assume from now on that I C pf q is of finite type. Denote the Γ-module associated to Hom Λ pt, I C pf qq by TC. Then we will show that the image of the classical tilting functor restricted to TC K, Hom ΛpT, TC Kq, is identified with K TC, where T C K denotes the category T K X C. Moreover,

3 RELATIVE THEORY IN SUBCATEGORIES 3 the Γ-module TC is cotilting. However, the Γ-module DT is not necessarily cotilting, and we give an example which shows that DT is not a direct summand of TC either. Nevertheless, we show that DT has a finite add TC -resolution. We also show that gl. dim F C, the relative global dimension of C, and gl. dim Γ, the global dimension of Γ, are related by the formula gl. dim F C pd F T gl. dim Γ gl. dim F C νpn, rq, where ν is a function of n and r. If the C-approximation dimension of mod Λ is infinite, then we have many examples where the Γ-module TC is not cotilting. However, it is not known that the C-approximation dimension of mod Λ being finite is necessary for TC to be cotilting. Consider the subfunctor F F X in C. Suppose T is an F -tilting F - cotilting module in C. In Section 5 we generalize the forementioned theorem from [19]. We show that the Γ-module TC is tilting and, that the tilting functor induces an equivalence between subcategories add { TC of C and add T C of mod Γ. Unless otherwise stated, throughout this paper Λ is a basic artin algebra and mod Λ denotes the category of all finitely generated left Λ-modules. Given a subcategory A of mod Λ, add A is the full subcategory of mod Λ containing all Λ-modules which are direct summands of finite direct sums of modules in A. Denote by D the duality between left and right modules as given in [5, II.3]. 1. Properties of homological finite subcategories In this section we recall some definitions from [6] and give some preliminary results. Among the results, we show that functorially finite subcategories C of mod Λ which are closed under extensions in mod Λ have enough Ext-projectives and Ext-injectives. Then we look at certain properties of covariantly and contravariantly finite subcategories of mod Λ which will be used, in the next section, to develop relative theory in subcategories. Let C be a subcategory of mod Λ. An exact sequence in C is an exact sequence in mod Λ with all the terms in C. A module Y in C is said to be Ext-injective if Ext 1 ΛpX, Y q for all X in C. We denote the subcategory of Ext-injective modules in C by IpCq. A subcategory C is said to have enough Ext-injectives if for all C is C there is an exact sequence Ñ C ÝÑ f I Ñ C 1 Ñ with I Ext-injective and C 1 in C. Note that if C has enough Ext-injectives and is closed under extensions in C, then any map g : C Ñ I 1 with I 1 in IpCq factors through f (i.e. there exists a map h: I Ñ I 1 such that g hf). The notions of Ext-projective module and enough Ext-projectives are defined dually. The subcategory of Extprojective modules in C is denoted by PpCq.

4 4 SOUD KHALIFA MOHAMMED Let D be a subcategory of mod Λ containing a subcategory C. Given a module M in D, a sequence Ñ Y Ñ C ÝÑ g M with C in C is said to be a right C-approximation of M if the sequence Ñ pc 1, Y q Ñ pc 1, Cq pc1,gq ÝÝÝÑ pc 1, Mq Ñ is exact in Ab for all C 1 in C. A right C-approximation is called a minimal right C-approximation if g is right minimal, that is, if every endomorphism s: C Ñ C satisfying g gs is an isomorphism. A minimal right C-approximation is unique up to isomorphism. A module has a right C-approximation if and only if it has a minimal right C- approximation [4]. We denote the minimal right C-approximation of M by g Ñ Y M Ñ C M M ÝÑ M. A subcategory of C of D is said to be contravariantly finite in D if every Λ-module in D has a right C-approximation. Dually, one defines the notions of left (minimal) C-approximation and covariantly finite subcategory of D. A subcategory C of D is said to be functorially finite in D if it is both contravariantly and covariantly finite in D. Let C be a contravariantly finite subcategory of mod Λ. Then by [6, Lemma 3.11] we have that C has a finite cocover, that is, there is some Y in add C such that C is contained in Sub Y, where Sub Y denotes the subcategory of mod Λ consisting of objects which are submodules of finite direct sums of copies of Y. Suppose C is closed under extensions in mod Λ. Then we have the following result which is an analog of [6, Lemma 3.11]. Proposition 1.1. Let C be a contravariantly finite subcategory of mod Λ which is closed under extensions. Then every X in C has an IpCq-coresolution. To prove Proposition 1.1 we need to show that the full subcategory E of mod Λ consisting of all Y such that Ext 1 ΛpX, Y q for all X in C is covariantly finite in mod Λ. To do this, we use the following proposition which is the dual of [4, Proposition 1.8]. Proposition 1.2. Suppose J is a subcategory of mod Λ which is closed under extensions such that Ext 1 Λp, Aq J is finitely generated for all A in mod Λ. Then the subcategory K ty P mod Λ Ext 1 ΛpJ, Y q u is covariantly finite in mod Λ. It is not difficult to see that when C is contravariantly finite in mod Λ, then Ext 1 Λp, Aq C is finitely generated for all A in mod Λ. Our subcategory C in Proposition 1.1 satisfies the conditions of Proposition 1.2. Hence the subcategory E is covariantly finite and contains the injective Λ-modules. Proof of Proposition 1.1. Let X be in C. Then we have a minimal left E- approximation Ñ X Ñ E X Ñ Z X Ñ of X, which is a monomorphism, since DΛ is in E. Then by [4, Corollary 1.7] we have that Z X is in C. Since C is closed under extensions, it implies that E X is in C X E IpCq. Then the result follows by induction.

5 RELATIVE THEORY IN SUBCATEGORIES 5 The following is a consequence of Propositions 1.1 and its dual. Corollary 1.3. Let C be functorially finite subcategory of mod Λ which is closed under extensions. Then (a) C has enough Ext-projectives and Ext-injectives. (b) The subcategory PpCq is contravariantly finite in C. (c) The subcategory IpCq is covariantly finite in C. We now want to find Ext-projective and Ext-injective modules in functorially finite subcategories. The following lemma is part (b) of [17, Lemma 2.1]. The result is a generalization of Wakamatsu s lemma [26]. Lemma 1.4. Let C be a contravariantly finite extensions-closed subcategory of mod Λ and let Z be a Λ-module. Then the natural transformation Ext 1 Λp, g Z q: Ext 1 Λp, C Z q C Ñ Ext 1 Λp, Zq C restricted to C is a monomorphism of contravariant functors. The following result, which is a consequence of [17, Theorem 3.4], gives us the Ext-injectives (the Ext-projectives are given dualy). Corollary 1.5. Let C be a contravariantly finite subcategory of mod Λ which is closed under extensions. Let Y be in mod Λ, and consider a succession of minimal right C-approximations Y 1 ãñ C Ñ Y, Y 2 ãñ C 1 Ñ Y 1,... Then for all i, C i is Ext-injective in C. Note that if Y I is an injective Λ-module, then C in Corollary 1.5 is Ext-injective in C [6, Lemma 3.5]. We recall the notions of a covariant and a contravariant defect of a short exact sequence [5]: Given a short exact sequence δ : Ñ L Ñ M Ñ N Ñ in mod Λ, the covariant defect δ and the contravariant defect δ of δ are the subfunctors of Ext 1 ΛpN, q and Ext 1 Λp, Lq respectively, defined by the exact sequences and Ñ Hom Λ pn, q Ñ Hom Λ pm, q Ñ Hom Λ pl, q Ñ δ Ñ Ñ Hom Λ p, Lq Ñ Hom Λ p, Mq Ñ Hom Λ p, Nq Ñ δ Ñ The next result given in [17], but we will give a different proof. Proposition 1.6. [17, Proposition 2.5(b)] Let C be a contravariantly finite subcategory of mod Λ which is closed under extensions. Let δ : Ñ L f ÝÑ M Ñ N Ñ be an exact sequence in C. For all Z in mod Λ, the morphism Hom Λ pl, g Z q: Hom Λ pl, Z C q Ñ Hom Λ pl, Zq induces an isomorphism δ pc Z q ÝÑ δ pzq. We have the following consequence of Proposition 1.6 which will be very useful for finding the relative injectives in subcategories in the next section.

6 6 SOUD KHALIFA MOHAMMED Corollary 1.7. Let Ñ A Ñ B Ñ C Ñ be exact in C, and let X be in mod Λ. Then the following are equivalent. (i) Hom Λ px, Bq Ñ Hom Λ px, Cq is an epimorphism. (ii) Hom Λ pb, C pdtr Xq q Ñ Hom Λ pa, C pdtr Xq q is an epimorphism. We recall the following definition from [9]. A subcategory X of C is said to be a generator for C if it contains PpCq. Dually one defines cogenerator subcategory for C. Lemma 1.8. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Let X be a contravariantly finite subcategory of C which is a generator for C. Consider a right X -approximation Ñ Y Ñ X g ÝÑ C Ñ of C in C. Then Y is in C. Proof. We know that C has enough Ext-projectives by Corollary 1.3. So, for any C in C, there is an exact sequence Ñ C 1 Ñ P ÝÑ p C Ñ with P in PpCq and C 1 in C. Therefore, we have the following exact commutative diagram C 1 C 1 Y Y ` P P Y X g C since g is a right X -approximation of C. But since C is closed under extensions and summands, it follows that Y is in C. 2. Subfunctors in Subcategories and their Properties Let C be a functorially finite subcategory of mod Λ which is closed under extensions. In this section we study subfunctors in C. We first recall some background on subfunctors in mod Λ from [9]. Then we study a special subfunctor F F X in C, where X is a contravariantly finite subcategory of C Background on Subfunctors. Let F be an additive sub-bifunctor of the additive bifunctor Ext 1 Λp, q: pmod Λq op mod Λ Ñ Ab, where pmod Λq op denotes the opposite category of mod Λ. Then F is said to be an additive subfunctor of Ext 1 Λp, q in mod Λ. A short exact sequence η : Ñ A Ñ B Ñ C Ñ is called an F -exact sequence if η is in F pc, Aq. Any pullback, pushout and Baer sum of F -exact sequences are again F - exact [9]. In particular, a subfunctor F determines a collection of short exact sequences which is closed under pushouts, pullbacks and Baer sums. p

7 RELATIVE THEORY IN SUBCATEGORIES 7 Conversely, given a collection of short exact sequences which is closed under pushouts, pullbacks and Baer sums, it gives rise to a subfunctor of Ext 1 Λp, q in the obvious way [9]. Let PpF q be a subcategory of mod Λ consisting of all Λ-modules P such that if Ñ A Ñ B Ñ C Ñ is F -exact, then the sequence Ñ pp, Aq Ñ pp, Bq Ñ pp, Cq Ñ is exact in Ab. The objects in PpF q are called projective modules of the subfunctor F or F -projectives. If PpΛq denotes the category of projective Λ-modules, then PpΛq is contained in PpF q. An additive subfunctor F is said to have enough projectives if for every A in mod Λ there exists an F -exact sequence Ñ A 1 Ñ P Ñ A Ñ with P in PpF q. The definitions of F -injectives and enough injectives are dual. Let Z be a subcategory of mod Λ. Define F Z pc, Aq t Ñ A Ñ B Ñ C Ñ pz, Bq Ñ pz, Cq Ñ is exactu for each pair of modules A and C in mod Λ. Dually one defines F Z pc, Aq t Ñ A Ñ B Ñ C Ñ pb, Zq Ñ pa, Zq Ñ is exactu for each pair of modules A and C in mod Λ. It is shown in [9, Proposition 1.7] that these constructions give (additive) subfunctors of Ext 1 Λp, q Subfunctors F in the Subcategory C. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Consider a subfunctor F in mod Λ. We want to look at this subfunctor when F - projectives and F -injectives are determined only by the F -exact sequences in C. In this case we say F is a subfunctor in C. Our aim is to study subfunctors F in C. First we want to find the subcategories of F -projectives and F -injectives in C. We denote these subcategories by P C pf q and I C pf q respectively. Let Ñ A Ñ B Ñ C Ñ be an exact sequence in C. Then by Corollary 1.7 we know that for all Z P mod Λ, the sequence pz, Bq Ñ pz, Cq Ñ is exact if and only if pb, C DTr Z q Ñ pa, C DTr Z q Ñ is exact. This gives the following proposition. Proposition 2.1. Let C be a functorially finite subcategory which is closed under extensions. Then (a) I C pf q C DTr PC pf q Y IpCq. (b) P C pf q C TrD I CpF q Y PpCq. Remark. Nothing can be said about the size of the subcategories P C pf q and I C pf q at the moment. But later we will see that if there exists an F -(co)tilting module in C, then P C pf q and I C pf q are of finite type. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. We now study some properties of the subfunctor F in C. A subfunctor F in C is said to have enough projectives if for each C in C

8 8 SOUD KHALIFA MOHAMMED there exists an F -exact sequence Ñ C 1 Ñ P Ñ C Ñ with P in P C pf q and C 1 in C. The notion of enough injectives is defined dually. Notation. Unless specified otherwise F denotes a subfunctor F X, where X is a generator subcategory of C. Consider the subfunctor F with enough projectives. Then the following proposition shows that C is closed under kernels of F -epimorphisms. Proposition 2.2. Let C be a functorially finite subcategory which is closed under extensions. Let F be a subfunctor in C with enough projectives in C. Then C is closed under kernels of F -epimorphisms. Proof. Let Ñ C 1 Ñ C 2 Ñ C 3 Ñ be an F -exact sequence with C 2, C 3 in C. Then, since F has enough projective in C, we have an exact sequence Ñ Y Ñ P ÝÑ C 3 Ñ with P P P C pf q and Y P C. From the following commutative diagram Y Y C 1 E P C 1 C 2 C 3 we have that E is in C. The exact sequence η 1 : Ñ C 1 Ñ E Ñ P Ñ is F -exact, and since P P P C pf q the sequence η 1 splits, so the claim follows. Now let F F X, and consider the subfunctor F I CpF q given by I C pf q. Let M be a Λ-module with a surjective C-approximation. Then we have g the F -exact sequence η : Ñ Y M ÝÑ CM Ñ M Ñ. If Y M is in C, then it is in I C pf q since IpCq is contained in I C pf q. Assume Y M is nonzero, then the identity map 1 YM does not factor through g. Therefore η is not F I CpF q -exact. Dually, given N in mod Λ, the exact sequence Ñ N Ñ C N Ñ Z N Ñ is not F -exact whenever Z N is a nonzero Λ-module in C. So outside C we may not have F F I CpF q. But inside C we have the following result. Corollary 2.3. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Then F C F I CpF q C. The following result shows that F has enough projectives and injectives under certain conditions. Proposition 2.4. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Then

9 RELATIVE THEORY IN SUBCATEGORIES 9 (a) If P C pf q is contravariantly finite in C, then F has enough projectives. (b) If I C pf q is covariantly finite in C, then F has enough injectives. Proof. (a) Follows from Lemma 1.8. (b) Suppose I C pf q is covariantly finite in C. Since I C pf q is a cogenerator for C, for each C in C there is, by the dual of Lemma 1.8, an exact sequence η : Ñ C Ñ I Ñ C 1 Ñ with I in I C pf q and C 1 in C, such that Ñ pc 1, I C pf qq Ñ pi, I C pf qq Ñ pc, I C pf qq Ñ is exact. Hence the sequence η is F I CpF q -exact. By Corollary 2.3 it follows that η is F -exact, since it is so in C. Thus F has enough injectives. Suppose I C pf q, where F F X, is covariantly finite in C. Then the following lemma, which is a dual of Lemma 2.2, shows that C is closed under cokernels of F I CpF q -monomorphisms. Proposition 2.5. Let Ñ C 1 Ñ C 2 Ñ C 3 Ñ be an F I CpF q -exact with C 1, C 2 in C. Assume I C pf q is covariantly finite in C. Then C 3 is in C. 3. Approximation Dimension Let C be a subcategory of mod Λ. In this section we define C-approximation dimension. Then we characterize subcategories C with C-approximation dimension equal to zero. Moreover, when we suppose that the C-approximation dimension of mod Λ is finite, then any long relative exact sequence in mod Λ with all the middle terms in C is eventually in C. This will be useful in the next section. Let C be a contravariantly finite subcategory of mod Λ. For any M in g mod Λ, consider a succession Ñ Y 1 Ñ C g ÝÑ M, Ñ Y2 Ñ C 1 1 ÝÑ Y1,... of minimal right C-approximations. Then, the complex g t g p q Ñ C t ÝÑ Ct 1 Ñ Ñ C 1 g 1 ÝÑ C ÝÑ M is called a right C-approximation resolution of M. In [16] this was defined in general for a contravariantly finite subcategory C in an additive category C 1 with kernels and cokernels. There, the right C-approximation resolution was called right C-resolution. Denote the Ker g i in p q by Y i 1. We write rc- app. dimpmq n if there exists a nonnegative integer n in a right C-approximation resolution of M such that Y n 1 and Y i for all i n. If no such integer exists, we write rc- app. dimpmq 8. We call rc-app. dimpmq the right C-approximation dimension of M. Then for mod Λ we define rc- app. dimpmod Λq suptrc- app. dimpmq M P mod Λu. Example 3.1. If C is closed under factor modules, then it is known that every right C-approximation is a monomorphism [6, Proposition 4.8]. Hence rc- app. dimpmod Λq.

10 1 SOUD KHALIFA MOHAMMED Dually, one can define left a C-approximation resolution of M, left C- approximation dimension of mod Λ, denoted by lc-app. dimpmod Λq, for a covariantly finite subcategory C of mod Λ. We have the following proposition relating the two approximation dimensions when C is of finite type [16, Corollary 1.1.2]. Proposition 3.2. Let C be a functorially finite subcategory of mod Λ. Then rc- app. dimpmod Λq is finite if and only if lc- app. dimpmod Λq is finite. Moreover, in this case they differ by at most 2. Let C be a functorially finite subcategory of mod Λ. The C-approximation dimension of mod Λ, C- app. dimpmod Λq, is defined to be C- app. dimpmod Λq maxtlc- app. dimpmod Λq, rc- app. dimpmod Λqu. The following is a nice corollary of Proposition 3.2. Corollary 3.3. Let C be a subcategory of mod Λ which is closed under factor modules. Then C- app. dimpmod Λq 2. Note: Let C be equal to mod Λ. Then C- app. dimpmod Λq. However, C- app. dimpmod Λq being zero does not necessarily mean that C mod Λ, as shown below. In general, A- app. dimpbq can be defined, where A is functorially finite subcategory of a category B with kernels and cokernels [16] Approximation Dimension Zero. In this section we want to characterize functorially finite subcategories C with C-approximation dimension zero. The following result shows that functorially finite subcategories with finite approximation dimension zero are the same as those which are closed under factor modules and submodules. Proposition 3.4. Let C be an additive functorially finite subcategory of mod Λ. Then C- app. dimpmod Λq if and only if C is closed under factor modules and submodules. Now we want to characterize subcategories of mod Λ closed under factor modules and submodules. But first we recall a well-known concept. Let C be a subcategory of mod Λ. Recall that the annihilator of C, ann Λ C, is equal to the intersection of the annihilator of the modules C P C, ann Λ pcq tλ P Λ λ C u. It is well-known that ann Λ C is an ideal of Λ. The following result shows that the subcategories of mod Λ which are closed under submodules and factor modules are abelian. Proposition 3.5. Let C be an additive subcategory of mod Λ which is closed under factor modules and submodules. Then C is equivalent to mod Λ{I, where I ann Λ C.

11 RELATIVE THEORY IN SUBCATEGORIES 11 Let C and I be as before and consider the algebra morphism ϕ: Λ Ñ Λ{I. Then ϕ induces an exact functor G ϕ : modpλ{iq Ñ mod Λ, which is an embedding. We have that Im G ϕ C. It is easy to see that G ϕ and its inverse preserve exact sequences and exact diagrams. Hence they preserve pushouts, pullbacks and Baer sums. Since these (pushouts pullbacks and Baer sums) determine subfunctors, it follows that G ϕ and its inverse preserve subfunctors too. Hence C and modpλ{iq have the same relative theory. Note that the factor category mod Λ{I, in Proposition 3.5, is not necessarily closed under extensions in mod Λ [3]. However, if C is closed under extensions, then mod Λ{I is also closed under extensions in mod Λ (by using the functor G ϕ above). Now, we combine Proposition 3.4 and 3.5 to get the following crucial result for subcategories C with C- app. dimpmod Λq. Corollary 3.6. Let C be an additive functorially finite subcategory of mod Λ which is closed under extensions. Assume the C- app. dimpmod Λq is zero. Then C is canonically equivalent to mod Σ, where Σ is a quotient algebra of Λ. Moreover, mod Σ inherits the relative theory in C and vice versa Approximation Dimension n. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Let X be a contravariantly finite generator subcategory of C. Consider the subfunctor F F X in C. In this subsection we look at some relationship between C and mod Λ which will be useful later. We show that any long F -exact sequence in mod Λ with the middle terms in C is eventually in C. The following lemma is very important. Lemma 3.7. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Consider a minimal right C-approximation g i s 1 g i 1 g resolution Ñ C i s 1 ÝÝÝÑ C i s Ñ Ñ C i 1 ÝÝÑ i Ci ÝÑ Mi of M i for some i. Denote Ker g i j by Y i j 1 for j and let M i Y i. Let Ñ M i j 1 Ñ T i j Ñ M i j Ñ be an F -exact sequence with T i j in C for j. Then there is a right C-approximation Ñ Y 1 i j 1 Ñ C 1 i j Ñ M i j with Y i j 1 Y 1 i j 1 for j. Proof. We prove this by induction on j. For j, we have M i Y i, so Y i 1 Y 1 i 1.

12 12 SOUD KHALIFA MOHAMMED For j 1, consider the following commutative F -exact diagram M i 1 M i 1 Y i 1 Y i 1 ` T i θ 1 : Y i 1 C i α T i M i and let X ÝÑ p C i be an epimorphism with X in X. Since Ñ M i 1 Ñ Y i 1 ` T α i ÝÑ C i Ñ is F -exact, we have that p factors through α. Moreover, since η : Ñ Y i 2 Ñ C i 1 ` T i ÝÝÝÝÝÝÑ Y i 1 ` T i is a right C- p g i 1 1 Ti q approximation of Y i 1 `T i, we have that p factors through f α p g i 1 Ti q. Hence f is onto, since p is onto. Then we use the F -exact sequence Ñ M i 1 Ñ Y i 1 ` T α i ÝÑ C i Ñ to construct the following commutative diagram Y i 2 Y i 2 C 1 i 1 g 1 i 1 C i 1 ` T i f p g i 1 Ti q M δ i 1 α Y i 1 ` T i N N C i C i By the earlier discussion, we have that the exact sequence Ñ C 1 i 1 Ñ C i 1 ` T i f ÝÑ Ci Ñ is F -exact. Then by Proposition 2.2, C 1 i 1 is in C. g 1 i 1 ÝÝÑ M i 1 is a right C- Our aim is to show that θ 2 : Ñ Y i 2 Ñ Ci 1 1 approximation of M i 1. If Ci 1 1 were a pullback of δ and p g i 1 Ti q, then by the universal property of pullbacks, θ 2 would be a right C-approximation, since η is a right C-approximation of Y i 1 ` T i. But it can be shown that, C 1 i 1 is indeed a pullback of δ and p g i 1 Ti q. Hence the sequence θ 2 is a right C-approximation, and we have Y 1 i 2 Y i 2. For j 1 we replace the sequence θ 1 in the first diagram by θ j and continue from there. Then the result will follow by induction.

13 RELATIVE THEORY IN SUBCATEGORIES 13 The following result, which is a consequence of Lemma 3.7, shows that any long F -exact sequence in mod Λ with the middle terms in C is eventually in C. This will be very useful in the next section. Corollary 3.8. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Assume C- app. dimpmod Λq n 8. Fix an integer t, and let Ñ M i 1 Ñ T i Ñ M i Ñ be F -exact in mod Λ with T i in C for all i t. Then M t n is in C. In general, M i is in C for all i t n. Proof. By Lemma 3.7 we have the following commutative exact diagram C 1 t n C i n ` T t n 1 C 1 t n 1 g 1 t n M t n Y t n ` T t n 1 C 1 t n 1 where gt 1 n is a right C-approximation of M t n. Since T t n maps onto M t n, we have that gt 1 n is an epimorphism, and hence an isomorphism. Therefore M t n is in C. Then by Lemma 2.2 M i is in C for all i t n. 4. relative theory, approximation and global dimension Throughout this section C is a functorially finite extension-closed subcategory of mod Λ and X is a contravariantly finite generator subcategory of C. Consider the subfunctor F F X in C. In this section we investigate a relative (co)tilting theory in C. Suppose T is an F -tilting module in C and let Γ End Λ pt q op. In 4.1 we show that the tilting functor Hom Λ pt, q induces an equivalence between the subcategories TC K of C and pt, T C Kq of mod Γ. Then we show that pp C pf q, T q is a tilting Γ op -module and we then use this to show that P C pf q is of finite type. In 4.2 we show that the image of the tilting functor restricted to TC K, pt, T C K q, is identified with the category K pt, I C pf qq. Moreover, we show that the Γ-module pt, I C pf qq is cotilting. In 4.3 we look at the relationship between the relative global dimension of C and the global dimension of Γ Relative Tilting in Subcategories. Consider the subfunctor F F X in C. We know that F has enough projectives in C (since P C pf q X ). Suppose I C pf q is covariantly finite in C. Then by Corollary 2.4 we have that F has enough injectives in C. So, from now on we assume that I C pf q is covariantly finite in C. First we define the concept of F -tilting in C. Definition. A Λ-module T is called F -tilting in C if (i) T is in C.

14 14 SOUD KHALIFA MOHAMMED (ii) Ext i F pt, T q for all i. (iii) pd F T 8. (iv) For all P in P C pf q there is an F -exact sequence Ñ P Ñ T Ñ T 1 Ñ Ñ T s Ñ with T i in add T. An F -cotilting module in C is defined dually. Let ω be a subcategory of mod Λ, then ω is said to be F -selforthogonal if Ext i F pω, ωq for all i. Let T be an F -selforthogonal Λ-module in C. Define T K to be the full subcategory of mod Λ consisting of all modules Y with Ext i F pt, Y q for all i. It has been shown in [1] that T K is F -coresolving in mod Λ. Denote T K X C by TC K. We then denote by YC T the full subcategory of all Λ-modules A in TC K such that there is an F -exact sequence f s f Ñ T s ÝÑ Ts 1 Ñ Ñ T 1 1 ÝÑ T Ñ A Ñ with T i in add T and Im f i in T K C. A subcategory J of C is said to be closed under F -extensions in C if for each F -exact sequence Ñ A Ñ B Ñ C Ñ in C with A and C in J, we have that B is in J. Then we have the following result which is a generalization of [4, Dual of Proposition 5.1]. Proposition 4.1. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. For an F -selforthogonal Λ-module T in C the subcategory YT C is closed under (a) F -extensions. (b) cokernels of F -monomorphisms. (c) direct summands. A subcategory Z of C is said to be F -resolving in C if it satisfies the conditions (a) it is closed under F -extensions, (b) if Ñ A Ñ B Ñ C Ñ is F -exact and B and C are in Z, then A is in Z and (c) it contains P C pf q. Dually, one defines F -coresolving in C Let Y be F -covariantly finite F -coresolving in C. Then the F -coresolution dimension of a Λ-module C with respect to Y is defined to be the minimum of all n including infinity such that there exists an F -exact sequence Ñ C Ñ Y Ñ Y 1 Ñ Ñ Y n 1 Ñ Y n Ñ where the Y i are in Y. We denote this dimension by Y-coresdim F M. If W is a subcategory of mod Λ, then Y- coresdim F pwq is defined to be supty- coresdim F Z Z P Wu. When our F -selforthogonal module T is F -tilting in C we have the following result, which is a generalization of [1, Dual of Theorem 3.2]. Denote add T X C by add TC.

15 RELATIVE THEORY IN SUBCATEGORIES 15 Proposition 4.2. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Let T be an F -tilting module in C. Then we have the following. (a) The subcategory YT C T C K is F -coresolving covariantly finite in C with YT C-coresdim F C finite. (b) The subcategory add TC K pyt C q X C is F -resolving contravariantly finite in C with pd F add TC finite. Proof. Here the proof is similar to [1, Dual of Theorem 3.2]. The only challenge is to get some of the modules involved in the proof into C. We do that by using Proposition 2.2. We restate [22, Lemma 2.2] for the relative theory in subcategories. The proof is similar, so it will not be given. We denote { add T X C by { add TC. Lemma 4.3. Let T be an F -tilting module in C. Then TC K X P C 8 pf q {add T C. Next we show that the tilting functor is fully faithful on the category Y C T. Let T be in C and Γ End Λ pt q op. Consider the tilting functor Hom Λ pt, q: mod Λ Ñ mod Γ. Then we have the following lemma which is an analog of [1, Dual of Lemma 3.3]. Lemma 4.4. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. If T is an F -tilting Λ-module in C, then the functor Hom Λ pt, q: YT C Ñ mod Γ is an F -exact fully faithful covariant functor. The following is a consequence of Lemma 4.4. Corollary 4.5. Let T be an F -tilting module in C and Γ End Λ pt q op. Then Hom Λ pt, q: Ext i F py, Y 1 q Ñ Ext i ΓppT, Y q, pt, Y 1 qq is an isomorphism for all Y and Y 1 in YT C functorial in both variables. Let T be a tilting module in mod Λ, Γ End Λ pt q op and DT the corresponding cotilting Γ-module. It is well known that the tilting functor pt, q: mod Λ Ñ mod Γ induces an equivalence between the categories T K ( Y T by the dual of [4, Theorem 5.4]) of mod Λ and pt, T K q of mod Γ, where the image pt, T K q is identified with the subcategory K DT. This was also established for relative tilting modules in mod Λ [1]. Let F be a subfunctor in mod Λ. Let T be an F -tilting module in mod Λ and denote End Λ pt q op by Γ. Then it can be shown (by using duality in [1]) that the tilting functor induces the same equivalence as in

16 16 SOUD KHALIFA MOHAMMED the standard case. But this time the image pt, T K q is identified with the category K pt, IpF qq, where pt, IpF qq is a cotilting Γ-module. Our aim is to show that this (in the above discussion) also holds for relative tilting modules T in subcategories. In the present subsection we prove the existence of an equivalence between the subcategory YT C of C and its image pt, YT C q in mod Γ. Assume that the C- app. dimpmod Λq is finite. In 4.2 we identify the subcategory which corresponds to the image pt, YT Cq of pt, q. Let T be an F -tilting Λ-module in C and Γ End Λ pt q op. We have seen that YT C T C K. Since Hom ΛpT, q: YT C Ñ mod Γ is a fully faithful functor by Lemma 4.4, we have that DY Hom Λ py, DΛq Hom Γ ppt, Y q, pt, DΛqq Hom Γ ppt, Y q, DT q for all Y in YT C. Applying the duality D to the above isomorphism we get the isomorphism Y D Hom Γ ppt, Y q, DT q T b Γ Hom Λ pt, Y q. Hence YT C T b Γ pt, YT Cq. Therefore YC T is equivalent to pt, YC T q in mod Γ. The following result, which summarizes the above discussion, shows that there is an equivalence between subcategories YT C of C and pt, YC T q of mod Γ. This is a generalization of the dual of [1, Corollary 3.6]. Theorem 4.6. Let C be a functorially finite subcategory of mod Λ which is closed under extensions. Let T be an F -tilting module in C and Γ End Λ pt q op. (a) The functor Hom Λ pt, q: C Ñ mod Γ induces an equivalence between Y C T and pt, YC T q. (b) The functor Hom Λ pt, q: C Ñ mod Γ induces an equivalence between I C pf q and pt, I C pf qq. If T is a standard tilting Λ-module, then we have that the Γ-modules pt, DΛ Λ q and DpΛ, T q coincide. But for relative tilting modules this is not always the case. We want to show that the Γ op -module pp C pf q, T q is a tilting Γ op -module. This will imply that the module DpP C pf q, T q is a cotilting Γ-module by duality. But first we need the following results. Lemma 4.7. For all W in add TC and all C in mod Λ the homomorphism Hom Λ p, T q: pc, W q Ñ Γ opppw, T q, pc, T qq is an isomorphism functorial in both variables. The following is a consequence of the above result, where the proof is similar to that of [1, Proposition 3.7]. Corollary 4.8. For W in add TC and C in K T C the homomorphism Hom Λ p, T q: Ext i F pc, W q Ñ Ext i ΓopppW, T q, pc, T qq for all i is an isomorphism functorial in both variables.

17 RELATIVE THEORY IN SUBCATEGORIES 17 Now we show that pp C pf q, T q is a tilting Γ op -module. Proposition 4.9. Let C be a subcategory of mod Λ which is closed under extensions. Let T be an F -tilting Λ-module in C with pd F T r. Denote End Λ pt q op by Γ. Then pp C pf q, T q is a tilting Γ op -module. Moreover, pp C pf q, T q is of finite type. Proof. Since P C pf q add TC K T C, we have Ext i F pp C pf q, P C pf qq Ext i Γ opppp CpF q, T q, pp C pf q, T qq for all i. Hence pp C pf q, T q is selforthogonal. Since T is F -tilting we infer that pd Γ oppp C pf q, T q is finite. Since pd F T is finite it is not difficult to see that Γ op is in addppc pf q, T q. Therefore pp C pf q, T q is a tilting Γ op -module. By the corollary to [21, Proposition 1.18] we have that, for all P in P C pf q, the module pp, T q is a direct summand of add rà i pp i, T q, where the P i are in P C pf q. Hence pp C pf q, T q is of finite type. Now we want to show that P C pf q is of finite type whenever there is an F -tilting module in C. But we need the following result which is an analog of [1, Proposition 5.4]. Lemma 4.1. Consider the functor Hom Λ p, T q: mod Λ Ñ mod Γ. Then (a) Hom Λ p, T q induces a duality between add TC and p add TC, T q. (b) Hom Λ p, T q induces a duality between P C pf q and pp C pf q, T q. The following result is a consequence of Proposition 4.9. Corollary The subcategory P C pf q is of finite type Relative Tilting and Finite Approximation Dimension. Consider the subfunctor F F X in C. Suppose T is an F -tilting module in C and let Γ End Λ pt q op. In this section we show that the image of the equivalence given in the previous section, namely pt, YT C q is identified with the subcategory K pt, I C pf qq. Moreover, we show that the Γ-module pt, I C pf qq is cotilting. Let C be a functorially finite subcategory of mod Λ which is closed under extensions and assume the C-approximation dimension of mod Λ is zero. Then, by Corollary 3.6, we have that C is canonically equivalent to mod Σ, where Σ is a quotient algebra of Λ. Moreover, we have that C and mod Σ have the same relative theory. Let T be an F -tilting module in C and denote End Λ pt q op by Γ. Then by the duals of [1, Proposition 3.8] and [1, Theorem 3.13] we have that pt, Y C T q K pt, I C pf qq and pt, I C pf qq is a cotilting Γ-module.

18 18 SOUD KHALIFA MOHAMMED For C- app. dimpmod Λq 8, we give examples which show that pt, I C pf qq is not always a cotilting Γ-module. Now assume that the C-approximation of mod Λ is greater than zero, but finite. Let T be an F -tilting module in C and denote End Λ pt q op by Γ. We want to show that the subcategory pt, Y C T q K pt, I C pf qq and pt, I C pf qq is a cotilting Γ-module. But first we need several preliminary results. The following result is an analog of [1, Dual of Lemma 2.9]. Lemma Let C be a functorially finite extensions-closed subcategory of mod Λ. Let T be an F -tilting module in C and let Γ End Λ pt q op. Then, the map Ψ: Hom Λ pw, T q b Γ Hom Λ pt, Y q Ñ Hom Λ pw, Y q given by ψpf b gq g f is an isomorphism for all W in add TC and Y in Y C T and is functorial in both variables. The following result is an analog of [1, Dual of Lemma 3.1]. Lemma Let C be a functorially finite subcategory of mod Λ which is closed under extensions. If T is F -tilting in C, then id Γ Dp add TC, T q pd F T, where Γ End Λ pt q op. In particular, id Γ DpPpCq, T q pd F T. We have the following nice corollary. Corollary Let C be a functorially finite subcategory of mod Λ and assume that C- app. dimpmod Λq n 8. Let T be an F -tilting module in C with pd F T r and let Γ End Λ pt q op. Then id Γ DT r n. Proof. We prove this by induction on n. For n, see Corollary 3.6 and the dual of [1, Lemma 3.1]. For n 1, we have a left C-approximation resolution (presentation) Λ ÝÑ f C f ÝÑ 1 C 1 Ñ of Λ. By the dual of Corollary 1.5 we have that C and C 1 are in PpCq. Applying Dp, T q to the sequence we get the exact sequence Ñ DpΛ, T q Ñ DpC, T q Ñ DpC 1, T q Ñ. By Lemma 4.13 we have that id Γ DpC i, T q r for i, 1. Hence, by [21, Lemma 2.1] (see also [24]) we have that id Γ DT r 1. Now suppose that n 1. Then we have a left C-approximation resolution Λ ÝÑ f C ÝÑ f 1 C 1 Ñ Ñ C n Ñ of Λ. Applying Dp, T q to the sequence we get the exact sequence Ñ DT Ñ DpC, T q Ñ DpC 1, T q Ñ Ñ DpC n, T q Ñ. Denote Ker Dpf i, T q by L i. Then by induction we have that id Γ L 1 r n 1. Again by [21, Lemma 2.1] it follows that id Γ DT r n. The following lemma will be very useful. Lemma Let C be a functorially finite subcategory of mod Λ which is closed under extensions and assume C- app. dimpmod Λq n 8. Let T be an F -tilting module in C with pd F T r. Let M be a Λ-module and consider a succession M 1 ãñ T Ñ M, M 2 ãñ T 1 Ñ M 1,... of minimal

19 RELATIVE THEORY IN SUBCATEGORIES 19 right add T -approximations. Then Ñ M i 1 Ñ T i Ñ M i Ñ is F -exact for i r n 1. Proof. Denote End Λ pt q op by Γ. From the complex Ñ T 2 Ñ T 1 Ñ T Ñ M we get a minimal projective resolution Ñ pt, T 1 q Ñ pt, T q Ñ pt, Mq Ñ of pt, Mq over Γ. We have that Ext j Γ ppt, M iq, Dp add TC, T qq for all j and i r, by Lemma So if one applies the functor Hom Γ p, DpW, T qq, for W P add TC, to the sequence Ñ pt, T r 1 q Ñ Ñ pt, T r q Ñ pt, M r q Ñ it remains exact. Let W P add TC. Then we have the following commutative diagram by the adjoint isomorphism and Lemma 4.12 ppt, M r q, DpW, T qq ppt, T r q, DpW, T qq ppt, T r 1 q, DpW, T qq DppW, T q b Γ pt, M r qq DppW, T q b Γ pt, T r qq DppW, T q b Γ pt, T r 1 qq DppW, M r qq DppW, T r qq DppW, T r 1 qq Since the middle row in the above diagram is exact, we have that the sequence (1) Ñ pw, M i 1 q Ñ pw, T i q Ñ pw, M i q Ñ is exact for i r P C pf q add TC. 1. In particular, (1) is exact for Q P P C pf q, since Now, since C-app. dimpmod Λq n, we have for any P P PpΛq a minimal left C-approximation resolution P ÝÑ f C ÝÑ f 1 C 1 Ñ Ñ C l 1 ÝÑ f l C l Ñ with l n. Denote Coker f i 1 by Z i for i l. Note that by the dual of Corollary 1.5 the C i are in P C pf q for i n. We want to show that the sequence Ñ pp, M i 1 q Ñ pp, T i q Ñ pp, M i q Ñ is exact for all i r n 1 by using induction on n. For n, it follows from Corollary 3.6 and the dual of [1, Propostion 3.8]. For n 1, we combine (1) and the resolution of P to get the following exact sequence of complexes.. pc 1, T r 2 q pc, T r 2 q pc 1, T r 1 q pc, T r 1 q pc 1, M r 1 q pc, M r 1 q. pp, T r 2 q pp, T r 1 q pp, M r 1 q

20 2 SOUD KHALIFA MOHAMMED By the long exact sequence (of complexes) [24], we have that the sequence Ñ pp, M i 1 q Ñ pp, T i q Ñ pp, M i q Ñ is exact for all i r 2. Therefore the sequence Ñ M i 1 Ñ T i Ñ M i Ñ is exact for i r 2. Then by (1), the sequence is F -exact. Suppose n 1. By induction and using (1) and the resolution of P, we get that the sequence Ñ pz n k, M i 1 q Ñ pz n k, T i q Ñ pz n k, M i q Ñ is exact for i r 1 k and k n. In particular, for k n, we get that the sequence Ñ M i 1 Ñ T i Ñ M i Ñ is exact for i r n 1. Then by (1) it is F -exact. Remark. Let B be in mod Γ and consider a projective resolution of B. Then the Γ-module Ω j ΓpBq has a preimage in mod Λ for j 2. However pbq does not necessarily has a preimage in mod Λ. Ω 1 Γ Now we show that pt, Y C T q K pt, I C pf qq for a functorially finite subcategory C of mod Λ which is closed under extensions and with the property that C- app. dimpmod Λq is finite. This result is a generalization of [1, Dual of Proposition 3.8]. Proposition Let C be a functorially finite extensions-closed subcategory of mod Λ and assume C- app. dimpmod Λq n 8. Let T be an F -tilting module in C with pd F T r and let Γ End Λ pt q op. Then, Ext i ΓpB, pt, I C pf qqq for all i if and only if B P Hom Λ pt, YT Cq. Proof. We have Ext i F py, I C pf qq Ext i ΓppT, Y q, pt, I C pf qqq for Y P Y C T, by Corollary 4.5. So pt, Y q B P K pt, I C pf qq. Conversely, let B be a Γ-module such that Ext i ΓpB, pt, I C pf qqq for i. Let Hom Λ pt, T 1 q pt,f 1q ÝÝÝÑ Hom Λ pt, T q Ñ B Ñ be a minimal projective presentation of B. By Lemma 4.4 the above sequence is induced f by T 1 1 ÝÑ T. Denote Ker f 1 by M 2. Let Ñ M 3 Ñ T 2 Ñ M 2, Ñ M 4 Ñ T 3 Ñ M 3,... be a succession of minimal left add T -approximations. Then f we get a complex Ñ T 4 f 4 ÝÑ 3 T3 ÝÑ T2 Ñ M 2 and the exact sequence (2) Ñ pt, T s q Ñ pt, T s 1 q Ñ Ñ pt, T 1 q Ñ pt, T q Ñ B Ñ is a minimal projective resolution of B over Γ. Denote Ω 1 Γ pbq by B 1. Applying Hom Γ p, pt, Iqq, with I P I C pf q, to the resolution of B, we get the following exact commutative diagram ΓpB, pt, Iqq Hom Λ pt b Γ B, Iq ΓppT, T q, pt, Iqq Hom Λ pt, Iq ΓppT, T 1 q, pt, Iqq Hom Λ pt 1, Iq by Lemma 4.4 and the adjoint isomorphism. The cohomology of the upper row is Ext i ΓpB, pt, I C pf qq for i. So the sequence (3) Ñ pt b Γ B, Iq Ñ pt, Iq Ñ Ñ pt r, Iq Ñ pt r 1, Iq Ñ is exact.

21 RELATIVE THEORY IN SUBCATEGORIES 21 On the other hand, since C-app. dimpipλqq n, we have, for all I P g IpΛq, a minimal right C-approximation resolution Ñ C l g l ÝÑ Ñ 1 C1 ÝÑ g C ÝÑ I with l n. Denote Ker gi by Y i 1 for i n. By Corollary 1.5 the modules C i are in IpF q for i n. Then by the adjoint isomorphism, we have the following commutative diagram pt b Γ B, C l q pt b Γ B, C q pt b Γ B, Iq pb, pt, C l qq pb, pt, C qq pb, pt, Iqq Ext 1 ΓpB, pt, Y 1 qq with l n. We then have that Ext 1 ΓpB, pt, Y 1 qq Ext n ΓpB, pt, C n qq since C n P I C pf q. So the top row in the above diagram is exact. Now, combining (3) and the resolution of I we get the following exact sequence of complexes pt b Γ B, C l q pt b Γ B, C q pt b Γ B, Iq pt, C l q pt, C q pt, Iq pt 1, C l q pt 1, C q pt 1, Iq. with l n. By the long exact sequence (of complexes) [24], we have that the sequence Ñ pt b Γ B, Iq Ñ pt, Iq Ñ Ñ pt r, Iq Ñ is exact for all I P IpΛq. Hence (4) Ñ M r 2n Ñ T r 2n 1 Ñ Ñ T Ñ T b Γ B Ñ is exact. By Lemma 4.15 we have that Ñ M i 1 Ñ T i Ñ M i Ñ is F -exact for all i r n 1. Then using Corollary 3.8 we get that M i P C for i r 2n 1. But, then by (3) we have that (4) is F I CpF q -exact. Hence by Proposition 2.5, M i for 2 i r 2n 1, T b Γ B 1 and T b Γ B are in C. But since F X C F I CpF q C by Corollary 2.3, we have that (4) is F -exact. We deduce from (2) and (4) that Ext 1 F pt, M i q for 2 i r 2n 1. The F -exact sequence Ñ M i 1 Ñ T i Ñ M i Ñ gives Ext j 1 F pt, M i 1q Ext j F pt, M iq for j and 2 i r 2n 1. By dimension shift, we have that Ext j F pt, M r 2n 1q for j r 1. Since pd F T r, it follows that M r 2n 1 P YT C T C K. By Proposition 4.2, the subcategory YT C is F -coresolving, hence, by using the fact that (4) is F -exact we have that T b Γ B, T b Γ B 1 and M i, for i 2,..., r 2n 1,..

22 22 SOUD KHALIFA MOHAMMED are in Y C T. Let V Ext1 F pt, T b Γ B 1 q. Then, from the commutative exact diagram pt, T 2 q pt, T 1 q pt, T q pt, T b Γ Bq V pt, T 2 q pt, T 1 q pt, T q B we have pt, T b Γ Bq B, since V. Therefore B is in pt, YT C q and the result follows. Remark. Note that C- app. dimpmod Λq being finite is sufficient but not necessary for the equality pt, Y C T q K pt, I C pf qq as illustrated below. Example Let Λ be an algebra given by the quiver α 1 β 1 β 2 2 with radical square-zero relations. Denote by P i, I i and S i the indecomposable projective, injective and simple Λ-modules corresponding to the vertex i (the notations are fixed throughout the paper). Let C FpΘq where Θ tp 1 {S 2, P 2 u. Note that C is closed under summands, so it is closed under extensions by [23]. C is functorially finite since it is of finite type. A right C-approximation resolution of S 1 is Ñ P 1 {S 2 Ñ P 1 {S 2 Ñ S 1 Ñ, then by Proposition 3.2 we have C- app. dimpmod Λq 8. We have PpCq IpCq C. Let F F PpCq. Then the only F -tilting module up to isomorphism is T P 1 {S 2 ` P 2. Let Γ End Λ pt q op and denote by Q i and J i the projective and injective Γ-module corresponding to the vertex i (the notations are fixed throughout the paper). It can be shown that pt, YT Cq pt, Cq K pt, I C pf qq. Next we want to show that pt, I C pf qq is a standard cotilting Γ-module. The following result will help us to achieve our goal. The result also shows that the pt, YT C q- coresdimpmod Γq is finite when C is a functorially finite subcategory of mod Λ which is closed under extensions and with the property that C- app. dimpmod Λq is finite. This result is a generalization of [1, Proposition 3.11]. Proposition Let C be a functorially finite subcategory of mod Λ which is closed under extensions and assume C- app. dimpmod Λq n 8. Let T be an F -tilting module in C with pd F T r and let Γ End Λ pt q op. Then pt, { Y C T q mod Γ and 2 n r pt, YT C q- resdimpmod Γq νpn, rq 3 2n r 1 r 2n 1 r 2

23 RELATIVE THEORY IN SUBCATEGORIES 23 Proof. Let pt, T 1 q Ñ pt, T 2 q Ñ B Ñ be a minimal projective presentation of B in mod Γ. By Lemma 4.4 the presentation is induced by T 1 f ÝÑ T 2. Denote Ker f by M, then we have that Ω 2 Γ pbq pt, M q. For r, we have that T P C pf q, so that YT C C. From the right C-approximation resolution of M, we have the sequence C l C 1 f 1 Y 1 f C T 1 T 2 f M with l n, since C-app. dimpmod Λq n. We then have the exact sequence Ñ pt, C l q Ñ Ñ pt, C q Ñ pt, T 1 q Ñ pt, T 2 q Ñ B Ñ. But since Y { T C C, it follows that pt, Y C T q mod Γ and pt, Y C T q- resdimpmod Γq 2 n. For r, let Ñ M 1 Ñ T Ñ M, Ñ M 2 Ñ T 1 Ñ M 1,... be a succession of minimal right add T -approximations. Then we get a complex Ñ T 2 Ñ T 1 Ñ T Ñ M and the exact sequence Ñ pt, T 1 q Ñ pt, T q Ñ pt, T 1 q Ñ pt, T 2 q Ñ B Ñ is a minimal projective resolution of B in mod Γ. Assume that r 2. Since C- app. dimpmod Λq n, it follows, by Lemma 4.15 that the sequence Ñ M i 1 Ñ T i Ñ M i Ñ is F -exact for all i r n 1. Then by Corollary 3.8, we have that M i P C for i r 2n 1. Moreover, by (1) in the proof of Lemma 4.15, we have that Ext 1 F p add TC, M i q for i r 2n 1. Using the fact that Ñ M i 1 Ñ T i Ñ M i Ñ is F -exact for i r 2n 1 and add TC K T, we have that Ext j F p add TC, M i q Ext j 1 F p add TC, M i 1 q for j and i r 2n 1. By dimension shift we have Ext i F p add TC, M 2r 2n 1 q for i r 1. Since add TC P r pf q we have that M 2r 2n 1 P p add TC q K YT C. But since YC T is F -coresolving and Ñ M i 1 Ñ T i Ñ M i Ñ is F -exact for i r 2n, we have that M i P YT C for r 2n 1 i 2r 2n 1. Hence pt, M r 2n 1 q Ω r 2n 1 Γ pbq P pt, YT Cq. Therefore pt, YT C q- resdimpmod Γq r 2n 1. If r 1, the proof of the case r 2 plus the remark after Lemma 4.15 can be used to show that M 2n 1 P YT C. Hence pt, M 2n 1q Ω 3 2n Γ pbq P pt, YT C q and we have that pt, YT C q- resdimpmod Γq 3 2n. Remark. C- app. dimpmod Λq being finite is sufficient for pt, { Y C T q mod Γ, but it is not known if the assumption is necessary. We are now in position to show that Hom Λ pt, I C pf qq is a cotilting module in mod Γ when C is a functorially finite subcategory of mod Λ