Tilting theory and functor categories I. Classical tilting

arXiv:1110.4785v1 [math.RT] 21 Oct 2011

TILTING THEORY AND FUNCTOR CATEGORIES I. CLASSICAL TILTING R. MART´INEZ-VILLA AND M. ORTIZ-MORALES Abstract. Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel [Ha] proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard [Ri] to develop a general Morita theory of derived categories. In the other hand, functor categories were introduced in representation theory by M. Auslander [A], [AQM] and used in his proof of the first BrauerThrall conjecture [A2] and later on, used systematically in his joint work with I. Reiten on stable equivalence [AR], [AR2] and many other applications. Recently, functor categories were used in [MVS3] to study the AuslanderReiten components of finite dimensional algebras. The aim of the paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod(modΛ ), with Λ a finite dimensional algebra.

1. Introduction and basic results Tilting theory traces back his history to the article by Bernstein, Gelfand and Ponomarev ([BGP] 1973), where they defined partial Coxeter functors. These functors were generalized by Auslander Platzeck and Reiten ([APR] 1979), but a major brake through was the article by Brenner and Butler ([BB] 1979). The notion of tilting was further generalized by Miyashita ([Mi] 1986) and Happel ([Ha] 1987). Happel showed that generalized tilting induces derived equivalence between the corresponding module categories. These results inspired Rickard [Ri] to develop his Morita theory of derived categories. This paper is the first one in a series of articles in which, having in mind applications to functor categories from subcategories of modules over a finite dimensional algebra to the category of abelian groups, we generalize tilting theory from modules to functor categories. The first paper is dedicated to classical tilting and it consists of four sections: In the first section we fix the notation and recall some notions from functor categories that will be used through the paper. In the second section, we generalize Date: December 15, 2010. 1991 Mathematics Subject Classification. 2000]Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. Classical Tilting, Functor Categories. The second author thanks CONACYT for giving him financial support during his graduate studies. This paper is in final form and no version of it will be submitted for publication elsewhere. 1

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R. MART´INEZ-VILLA AND M. ORTIZ-MORALES

Bongartz´s proof [B] of Brenner-Butler´s theorem [BB] to arbitrary functor categories. In the third section we deal with two special cases: first we extend tilting from subcategories to full categories, and second we apply the theory so far developed, to study tilting for infinite quivers with no relations, and apply it to compute the Auslander-Reiten components of infinite Dynkin quivers. In the last section we recall results from [MVS1], [MVS2], [MVS3] on graded and Koszul categories and we apply our results from the third section, to compute the Auslander-Reiten components of the categories of Koszul functors over a regular Auslander-Reiten component of a finite dimensional algebra. These results generalize previous work by [MV] on the preprojective algebra. 1.1. The Category Mod(C). In this section C will be an arbitrary skeletally small pre additive category, and Mod(C) will denote the category of contravariant functors from C to the category of abelian groups. Following the approach by Mitchel [M], we can think of C as a ring ”with several objects” and Mod(C) as a category of C-modules. The aim of the paper is to show that the notions of tilting theory can be extended to Mod(C), to obtain generalizations of the main theorems on tilting for rings. To obtain generalizations of some tilting theorems for finite dimensional algebras, we need to add restrictions on our category C, like: existence of pseudo kernels, Krull-Schmidt, Hom-finite, dualizing, etc.. To fix the notation, we recall known results on functors and categories that we use through the paper, referring for the proofs to the papers by Auslander and Reiten [A], [AQM], [AR]. 1.2. Functor Categories. Let C be a pre additive skeletally small category. By Mod(C) we denote the category of additive contravariant functors from C to the category of abelian groups. Then, Mod(C) is an abelian category with arbitrary sums and products, in fact it has arbitrary limits and colimits, and filtered limits are exact (Ab5 in Grothendiek terminology). It has enough projective and injective objects. For any object C ∈ C, the representable functor C( , C) is projective, arbitrary sums of representable`functors are projective, and any object M ∈ Mod(C) is covered by an epimorpism i C( , Ci ) → M → 0. We say ` that an object M in Mod(C) is finitely generated if there exists an epimorphism i∈I C( , Ci ) → M → 0, with I a finite set. ` Given a finitely generated functor M and and an arbitrary sum of functors j Nj , ` ` there is a natural isomorphism HomC (M, j Nj ) ∼ = j HomC (M, Nj ) (finitely generated are compact). An object P in Mod(C) is projective (finitely generated projective) if and only ` if P is summand of i∈I C( , Ci ) for a family (finite) {Ci }i∈I of objects in C. The subcategory p(C) of Mod(C), of all finitely generated projective objects, is a skeletally small additive category in which idempotents split, the functor P : C → p(C), P (C) = C( , C), is fully faithful and induces by restriction res : Mod(p(C)) → Mod(C), an equivalence of categories [A]. For this reason we may assume that our categories are skeletally small additive categories such that idempotents split, they were called annuli varieties in [A]. Definition 1. [AQM] Given a preadditive skeletally small category C, we say C has pseudokernels, if given a map f : C1 → C0 there exists a map g : C2 → C1 ( ,g)

( ,f )

such that the sequence: C( C2 ) → C( C1 ) → C( , C0 ) is exact.

CLASSICAL TILTING

3

A functor M is finitely presented if there exists an exact sequence C( C1 ) → C( , C0 ) → M → 0. We denote by mod(C) the full subcategory of Mod(C) consisting of finitely presented functors. It was proved in [AQM] mod(C) is abelian if and only if C has pseudokernels. We will say indistinctly that M is an object of Mod(C) or that M is a C-module. A representable functor C( , C) will be sometimes denoted by ( , C). 1.3. Change of Categories. According to [A], there exists a unique up to isomorphism functor − ⊗C − : Mod(C op ) × Mod(C) → Ab, called the tensor product, with the following properties: (i) For each C-module N , the functor ⊗C N : Mod(C op ) → Ab, given by (⊗C N )(M ) = M ⊗C N is right exact. (ii) For each C op -module M , the functor M ⊗C : Mod(C) → Ab, given by (M ⊗C )(N ) = M ⊗C N is right exact. (b) The functors M ⊗C − and − ⊗C N preserve arbitrary sums. (c) For each object C in C M ⊗C ( , C) = M (C) and (C, ) ⊗C N = N (C). (a)

Given a full subcategory C ′ of C. The restriction res : Mod(C) → Mod(C ′ ) has a right adjoint, called also the tensor product, and denoted by C⊗C ′ : Mod(C ′ ) → Mod(C). This functor is defined by: (C ⊗C ′ M )(C) = (C, )|C ′ ⊗C ′ M , for any M in Mod(C ′ ) and C in C. The following proposition is proved in [A Prop. 3.1]. Proposition 1. Let C ′ be a full subcategory of C. The functor C⊗C ′ : Mod(C ′ ) → Mod(C) satisfies the following conditions: (a) C⊗C ′ is right exact and preserves arbitrary sums. C⊗

res

′

(b) The composition Mod(C ′ ) −−−C→ Mod(C) −−→ Mod(C ′ ) is the identity in Mod(C ′ ). (c) For each object C ′ in C ′ , C ⊗C ′ C ′ ( , C ′ ) = C( , C ′ ). (d) C⊗C ′ is fully faithful. (e) C⊗C ′ preserves projective objects. ′ The functor M ` in Mod(C) is called ` projectively presented over C , if there exists an exact sequence i∈I C( , Ci′ ) → j∈J C( , Cj′ ) → M → 0, with Ci′ , Cj′ ∈ C ′ . The category C ⊗C ′ Mod(C ′ ) is the subcategory of Mod(C) whose objects are the functors projectively presented over C ′ . The functor res and C⊗C ′ induces an equivalence between Mod(C ′ ) and C ⊗C ′ Mod(C ′ ). α

β

→ M → 0 is a minimal projective → P0 − We say that an exact sequence P1 − β → M and P1 → Imα, are presentation of M if and only if the epimorphisms P0 − minimal projective covers (in the sense of [AF]). It is of interest to know under what conditions minimal projective presentations exist. Theorem 1 (A Theor. 4.12). The following conditions are equivalent: (a) Every object in mod(C) has a minimal projective presentation. (b) For each object C in C, every finitely presented End(C)op -module has a minimal projective presentation. A ring R is semiperfect, if every finitely generated R-module has a projective cover.


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Corollary 1. If C is a category, such that for every C in C, End(C)op is semiperfect, then every object in mod(C) has a minimal projective presentation. 1.4. Krull-Schmidt Categories. We start giving some definitions from [AR]. Definition 2. Let R be a commutative artin ring, an R-category C, is a pre additive category such that C(C1 , C2 ) is an R-module and composition is R-bilinear. Under these conditions Mod(C) is a R-category which we identify with the category of functors (C op , Mod(R)). An R-category C is Hom-finite, if for each pair of objects C1 , C2 in C the Rmodule C(C1 , C2 ) is finitely generated. We denote by (C op , mod(R)), the full subcategory of (C op , Mod(R)) consisting of the C-modules such that; for every C in C the R-module M (C) is finitely generated. The category (C op , mod(R)) is abelian and the inclusion (C op , mod(R)) → (C op , Mod(R)) is exact. The category mod(C) is a full subcategory of (C op , mod(R)). The functors D : (C , mod(R)) → (C, mod(R)), and D : (C, mod(R)) → (C op , mod(R)), are defined as follows: for any C in C, D(M )(C) = HomR (M (C), I(R/r)), with r the Jacobson radical of R, and I(R/r) is the injective envelope of R/r. The functor D defines a duality between (C, mod(R)) and (C op , mod(R)). If C is an Hom-finite R-category and M is in mod(C), then M (C) is a finitely generated R-module and is therefore in mod(R). op

Definition 3. An Hom-finite R-category C is dualizing, if the functor D : (C op , mod(R)) → (C, mod(R)) induces a duality between the categories mod(C) and mod(C op ). It is clear from the definition that for dualizing categories mod(C) has enough injectives. To finish, we recall the following definition: Definition 4. An additive category C is Krull-Schmidt, if every object in C decomposes in a finite sum of objects whose endomorphism ring is local. 1.5. The radical of a category. The notion of the Jacobson radical of a category was introduced in [M] and [A], it is defined in the following way: Definition 5. The Jacobson radical of C, radC ( , ), is a subbifunctor of HomC ( , ) defined in objects as: radC (X, Y ) = {f ∈ HomC (X, Y ) |for any map g : Y → X, 1 − gf is invertible }. If M is a C-module , then we denote by radM the intersection of all maximal subfunctors of M . Proposition 2. [A], [BR], [M] Let C be an additive category and radC ( , ) the Jacobson radical of C. Then: (a) For every object C in C radC (C, C) is just the Jacobson radical of EndC (C). (b) If C and C ′ are indecomposable objects in C, then the radical radC (C, C ′ ) consists of all non isomorphisms from C to C ′ . (c) For every object C in C, radC (C, ) = radC(C, ) and radC ( , C) = radC( , C). (d) For every pair of objects C and C ′ in C, radC (C ′ , )(C) = radC ( , C)(C ′ ). Definition 6. By an ideal of the additive category C we understand a sub bifunctor of HomC ( , ).

CLASSICAL TILTING

5

Given two ideals of I1 and I2 of C we define I1 I2 as follows: f ∈ I1 I2 (C1 , C3 ), h

g

→ C3 , with h ∈ I1 (C1 , C2 ) → C2 − if and only if, f is a finite sum of morphisms C1 − and g ∈ I2 (C2 , C3 ). Given an ideal I of C and a C-module M, we define a C-sub module IM of M by IM (C) = Σf ∈I(C,C ′ ) ImM (f ), with C in C. We say a C-module S is simple, if it does not have proper C-sub modules. Lemma 1. [MVS1 Lemma 2.5], [BR] Let C be a Krull-Schmidt K-category. Then the following statements are true: (a) Any simple functor in Mod(C) is of the form C( , C)/radC ( , C), for some indecomposable object C in C. (b) For all finitely generated functors F in Mod(C), the radical of F is isomorphic to radC F. (c) All finitely generated functors F in Mod(C) have a projective cover. 1.6. A Pair of Adjoint Functors. Let C be a skeletally small additive category, and T a skeletally small full subcategory of Mod(C). Let’s define the following functor φ : Mod(C) → Mod(T ), φ(M ) = Hom( , M )T . Our aim in this subsection is to prove φ has a left adjoint. Since Mod(C) is abelian, it has equalizers, and it was remarked above it is Ab5. In this way Mod(C) is complete [SM V.2 Theo.1 ]. Since the functor HomC (T, −) preserve limits, for every T in T it follows that the functor φ preserve limits. Lemma 2. The category Mod(C) has an injective cogenerator. Proof. Denote by D the functor D : Mod(C) → Mod(C op ), D(F )(C) =` (F (C), Q/Z), and let F be a functor in Mod(C op ). Then we have an epimorphism i C(Ci , ) → DF and hence, a monomorphism η Df Y F − → D2 F −−→ DC(Ci , ), i∈I

where ηC (x)(f ) = f (x). Moreover, D(Ci , ) is injective. Hence {DC(C, )}C∈C is a small cogenerator set consisting of injective objects in Mod(C). We need the following fi

Definition 7. Let A be an arbitray category and X → Y = {Xi −→ Y }i∈I a family pi of morphisms. A fibered product of X → Y is a pair ({P −→ Xi }i∈I , L), where L : P → Y is a morphism such that fi pi = L, and satisfies the following universal property: qi If ({Q −→ Xi }i∈I , L′ ) is a pair such that for each i ∈ I, fi qi = L′ , then there exists a unique morphism η : Q → P , such that Lη = L′ , and for each i ∈ I, pi η = qi . Using the fact Mod(C) is Ab5 and φ is left exact, the reader can verify the following fi

Proposition 3. In Mod(C) any family of monomorphisms X → Y = {Xi −→ Y }i∈I has a fibered product. Moreover, φ preserves fibered products of monomorphisms.


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Freyd’s special adjoint functor theorem [MS V.7 Theo. 2] justifies the following assertion. Theorem 2. Let C be a skeletally small pre additive category. Then the following is true: (a) The category Mod(C) is complete-small, has a small cogenerator, and any set of subobjects of a functor M has a fibered product. (b) Let T be a small full subcategory of Mod(C). Then the functor φ : Mod(C) → Mod(T ) preserves small limits and fibered products of families of monomorphisms. (c) The functor φ : Mod(C) → Mod(T ) has a left adjoint. Remark 1. Denote by −⊗T the left adjoint of φ, it follows from Yoneda’s Lemma, that for any pair of objects T in T and M in Mod(C), there are natural isomorphisms: ∼ HomT (( , T )T , φ(M )) ∼ HomC (( , T )T ⊗ T , M ) = = HomC (T, M ). By Yoneda’s Lemma again, ( , T )T ⊗ T = T. Since there are enough projective and enough injective objects in Mod(C), and Mod(T ), we can define, for any integer n, the nth right derived functors of the functors HomC (M, ), HomC ( , N ), which will be denoted by ExtnC (M, ) and ExtnC ( , N ) respectively. Analogously, the nth left derived functors of M ⊗C − and − ⊗ N , can be defined. They will be denoted by TorCn (M, ) and TorCn ( , N ), respectively. In the same way, the right derived functors of the functor φ can be defined, they will be denoted by ExtnC ( , −)T : Mod(C) → Mod(T ), and they are defined as ExtnC ( , −)T (M ) = ExtnC ( , M )T . Of course, we can also define the left derived functors of the functor − ⊗ T , they will be denoted by TorTn ( , T ) : Mod(T ) → Mod(C). We will see below relations among these functors. Proposition 4. If M is finitely presented, then the functors Ext1C (M, ) commute with arbitrary sums. Proof. There is an exact sequence α

0 → Ker(α) → C( , C) − →M →0

(1.1)

with Ker(α) finitely generated and C an object ` in C. Let {Ni }i∈I be a family of objects in Mod(C). After applying HomC ( , i∈I Ni ) to (1.1), it follows the existence of a isomorphism η, such that the following diagram commutes a a a 1 Ni ) Ni ) Ni ) (C( , C), / ExtC (M, / 0 / (Ker(α), i∈J

i∈I

∼ =

a (C( , C), Ni ) i∈I

i∈I

∼ =

a (Ker(α), Ni ) / i∈I

/

η

a i∈I

Ext1C (M, Ni )

/ 0

Corollary 2. If T consists of finitely presented functors, then the functors φ and Ext1C ( , −)T commute with arbitrary sums.

CLASSICAL TILTING

7

2. Brenner-Butler´s Theorem 2.1. The main theorem. In this section we introduce the notions of tilting categories and we show, that with slights modifications, Bongartz’s proof [B] of BrennerButler’s theorem [BB], extends to tilting categories over arbitrary skeletally small pre additive categories. We also have the corresponding theorems on the invariance of Grothendieck groups under tilting, and the relations between the global dimension of a category and the global dimension of the tilted category. Then we prove, that under mild conditions, tilting functors restrict to the categories of finitely presented functors. For dualizing categories we have theorems analogous to classical tilting for finite dimensional algebras. The first definition is a natural generalization of the classical notion of a tilting object. Definition 8. Let C an annuli variety, a subcategory T of Mod(C) is a tilting category, if every object in T is finitely presented and the following conditions are satisfied: (i) pdimT ≤ 1. (ii) We have Ext1C (Ti , Tj ) = 0, for every pair of objects Ti , Tj in T . (iii) For every object C in C, the representable functor C( , C) has a resolution 0 → C( , C) → T1 → T2 → 0, with T1 , T2 in T . Definition 9. Given a skeletally small pre additive category C, we introduce the following notion of tensor product: given an abelian group G and an object M in Mod(C) the tensor product G ⊗Z M , is the functor defined in objects as (G ⊗Z M )(C) = G ⊗Z M (C). Given a subcategory T of Mod(C), we define for every M in Mod(C) and every (T,M) object T of T the evaluation map e(T,M) : C(T, M ) ⊗Z M → M , eC (f ⊗ m) = fC (m), where, C ∈ C, m ∈ M (C) and f = {f } ∈ C(T, M ). Define the trace C C∈C ` ` of T in M , as the image of the sum e(Ti ,M) : i∈I C(Ti , M ) ⊗Z Ti → M , where {Ti }i∈I is a set of representatives of the isomorphism classes of objects in T . Let Ti be an object in {Ti }i∈I , and Xi = {fji }j∈Ji a set of generators of the abelian group C(Ti , M ), let Z(Xi ) be the free abelian group with basis Xi and X = ∪Xi . Then the epimorphism of abelian groups ϕ : Z(Xi ) → C(Ti , M ) → 0 Induces an epimorphism a (X ) a a ψ: Ti i ∼ Z(Xi ) ⊗Z Ti → C(Ti , M ) ⊗Z Ti → 0 = i∈I

i∈I

i∈I

composing with the sum of the evaluation maps we obtain a map a a (X ) e(Ti ,M) ψ : Ti i → M i∈I

` ` (X ) Let’s denote i∈I Ti i by T (X) , then the map: ΘM = e(Ti ,M) ψ : T (X) → M has the following property:


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Proposition 5. Let M be in Mod( C), and ΘM : T (X) → M the map given above. Then for every object T ′ ∈ T and every map η : T ′ → M , there exists a map γ : T ′ → T (X) such that ΘM γ = η. Given a skeletally small pre additive category C and T a tilting subcategory of Mod(C), we denote by T the full subcategory of Mod(C) whose objects are epimorphic images of objects in T . We want to prove that T is a torsion class of Mod(C). The above proposition implies the following: Proposition 6. Let M be an object in Mod(C). Then M is in T , if and only if, ` e(Ti ,M) is an epimorphism. We can prove now:

Proposition 7. If M is in T , then, for every object T ∈ T , Ext1C (T, M ) = 0. Proof. Let’s assume M is in T . Then there exists a short exact sequence a α (2.1) 0 → Ker(α) → → M → 0, Tj − j∈J

where {Tj }j∈J is a family of objects in T . Since T is finitely presented, it follows Ext1C (T, ) commutes with arbitrary sums. By hypothesis, Ext1C (T, Tj ) = 0 for every j ∈ J, and T of projective dimension one, implies for every object K, Ext2C (T, K) = 0. Applying HomC (T, ) to the above exact ` sequence, it follows from the long homology sequence, that the sequence Ext1C (T, j∈J Tj ) → Ext1C (T, M ) → Ext2C (T, K) is exact. Finally we have Ext1C (T, M ) = 0. Proposition 8. The category T is closed under extensions, direct sums and epimorphic images. This is: T is a torsion class of a torsion theory of Mod(C). Proof. We only need to see it is closed under extensions. Consider the exact sequence 0 → M1 → M2 → M3 → 0, with M1 , M3 in T . Then Ext1C (Ti , M1 ) = Ext1C (Ti , M3 ) = 0, by Proposition 7. For every object Ti ∈ {Ti }i∈I , we apply the functor C(Ti , ), to the above exact sequence to obtain, by the long exact sequence, an exact sequence of abelian groups 0 → C(Ti , M1 ) → C(Ti , M2 ) → C(Ti , M3 ) → Ext1C (Ti , M1 ) = 0. Applying the tensor product ⊗Z Ti to the sequence, and adding the exact sequences, we obtain the exact sequence: a a a C(Ti , M1 ) ⊗Z Ti → C(Ti , M2 ) ⊗Z Ti → C(Ti , M3 ) ⊗Z Ti → 0, i∈I

i∈I

i∈I

which induces a commutative diagram: a a a C(Ti , M1 ) ⊗ Ti C(Ti , M1 ) ⊗ Ti / C(Ti , M1 ) ⊗ Ti → 0 / i∈I ` (T ,M ) e i 1

0

/ M1

i∈I ` (M ,F ) e i 2

/ M2

i∈I `

e(Ti ,M3 )

/ M3 → 0

` (Ti ,M1 ) ` (Ti ,M3 ) The morphisms e and e are epimorphisms, by Proposition 6. ` (T Then it follows, by the Snake lemma, e i ,M2 ) is an epimorphism. Again by Proposition 6, M2 is in T .

CLASSICAL TILTING

9

We will prove next that the trace of T in M , denoted by τT (M ), is the idempotent radical corresponding to the torsion theory with torsion class T . Proposition 9. For any M in Mod(C) and T ′ in T , HomC (T ′ , M/τT M ) = 0. In particular τT (M/τT (M )) = 0. Proof. Consider the exact sequence j

p

0 → τT (M ) − →M − → M/τT (M ) → 0

(2.2)

and a natural transformation η : T ′ → M/τT (M ). Taking the pull back of the maps η and p we get the following commutative exact diagram: 0

/ τT (M )

/ T′

/ W

/ 0

η

0

/ τT (M )

j

/ M

p

/ M/τT (M )

/ 0

By Proposition 7, the top exact sequence splits, hence there exists a map f : T ′ → M , such that pf =η. By the properties of the trace, f (T ′ ) ⊂ τT M and η = 0. We have the following characterization of the torsion class: Proposition 10. Let T be a tilting category in Mod(C). Then: T = {M ∈ Mod(C)|Ext1C (T, M ) = 0, T ∈ T }. Proof. Let’s assume M ∈ Mod(C) and for each T in T , Ext1C (T, M ) = 0. Then applying the functor HomC (T, −) to the sequence (2.2), it follows from the long homology sequence and the fact pdimT = 1 that 0 = Ext1C (T, M ) → Ext1C (T, M/τT (M )) → Ext2C (T, τT (M )) = 0 is exact, and in consequence, Ext1C (T, M/τT (M )) = 0. Let C ∈ C. Then we have an exact sequence 0 → C( , C) → T0 → T1 → 0, with Ti ∈ T , i = 0, 1. After applying HomC ( , M/τT (M )) to the sequence, we get, by the long homology sequence, the exact sequence 0 = Hom(T0 , M/τT (M )) → Hom(( , C), M/τT (M )) → Ext1 (T1 , M/τT (M )) = 0. By Yoneda’s Lemma, 0 = Hom(( , C), M/τT (M )) = M (C)/τT M (C), this is: M = τT (M ) is in T . The other inclusion is already proved in Proposition 7. The torsion class T induces a torsion pair (T , F ), where F = {N ∈ Mod(C)|Hom(M, N ) = 0, M ∈ T }, [see S]. Proposition 11. F = {N ∈ Mod(C)}|Hom(T, N ) = 0, T ∈ T }. ` Proof. Let M be an object in T . Then there is an epimorphism j∈J Tj → M → 0, with {Tj }i∈J a family of objects in T , and let N ∈ {N ∈ Mod(C)|Hom(T, N ) = 0, T ∈ T }. Then there is a monomorphism a Y 0 → HomC (M, N ) → HomC ( Tj , N ) ∼ HomC (Tj , N ) = 0, = j∈J

and M is in F . The other inclusion is clear.

j∈J


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Let T be a tilting subcategory of Mod(C), we define AddT as the full subcategory of Mod(C) whose objects are direct summands of arbitrary sums of objects in T . Let φ : Mod(C) → Mod(T ) be the functor defined as φ(M ) = Hom( , M )T . We adapt Bongartz ’s proof of Brenner-Butler’s theorem to our situation. We start with a version of [B Prop. 1.4]: f

g

h

→ M0 is an exact sequence in → M1 − → M2 − Proposition 12. (i) If M3 − Mod(C), such that Mi belongs to T , then the sequence φ(M2 ) → φ(M1 ) → φ(M0 ) is exact. (ii) For each M ∈ T there is an exact sequence tn+1

t

t

0 1 M →0 T 0 −→ · · · −−−→ T n → · · · → T 1 −→

δ

n Imtn → 0, have the following with T i in AddT , such that the maps T n → property: given T in T and a map f : T → Imtn , there exist a map h : T → T n , such that δn h = f. (iii) For each M in T there exists an isomorphism

φ(M ) ⊗ T → M . (iv) TorT1 (φ(M ), T ) = 0. (v) For any pair of objects M, N in T we have an isomorphism ExtiC (M, N ) = ExtiT (φ(M ), φ(N )). Proof. (i) Since T is closed under epimorphic images, then Im(g), Im(f ) and Coker(f ) are in T . From the exact sequences: 0 → Imh → M2 → Img → 0, 0 → Img → M1 → Imf → 0, 0 → Imf → M0 → Cokerf → 0, we obtain exact sequences in Mod( T ) : 0

→ ( , Imh)T → ( , M2 )T → ( , Img)T → Ext1 ( , Imh)T = 0,

0

→ ( , Img)T → ( , M1 )T → ( , Imf )T → Ext1 ( , Img)T = 0,

0

→ ( , Imf )T → ( , M0 )T → ( , Cokerf )T → Ext1 ( , Imf )T = 0.

Gluing the sequences, the result follows. (ii) By the Propositions 5 and 6, for each M ∈ T there is an exact sequence (2.3)

η0

0 → K0 → T (X0 ) −→ M → 0

such that; every map η : T ′ → M factors through η0 . (T ′ ,η0 )

It follows that HomC (T ′ , T (X0 ) ) −−−−→ HomC (T ′ , M ) is an epimorphism. Since Ext1C (T ′ , T (X0 ) ) = 0, after applying HomC (T ′ , ) to (2.3), it follows by the long homology sequence, that Ext1C (T ′ , K0 ) = 0, and K0 is in T . The claim follows by induction. (iii) By part (ii) there exists an exact sequence · · · → T 1 → T 0 → M → 0. After applying φ we obtain an exact sequence · · · → φ(T 1 ) → φ(T 0 ) → φ(M ) → 0. Since φ and −⊗T preserve arbitrary sums, for i ≥ 0, we have, φ(T i )⊗T = ( , T i )T ⊗T ∼ = Ti . It follows, the existence of an isomorphism η, such that the following diagram

CLASSICAL TILTING

11

commutes: φ(T 1 ) ⊗ T ∼ =

/ φ(T 0 ) ⊗ T ∼ =

T1

/ φ(M ) ⊗ T

/ 0

η

/ M

/ T0

/ 0

(iv) It is also clear from the above diagram TorT1 (φ(M ), T ) = 0. (v) Using the sequence constructed in (ii) and the relation ExtjC (T n , N ) = 0 for all j ≥ 0 and n ≥ 0, we get by dimension-shift ExtjC (M, N ) ∼ = Ext1C (Im(tj−1 ), N ), for j ≥ 1. The exact sequences T j+1 → T j → Im(tj ) → 0, 0 → Im(tj ) → T j−1 → Im(tj−1 ) → 0, induce the following commutative diagram: 0 HomC (q,N ) / HomC (Im(tj ), N ) HomC (T j−1 , N ) RRR RRHom RRRC (tj ,N ) RRR HomC (p,N ) RRR ) HomC (T j , N )

/ Ext1 (Im(tj−1 ), N ) C

/0

HomC (tj+1 ,N )

HomC (T j+1 , N ) Hence, there are isomorphisms: (2.4) ExtjC (M, N ) ∼ = Ext1C (Im(tj−1 ), N ) ∼ = H j (HomC (T., N )). Applying φ to the exact sequence in (ii), we obtain the following projective resolution of φ(M ) ( , T.) → ( , M )T : · · · → ( , T 1 ) → ( , T 0 ) → ( , M )T → 0. By Yoneda’s Lemma, HomT (( , T. ), ( , N )T ) ∼ = HomC (T. , N ), therefore: (2.5)

ExtjT (φ(M ), φ(N )) = H j (( , T.), φ(N )) = H j (HomC (T., N )).

The claim follows from (2.5) and (2.4).

We will see next how to compute − ⊗ T , and its left derived functors. Proposition 13. For each C in C, and M ∈ Mod(T ) there is a natural isomorphism in Mod( C), (M ⊗ T )(C) ∼ = (C( , C), )T ⊗T M . Proof. Let’s consider the following presentation of M a (( ,fij )) a (2.6) ( , Ti ) → ( , Tj ) → M → 0. i∈I

j∈J

Applying ⊗T and evaluating in C, we obtain the following short exact sequence, a (fij )C a (2.7) Ti (C) −−−−→ Tj (C) → M ⊗ T (C) → 0. i∈I

j∈J


12

By the properties of the tensor product, and Yoneda’s Lemma, we have isomorphisms: a a a a (C( , C), ) ⊗T ( , Ti ) ∼ (C( , C), ) ⊗T ( , Ti ) ∼ (C( , C), Ti ) ∼ Ti (C) = = = i∈I

i∈I

i∈I

i∈I

Since (C( , C), )⊗T , is right exact, when we apply it to the presentation (2.6), and compare the exact sequence we get, with the exact sequence (2.7), we obtain the desired isomorphism. It is easy to verify that this isomorphism is natural. Proposition 14. For each C in C, and M ∈ Mod(T ), and any non negative integer n, there exists an isomorphism ∼ TorT (M, T )(C), (2.8) TorT ((C( , C), )T , M ) = n

n

which is natural in C and M. Proof. Let’s consider an exact sequence 0 → ΩM → ( , T ) → M → 0,

(2.9)

where T is a sum of objects in T . Applying ⊗T to (2.9) we get the exact sequence 0 → TorT1 (M, T ) → ΩM ⊗ T → ( , T ) ⊗ T , and applying (C( , C), ) ⊗T − to (2.9), we get the exact sequence: 0 → TorT1 ((( , C), ), M ) → (( , C), ) ⊗ ΩM → (( , C), ) ⊗ ( , T ). By Proposition 13, there exists an isomorphism γC , such that the following diagram commutes / TorT ((( , C), ), M ) / (( , C), ) ⊗ ΩM / (( , C), ) ⊗ ( , T ) 0 1 ∼ =

γC

∼ =

/ ΩM ⊗ T (C) / T (C) / TorT (M, T )(C) 0 1 We can easily verify that γ = {γC }C∈C is a natural transformation. We leave to the reader the proof of the naturality in M .

′

Remark 2. Let C an C be two objects in C, and T a tilting subcategory of Mod(C). g Then we have two exact sequences: 0 → C( , C) − → T0 → T1 → 0 and 0 → g′

C( , C ′ ) −→ T0′ → T1′ → 0, with Ti and Ti′ in T , for i = 0, 1. Let f : C → C ′ be a morphism. Then, taking the push out, we have the following commutative exact diagram: g / T1 / 0 0 / C( , C) / T0 g′ (C( ,f ))

0

/ T′ 0

/ W

/ T1

/ 0

Since Ext1C (T1 , T0′ ) = 0, the lower exact sequence in the diagram splits. Hence, there exist morphisms u, v, such that the following exact diagram commutes: / C( , C) C( ,f ) 0 / C( , C ′ ) 0

(2.10)

g

/ T0 u g′ / T′ 0

/ T1 v / T′ 1

/ 0 / 0

CLASSICAL TILTING

13

Proposition 15. Let T be a tilting subcategory of Mod(C). Then for each M in Mod(C), there is an exact sequence 0 → φ(M ) ⊗ T → M → TorT1 (Ext1C ( , M )T , T ) → 0. Moreover, Ext1C ( , M )T ⊗ T = 0. Proof. For each C in C we have an exact sequence 0 → C( , C) → T0 → T1 → 0,

(2.11)

which induces the following exact sequence 0 → HomC (T1 , )T → HomC (T0 , )T → HomC (C( , C), )T → 0.

(2.12)

After applying − ⊗T Ext1C ( , M )T to (2.12), we get the following exact sequence (2.13)

0

→ TorT1 ((C( , C), )T , Ext1C ( , M )T ) → Ext1C (T1 , M ) → → Ext1C (T0 , M ) → Hom(C( , C), )T ⊗T Ext1C ( , M )T → 0.

Applying HomC ( , M ) to (2.11), we get from the long homology sequence, the exact sequence 0 → HomC (T1 , M ) → HomC (T0 , M ) → HomC (C( , C), M ) →

(2.14)

→ Ext1C (T1 , M ) → Ext1C (T0 , M ) → Ext1C (C( , C), M ) = 0. Then, for each C in C we have the following isomorphisms: ∼ (C( , C), )T ⊗T Ext1 ( , M )T , by Proposition 13 (Ext1C ( , M )T ⊗ T )(C) = C ∼ = Ext1 (C( , C), M )T = 0, by (2.13) and (2.14), C

which proves the second part of the Proposition. Now, by Proposition 14 we know that ∼ TorT ((C( , C), )T , Ext1 ( , M )T ). (2.15) TorT1 (Ext1C ( , M )T , T )(C) = C 1 It follows by 2.15, and the isomorphism HomC (C( , C), M ) ∼ = M (C), together with (2.13), and (2.14), that there exist a isomorphisms γC such that the following diagram commutes: / Ext1 (T1 , M ) M (C) C γC 1 / TorT (Ext1 ( , M )T , T )(C) / Ext1 (T1 , M ) C 1 C

0

/ Ext1 (T0 , M ) → 0 C 1 / Ext1 (T0 , M ) C

Moreover, the Snake Lemma implies γC is epimorphisms. Using Remark 2.10, the reader can check that γ = {γC }C∈C is a natural transformation. After applying − ⊗T φ(M ) to the exact sequence (2.12) we obtain the following exact sequence φ(M )(T1 ) → φ(M )(T0 ) → Hom(C( , C), ) ⊗T φ(M ) → 0. By the isomorphism φ(M ) ⊗ T (C) ∼ = Hom(C( , C), )T ⊗T φ(M ), and the exact sequence (2.14), there exist a morphisms ηC such that the following diagram commutes:

0

φ(M )(T1 ) ∼ = / Hom(T1 , M )

/ φ(M )(T0 ) / φ(M ) ⊗ T (C) ∼ ηC = / Hom(T0 , M ) / Hom(C( C), M )

/ 0


14

Using the Snake Lemma again, the map ηC is monomorphism. The reader can check η = {ηC }C∈C is natural transformation. In this way we have the following exact sequence η

γ

0 → φ(M ) ⊗ T − →M − → TorT1 (Ext1C ( , M ), T ) → 0. Proving the proposition.

Proposition 16. Let T be a tilting subcategory of Mod(C). Then for each N in Mod(T ) there exists an exact sequence 0 → Ext1C (−, TorT1 (N, T )) → N → φ(N ⊗ T ) → 0. In addition φ(TorT1 (N, T )) = 0. Proof. We choose the projective resolutions (2.16) L. → N (2.17)

(−,h2 )

(−,h1 )

: · · · → (−, T2 ) −−−−→ (−, T1 ) −−−−→ (−, T0 ) → N → 0 0 → P1 → P0 → T → 0

such that for i ≥ 0, Ti is a sum of objects in T . Applying − ⊗ T to the complex L., we obtain the complex L. ⊗ T , whose objects are in AddT , and we have Ext1C (T, L. ⊗ T ) = 0. In this way, we obtain the following exact sequence of complexes: (2.18)

0 → HomC (T, L. ⊗ T ) → HomC (P0 , L. ⊗ T ) → HomC (P1 , L. ⊗ T ) → 0.

Observe that L. (T ) and HomC (T, L. ⊗ T ) are isomorphic, hence the exact sequence (2.18) becomes 0 → L. (T ) → HomC (P0 , L. ⊗ T ) → HomC (P1 , L. ⊗ T ) → 0. By the above sequence, and the long homology sequence, we get an exact sequence: (2.19) 0 = H1 (L. (T )) → H1 (HomC (P0 , L. ⊗ T )) → H1 (HomC (P1 , L. ⊗ T )) → → H0 (HomC (L. (T ))) → H0 (HomC (P0 , L. ⊗ T )) → H0 (HomC (P1 , L. ⊗ T )) → 0. Since Pi , i = 0, 1 are projective, there exists an isomorphism H1 (HomC (Pi , L. ⊗ T )) ∼ = HomC (Pi , H1 (L. ⊗ T )) = HomC (Pi , TorT1 (N, T )). Finally, the exact sequence (2.19) can be written as: (2.20)

0 → HomC (P0 , TorT1 (N, T )) → HomC (P1 , TorT1 (N, T )) → N (T ) → HomC (P0 , N ⊗ T ) → HomC (P1 , N ⊗ T ) → 0.

After applying HomC ( , TorT1 (N, T )) to the projective resolution of T , we obtain the following exact sequence: (2.21)

0 → HomC (T, TorT1 (N, T )) → HomC (P0 , TorT1 (N, T )) → → HomC (P1 , TorT1 (N, T )) → Ext1C (T, TorT1 (N, T )) → 0.

Comparing (2.20) and (2.21), we get φ(TorT1 (N, T ))(T ) = 0, which proves part of the Proposition. In addition (2.20) and (2.21), imply the existence of a morphism ηT , such that the following diagram commutes:

CLASSICAL TILTING

(P0 , TorT1 (N, T )) ∼ = 0 → (P0 , TorT1 (N, T ))

15

/ (P1 , TorT (N, T )) / Ext1 (T, TorT (N, T ) → 0 1 C 1 ∼ ηT = / (P1 , TorT (N, T )) / N (T ) 1

By the Snake Lemma, ηT is mono. After applying HomC ( , N ⊗ T ) to the projective resolution of T , we obtain the following exact sequence: (2.22)

0

→ HomC (T, N ⊗ T ) → HomC (P0 , N ⊗ T ) → → HomC (P1 , N ⊗ T ) → ExtC (T, N ⊗ T ) → 0.

The sequences (2.20), and (2.22), imply the existence of a morphism γT : N (T ) → HomC (T, N ⊗ T ) = φ(N ⊗ T ). It follows by the Snake Lemma that γT is epimorphism. Moreover, Ext1C ( , N ⊗ T ) = 0.

(2.23)

It’s not hard to check that η = {ηT }T ∈T and γ = {γT }T ∈T are natural transformations. We leave the details to the reader. In this way the following sequence of functors η γ →N − → φ(N ⊗ T ) → 0 0 → Ext1C (−, TorT1 (N, T )) − is exact, proving the Proposition.

We call F, F ′ : Mod(C) → Mod(T ) to the functors defined by: F (M ) = ′

F (M ) =

φ(M ), Ext1C ( , M )T ,

and, G, G′ : Mod(T ) → Mod(C) to the functors: G(N ) ′

=

G (N ) =

N ⊗T, TorT1 (N, T ).

Let T be a tilting subcategory of Mod(C), and (T , F ) the torsion theory considered above. We look to the full subcategories X and Y of Mod(T ), defined by: X

=

{N ∈ Mod(T )|N ⊗ T = 0},

Y

=

{N ∈ Mod(T )|TorT1 (N, T ) = 0}.

From the previous results, the main theorem of the section follows: Theorem 3 (Brenner-Butler). With the above notation, the following statements are true: (i) F and G induce an equivalence between T and Y . (ii) F ′ and G′ induce an equivalence between F and X . (iii) The following equations F G′ = F ′ G = 0 and G′ F = GF ′ = 0 hold.. Corollary 3. The pair of subactegories (X , Y ) form a torsion theory in Mod(T ). Proof. It is easy to check that for pair of objects X in X , and Y in Y , HomT (X, Y ) = 0. For every N in Mod(T ) there is an exact sequence:


16

η

γ

0 → Ext1C (−, TorT1 (N, T )) − →N − → φ(N ⊗ T ) → 0. 1 By condition (iii) of the Theorem, ExtC (−, TorT1 (N, T )) is in X and φ(N ⊗ T ) is in Y , which implies the pair (X , Y ) is a torsion theory. We want next to generalize the following result on tilting of finite dimensional algebras: Let A be a finite dimensional K-algebra and A mod the category of finitely generated left A-modules. Let’s suppose that A T is a tilting A-module and B = EndA (T ). Tilting theorem [B] proves TB is a right tilting B-module and Aop is isomorphic to EndB (TB ). Proposition 17. Let T be a tilting subcategory of Mod(C). Let’s assume each T in T has a projective resolution of finitely generated projectives, 0 → P1 → P0 → T → 0. Then the following statements hold: (a) The full subcategory θ of Mod(T op ), with objects {(C( , C), )T }C∈C is a tilting subcategory in Mod(T op ). (b) The category θ = {(C( , C), )T }C∈C is equivalent to C op . Proof. (a)(i) For each object C in C there is a resolution f

→ T1 → 0, 0 → C( , C) → T0 −

(2.24)

with Ti ∈ T , for i = 0, 1. By the long homology sequence, there is an exact sequence of objects in Mod(C), (2.25)

(f, )

0 → (T1 , )T −−−→ (T0 , )T → (C( , C), )T → Ext(T1 , )T = 0,

this is; pdim(C( , C), )T ≤ 1. (ii) Let C ′ be another object in C. After applying the functor HomT op ( , (C( , C ′ ), )T ) to (2.25), we get an exact sequence: (2.26)

0 → ((C( , C), )T , (C( , C ′ ), )T ) → ((T0 , ), (C( , C ′ ), )T ) →

→ ((T1 , ), (C( , C ′ ), )T ) → Ext1T op ((C( , C), )T , (C( , C ′ ), )T ) → 0. By Yoneda’s Lemma the following diagram: ((T0 , ), (C( , C ′ ), )T )

/ ((T1 , ), (C( , C ′ ), )T )

∼ =

(C( , C ′ ), T0 )

∼ = (C( ,C ′ ),f )

∼ =

T0 (C ′ )

/ (C( , C ′ ), T1 ) ∼ =

fC ′

/ T1 (C ′ )

commutes. Evaluating (2.24) in C ′ , and using Yoneda’s Lemma, together with (2.26), there exist the following isomorphisms of abelian groups: ((C( , C), )T , (C( , C ′ ), )T ) ∼ (2.27) = C(C ′ , C) Ext1T op ((C( , C), )T , (C( , C ′ ), )T ) ∼ =

0.

CLASSICAL TILTING

17

(iii) The sequence 0 → (T, )T → (P0 , )T → (P1 , )T → Ext1T op (T, )T = 0 is exact, and each (Pi , )T is in add{(C( , C), )T }C∈C . (b) We define the functor: α : C op → θ, α(C) = (C( , C), )T Which by (2.27), is full and faithful, giving the desired equivalence.

2.2. The Grothendieck Groups K0 (C) and K0 (T ). Given a ring A and a tilting A-module T it is a classical theorem [B] that the Grotendieck groups of A and B = EndA (T )op are isomorphic. In this subsection we will prove that there is also an isomorphism between the Grotendieck group, K0 (C), of an arbitrary skeletally small pre additive category C, and the Grotendieck group, K0 (T ), of a tilting subcategory T of Mod(C). The proof will follow closely [CF]. Definition 10. Let C a skeletally small pre additive category C and let T be a tilting subcategory of Mod(C). Let’s denote by |Mod(C)| the set of isomorphism classes of objects in Mod(C). Let A be the free abelian group generated by |Mod(C)| and R the subgroup of A generated by relations M − K − L such that 0 → K → M → L → 0 is a short exact sequence in Mod(C). Then, the Groethendiek group of C is K0 (C) = A/R. Proposition 18. The Groethendieck groups K0 (C) and K0 (T ) are isomorphic. ˆ ) = Proof. We define the group homomorphism φˆ : A → K0 (T ), sending φ(M ′ ′ |F (M )| − |F (M )|, where F and F are the functors given in Brenner-Buttler’s theˆ In fact, let M − K − L be a orem. We claim R is contained in the kernel of φ. generator of R. Then there exists an exact sequence in Mod(T ), 0 → ( , K)T → ( , M )T → ( , L)T → Ext1C ( , K)T → (2.28)

→ Ext1C ( , M )T → Ext1C ( , L)T → Ext2C ( , K)T = 0,

hence, the alternating sum, −(|F (K)| − |F ′ (K)|) − (|F (L)| − |F ′ (L)|) + (|F (M )| + |F ′ (M )|) = 0. ˆ Which means φ(M − K − L) = 0. Then, there is a unique map φ : K0 (C) → K0 (T ), given by φ(|M |) = |F (M )| − |F ′ (M )|. For each object C in C, there is a short exact sequence 0 → C( , C) → T 0 → 1 T → 0 with T i in T , for i = 0, 1, which induces the exact sequence in Mod(T op ): 0 → (T1 , )T → (T0 , )T → (C( , C), )T → Ext1C (T1 , ) = 0. Therefore: for n > 1, and any object N in Mod(T ), pdim(C( , C), )T ≤ 1 implies TorTn ((C( , C), )T , N ) = 0. For an exact sequence 0 → K → N → L → 0 in Mod(T ), and any object C in C, there is an exact sequence: 0

→ TorT1 (C( , C), ), K) → TorT1 (C( , C), ), N ) → TorT1 (C( , C), ), L) → → (C( , C), ) ⊗ K → (C( , C), ) ⊗ N → (C( , C), ) ⊗ L → 0.

Which can be re written as follows: 0 (2.29)

→ TorT1 (K, T ) → TorT1 (N, T ) → TorT1 (L, T ) → → K ⊗T →N ⊗T →L⊗T →0


18

In an analogous way, there is a group homomorphism ψ : K0 (T ) → K0 (C), given by ψ(|N |) = |G(N )| − |G′ (N )|. By Brenner-Butler’s theorem, there are isomorphisms: ψφ(|M |)

= ψ(|F (M )| − |F ′ (M )|) = ψ(|F (M )|) − ψ(|F ′ (M )|) = (|GF (M )| − |G′ F (M )|) − (|GF ′ (M )| − |G′ F ′ (M )|) = |GF (M )| − |G′ F ′ (M )|,

and from the exact sequence 0 → GF (M ) → M → G′ F ′ (M ) → 0 it follows: |M | = |GF (M )| − |G′ F ′ (M )|, this is: ψφ = 1K0 (C) . With a similar argument we prove φψ = 1K0 (T ) . 2.3. Global Dimension and Tilting. In this subsection we will compare the global dimensions of a category C and its tilting category T , obtaining results similar to the ring situation. The proof given here uses the same line of arguments as in [ASS]. Let C be a skeletally small pre additive category and T a tilting subcategory of Mod(C). Let T be the torsion class of Mod(C), whose objects are epimorphic images of arbitrary sums of objects in T . We proved T = {M ∈ Mod(C)|Ext1 (T , M ) = 0}. We use this fact in the following: Lemma 3. Let M be an object in T and assume Ext1C (M, )T = 0. Then M is in AddT . Proof. Let M be an object in T . By Proposition 12 there is a short exact sequence a α 0 → Ker(α) → → M → 0, Ti − i∈I

with Ker(α) in T . Therefore: the sequence splits and M is in AddT .

Proposition 19. If M is in T , then pdimHomC ( , M )T ≤ pdimM . Proof. By induction in pdimM . If` pdimM = 0, then M is projective, and since M is in T there is an epimorphism f : i∈I Ti → M → 0, with Ti in T , which splits and ` T . It follows M is in AddT , and HomC ( , M )T is sumM is a summand of i i∈I ` mand of i∈I ( , Ti ), this is, HomC ( , M )T is projective and pdimHomC ( , M ) = 0. Let’s assume pdimM = 1. There is an exact sequence (2.30)

0 → L → T0 → M → 0,

with T0 in AddT and L in T . The sequence (2.30) induces an exact sequence: 0 → ( , L)T → ( , T0 )T → ( , M )T → Ext1C ( , L)T = 0. Since pdM = 1, then Ext2C (M, )T = 0, and from (2.30) and the long homology sequence, it follows the sequence 0 = Ext1C (T0 , )T → Ext1C (L, )T → Ext2C (M, )T = 0 is exact. In consequence, Ext1C (L, )T = 0, and by Lemma 3, L is in AddT . Therefore HomC ( , L)T is projective and HomC ( , M )T , has projective dimension less or equal to one.

CLASSICAL TILTING

19

Suppose n ≥ 2 and the claim is true for all objects in T with projective dimension less than n. Let M be an object in T with pdimM = n. Then from (2.30) and the long homology sequence we get an exact sequence 0 = ExtnC (T0 , ) → ExtnC (L, ) → Extn+1 (M, ) = 0. C Then pdimL ≤ (n − 1). By induction hypothesis pdHomC ( , L)T ≤ n − 1 and HomC (, T0 )T is projective. Applying the contravariant functor HomT (−, ) to the exact sequence (2.30), we obtain by the long homology sequence the inequalities: pdimHomC ( , M )T ≤ pdimHomC ( , L)T + 1 ≤ 1 + (n − 1) Theorem 4. With the same assumptions as above, we have the inequality: gdim(T ) ≤ 1 + gdim(C). Proof. Let X be an object in Mod(T ), and cover it with a projective object, to get an exact sequence a (2.31) 0→Y → ( , Ti ) → X → 0. i∈I

` The functor φ : T → Y is an equivalence, and i∈I ( , Ti ) is in Y , and Y is closed under sub-objects, hence Y is in Y . Since φ is dense, there exists an object M in T , such that φ(M ) = HomC ( , M )T ∼ = Y , and pdimY ≤ pdimM . From the exact sequence (2.31) we have the following inequalities: pdimX ≤ 1 + pdimY ≤ 1 + pdimM ≤ 1 + gdim(C) and gdim(T ) ≤ 1 + gdim(C).

2.4. Brenner-Butler’s theorem for categories of finitely presented functors. In this subsection we will prove that, under mild assumptions on the categories C and T , Brenner-Butler’s theorem holds in the categories of finitely presented functors. To prove it we need to see under which conditions the functor φ : Mod(C) → Mod(T ) restricts to the categories of finitely presented functors, φ : mod(C) → mod(T ). It was recalled in Section 1, that the category of finitely presented functors mod(C) is abelian, if and only if, C has pseudokerneles [AR]. Hence; it is natural to assume C and T have pseudokerneles. Under these conditions we have the following. Proposition 20. Let’s assume C and T have pseudokerneles. Then the functor φ|mod(C) : mod(C) → Mod(T ), has image in mod(T ). Proof. (a) For each object C, the functors ( , C( , C))T and Ext1C ( , C( , C))T are in mod(T ). To see this, consider the exact sequence 0 → C( , C) → T0 → T1 → 0, with T0 , T1 in T . From the above exact sequence, and the long homology sequence, we get an exact sequence: 0

→ ( , C( , C))T → ( , T0 )T → ( , T1 )T → → Ext1C ( , C( , C))T → Ext1C ( , T0 )T = 0.


20

The claim follows from the fact mod(T ) is abelian. (b) Let M be in mod(C). Since C has pseudokernels, then M has a projective resolution ( ,f2 )

( ,f1 )

( ,f0 )

· · · → C( , C3 ) −−−−→ C( , C2 ) −−−−→ C( , C1 ) −−−−→ C( , C0 ) → M → 0 Let Ki be Im( , fi ). Then for all i ≥ 0, the functors Ext1C ( , Ki )T and ( , Ki )T are finitely presented. Indeed, from the exact sequences, ki+1

pi

0 → Ki+1 −−−→ C( , Ci ) −→ Ki → 0, the long homology sequence, and the fact pdimT ≤ 1, we obtain for each i an exact sequence: ( ,ki+1 )

( ,pi )

∂

i Ext1C ( , Ki+1 )T → 0 → ( , Ki+1 )T −−−−−→ ( , C( , Ci ))T −−−−→ ( , Ki )T −→

Ext1 ( ,ki+1 )

Ext1 ( ,pi )

−−−−−−−−→ Ext1 ( , C( , Ci ))T −−−−−−−→ Ext1 ( , Ki )T → Ext2 ( , Ki+1 )T = 0. By (a), for all i ≥ 0, the functor Ext1C ( , C( , Ci ))T is finitely presented. Hence; each Ext1 ( , Ki )T is finitely generated . From the exact sequence Ext1 ( ,pi )

Ext1 ( ,ki+1 )

C −−−→ Ext1C ( , Ki )T → 0, Ext1C ( , Ki+1 )T −−−C−−−−−→ Ext1C ( , C( , Ci ))T −−−−

it follows Ext1C ( , Ki )T is actually finitely presented. Since mod(T ) is abelian, the Ext1 ( ,ki+1 )

kernel of Ext1C ( , Ki+1 )T −−−C−−−−−→ Ext1C ( , C( , Ci )) T is finitely presented. In a similar way, each ( , Ki )T is finitely generated, and it follows that the ( ,ki+1 )

cokernel of the map 0 → ( , Ki+1 )T −−−−−→ ( , C( , Ci ))T is finitely presented. We have proved each ( , Ki )T is an extension of two finitely presented functors, therefore: it is finitely presented. k0 C( , C0 ) → M → 0, and the long (c) From the exact sequence 0 → K0 −→ homology sequence, we have an exact sequence 0 → ( , K0 )T → ( , C( , C0 ))T → ( , M )T → Ext1C ( , K0 )T → Ext1C ( , C( , C0 )T . Using again mod(T ) is abelian, it follows φ(M ) = ( , M )T is finitely presented. Corollary 4. Assume C and T have pseudokernels. Then T is contravariantly finite in mod(C). Proposition 21. Assume C and T have pseudokernels. Then the following statements hold: (i) The functors F, F ′ , G, G′ in Brenner-Buter’s theorem restrict to the subcategories of finitely presented functors. (ii) Given a functor M in T ∩ mod(C), there exists a resolution t

t

t

t

0 1 2 n M →0 T0 → T1 → · · · → T2 → → Tn →

δ

n Imtn is a T -approximation of such that, each Ti is in addT , and Tn → Imtn . (iii) If M is a functor in mod(C), then the the trace τT (M ) of T in M , and M/τT (M ) are finitely presented. (iv) Denote by tX the radical of the torsion theory (X , Y ) of Mod(T ). Then for any functor N in mod(T ), tX (N ) and N/tX (N ) are finitely presented.

CLASSICAL TILTING

21

(v) For any pair of finitely presented functors M, N in T , we have an isomorphism ExtiC (M, N ) = ExtiT (φ(M ), φ(N )). Proof. (i) We proved F preserves finitely presented functors. If ( , T1 ) → ( , T0 ) → N → 0 is a presentation of N ∈ mod(T ) , T0 , T1 ∈ addT , then tensoring the exact sequence with T , we obtain an exact sequence T0 → T1 → T ⊗ N → 0. Since T0 , T1 are finitely presented, and mod(C) is abelian, G(N ) = T ⊗ N is finitely presented. We left to the reader to prove that the functors F ′ and G′ , preserve finitely presented functors. (ii) Let M be in mod(C) and a map δ : T 0 → M with T 0 in Add(T ), as in Proposition 5, its image is τT (M ). If f : T0 → M is a T -approximation, then f factors through δ, and δ factors through f . In consequence, Imf = τT (M ). In particular, if M ∈ T , then f is an epimorphism. Using again the fact mod(T ) is abelian, the kernel of f , K0 , is finitely presented. From the exact sequence 0 → K0 → T0 → M → 0, the long homology sequence, and the fact f : T0 → M is a T -approximation, it follows Ext1C ( , K0 )T = 0. Hence, K0 is in T , and the claim follows by induction. (iii) Let M be in mod(C). Then by the proof of (ii), τT (M ) finitely generated, and M finitely presented, implies M/τT (M ) is finitely presented, and mod(C) abelian, implies τT (M ) is finitely presented. (iv) Assume N in mod(T ). Since the functors: F, F ′ , G, G′ preserve finitely presented functors, all terms in the exact sequence: η

γ

0 → Ext1C (−, TorT1 (N, T )) − →N − → φ(N ⊗ T ) → 0 are finitely presented. The claim follows by observing the isomorphisms: tX (N ) ∼ = Ext1C (−, TorT1 (N, T )) and N/ tX (N ) ∼ = φ(N ⊗ T ). (v) Follows as in Proposition 12. We denote by (T˜ , F˜ ) and (X˜ , Y˜ ) the intersection of the torsion theories (T , F ) and (X , Y ) with the categories of finitely presented functors, mod(C) and mod(T ), respectively. From the previous proposition we obtain the following: Theorem 5 (Brenner-Butler). Let T be a tilting subcategory of mod(C) and assume C and T have pseudokernls. With the above notation the following statements hold: (i) The functors F and G induce an equivalence between T˜ and Y˜ . (ii) The functors F ′ and G′ induce an equivalence between F˜ and X˜ . (iii) We also have: F G′ = F ′ G = 0 and G′ F = GF ′ = 0. Proof. The proof is clear from (i) in the above proposition.

2.5. Classical tilting for dualizing varieties. In order to have a complete analogy with tilting theory for finite dimensional algebras, we need to add more restrictions in our categories, in particular, we need the existence of duality. We will assume in this subsection that C and T are dualizing varieties. It was proved above that the category θ = {θC }C∈C , where θC = (C( , C), )T , is a tilting subcategory of mod(T op ). Then by Brenner-Butler’s theorem, there are torsion pairs (T (θ), F (θ)) and (X (θ), Y (θ)) in mod(T op ) and mod(θ), respectively, and equivalence of categories


22

T (θ) F (θ) ?? ?? ? φθ ? Ext1T op ( ? _?? Torθ1 ( ,θ) ??⊗θ ?? X (θ)

,−)θ

Y (θ)

By Proposition 17, there is an equivalence of categories α(C) : C op → θ, α(C) = θC = (C( , C), )T , which induces an equivalence α∗ : mod(θ) → mod(C op ) given by: α ∗ (H)(C) = H(α(C)) = H(θC ) = H((C( , C), )|T ), for each C in C, and H in mod(θ). Lemma 4. Let N be an object in mod(T ) and C one object in C. Then the following statements hold: (a) There is an isomorphism ((C( , C), )T , DN ) ∼ = D(N ⊗ T )(C), such that the following square mod(C) o

⊗T

mod(T )

D

mod(C op ) o

D α∗φθ

mod(T op )

commutes. (b) There is an isomorphism Ext1T op ((C( , C), )T , DN ) ∼ = D(TorT1 (N, T ))(C), such that the following square T

Tor1 ( mod(C) o

,T )

mod(T )

D

α∗Ext1 ( mod(C op ) o

D

mod(T op )

,−)|θ

commutes. (c) We have the following equivalences of categories: (i) D(X (T )) ∼ = F (θ), D(Y (T )) ∼ = T (θ) (ii) D(T (T )) ∼ = Y (θ), D(F (T )) ∼ = X (θ) Proof. Applying the functor ( , DN ) to the exact sequence (2.32)

0 → (T 1 , ) → (T 0 , ) → (C( , C), )T → 0.

We obtain by the long homology sequence, and Yoneda’s Lemma, the following exact sequence: (2.33) 0 → (( , C), )T , DN ) → DN (T 0 ) → DN (T 1 ) → Ext1 (( , C), )T , DN ) → 0,

CLASSICAL TILTING

23

and applying ⊗N to (2.32), by the long homology sequence, we get an exact sequence (2.34)

0 → TorT1 (N, T )(C) → N (T 1 ) → N (T 0 ) → (N ⊗ T )(C) → 0.

Dualizing (2.34), and comparing it with (2.33), we obtain the isomorphisms in (a) and (b). To see that the first square commutes, let N be in mod(T ). Then there are equalities: α ∗ (φθ (DN ))(C)

=

α ∗ (( , DN )|θ)(C)

=

((( , C), ), DN ) = D(N ⊗ T )(C).

The equalities α ∗ (Ext1 ( , −)|θ(DN ))(C)

= α ∗ (Ext1 ( , DN )|θ)(C) = ExtT1 ((( , C), ), DN ) = D(TorT1 (N ⊗ T ))(C),

imply, the second square commutes. It only remains to prove (c). By (a) it follows T (θ) = {N ∈ mod(T op )|Ext1 (θC , N ) = 0} = D(Y (T )). By Brenner-Butler’s Theorem, there are equivalences of categories: φθ −⊗T

: F (θ) → X (θ) : Y (T ) → T (T ).

Then we have a commutative square T (T ) o D Y (θ) o

⊗T

α∗φθ

Y (T ) D T (θ)

By part (b), it follows F (θ) = {N ∈ mod(T op )|Hom(θC , N ) = 0} = D(X (T )) From the equivalence of categories given in Brenner-Butler’s Theorem: TorT1 ( , T ) : X (T ) → F (T ) Ext1T op ( , −)θ

: T (θ) → Y (θ)

We have a commutative square: T

Tor1 ( F (T ) o D α∗Ext1 ( X (θ) o

,T )

X (T ) D ,−)|θ F (θ)


24

Definition 11. [ASS] Let C be Krull-Schmidt. A torsion theory (T , F ) in mod(C) splits, if every indecomposable M ∈ mod(C) is, either in T or in F . Proposition 22 (ASS Prop. 1.7). Let C be dualizing category and (T , F ) a torsion pair in mod(C). Then the following conditions are equivalent: (a) The torsion theory (T , F ) splits. (b) Let τ be the radical of the torsion theory. Then for any M en mod(C), the exact sequence: 0→ τ (M ) → M → M/τ (M ) → 0, splits. (c) For any N in F and any M in T , Ext1C (N, M ) = 0. (c) If M ∈ T , then TrDM ∈ T . (d) If N ∈ F , then DTrN ∈ F . We say that a tilting subcategory T of mod(C) separates, if the torsion theory (T (T ), F (T )) in mod(C) splits, and we say it splits, if the torsion theory (X (T ), Y (T )) in mod(T ) splits. Lemma 5. Let T be a tilting subcategory of mod(C) that splits. Then the following statements hold: f

g

(a) Let 0 → M − → E − → TrD(M ) → 0 be an almost split sequence, with M in T (T ). Then the three terms are in T (T ), and 0 → φ(M ) → φ(E) → φ(TrD(M )) → 0 is an almost split sequence, whose terms are in Y (T ). (b) Let 0 → DTr(M ) → E → M → 0 be an almost split sequence, with M in F (T ). Then the three terms are in F (T ), and 0 → Ext1C ( , Dtr(M ))T → Ext1C ( , E)T → Ext1C ( , M )T → 0 is an almost split sequence, whose terms are in X (T ). Proof. We will prove only (a), being (b) similar. By previous Lemma, TrD(M ) ∈ T (T ), therefore: E ∈ T (T ). By the long homology sequence, we get the exact sequence (2.35)

( ,f )

( ,g)

0 → ( , M )T −−−→ ( , E)T −−−→ ( , TrD(M ))T → Ext1C ( , M )T = 0,

whose terms are in Y (T ). The morphism ( , g)T is right minimal. Let η : ( , E)T → ( , E)T be an endomorphism, such that ( , g)T η˜ = ( , g)T . Since φ : T (T ) → Y (T ) is an equivalence of categories we have (( , E)T , ( , E)T ) ∼ = (E, E) and η˜ = ( , η)T , with η : E → E. Then gη = g, and since g is right minimal, it follows η is an isomorphism, hence, η˜ is an isomorphism. Let N be indecomposable and γ˜ : N → ( , TrD(M )) a non isomorphism and non zero map. Then, either N ∈ X (T ) or N ∈ Y (T ). If N ∈ X (T ), then γ˜ ∈ Hom(X (T ), Y (T )) = 0, hence, N = ( , H)T , with H ∈ T (T ) and γ˜ = ( , γ)T : ( , H)T → ( , TrDM )T , where γ is non isomorphism and non zero. Then η factors through E, and η˜ factors through ( , E)T . As a consequence of Lemma 5, and Proposition 21, we have: Lemma 6 ( ASS Lemma 5.5). Let T be a tilting subcategory of mod(C). If M ∈ T (T ) and N ∈ F (T ), then for any j ≥ 1, there is an isomorphism j−1 Extj (M, N ) ∼ = Ext (φ(M ), Ext1 ( , N )T ). C

T

C

From the fact θ = {θC = (( , C), )T }C∈C is a tilting subcategory in mod(T op ), we obtain the following:

CLASSICAL TILTING

25

Theorem 6. Let T be a tilting subcategory of mod(C).Then: (a) T separates, if and only if, for any Y ∈ Y (T ), pdimY = 1. (b) T splits, if and only if, for any N ∈ F (T ), idimN = 1. Proof. (b) Is proved in [ASS Theo. 5.6], but, for he benefit of the reader, we repeat the proof here. First, the sufficiency of the condition. Assume that for every N ∈ F (T ), we have idimN = 1. Let X ∈ X (T ) and Y ∈ Y (T ). Then there exist M ∈ T (T ) and N ∈ F (T ) such that X ∼ = Ext1C ( , N )T and Y ∼ = φ(M ), by Brenner-Buttler theorem. Hence, by the above Lemma Ext1 (Y, X) ∼ = Ext1 (φ(M ), Ext1 ( , N )T ) ∼ = Ext2 (M, N ) = 0, T

T

C

C

Conversely, assume that (X (T ), Y (T )) is spplitting and let N ∈ F (T ). Take an injective resolution of N d0

d1

d2

0 → N −→ I 0 −→ I 1 −→ I 2 → · · · Let L0 = Imd1 and L1 = Imd2 . Since T (T ) = KerExt1C (T , ) contais the injective objects and it is closed under epimorphic images, and N ∈ F (T ), it follows L1 ∈ T (T ). Then, by the above Lemma, we have: Ext1C (L1 , L0 ) ∼ = Ext2C (L1 , N ) ∼ = Ext1T (φ(L1 ), Ext1C ( , N )T ) But, φ(L1 ) ∈ Y (T ), Ext1C ( , N )T ∈ X (T ) and (X (T ), Y (T )) is splitting, by hypothesis. By Proposition 22, this implies Ext1T (φ(L1 ), Ext1C ( , N )T ) = 0, proving that the exact sequence 0 → L0 → I 1 → L1 → 0 splits . Therefore, L0 is injective and idimN ≤ 1. Finally, N ∈ F (T ), implies N is not injective, thus idimN = 1. We prove that (a) follows from (b): By definition, T separates, if and only if, the torsion pair (T (T ), F (T )) in mod(C) splits. By duality, this occurs if and only if, the pair (X (θ), Y (θ)) = (DF (T ), DT (T )), is a splitting torsion theory in mod(C op ) ∼ = mod(θ). By (b), θ is a tilting subcategory that splits in mod(T op ), if and only if for all N ∈ F (θ), idimN = 1. But F (θ) = DY (T ), and (a) follows. Corollary 5. If gdimC ≤ 1, then any tilting subcategory T ⊂ mod(C) splits. As a consequence of the previous theorem and Lemma 4, we have the following: Corollary 6. If C and T are hereditary, then T splits and separates The last theorems in this subsection will have important implications in the hereditary case, that we will consider in next section. 3. Infinite quivers We begin this section showing that there exist natural examples of the notions discussed in the previous section. We prove first that for a finite dimensional algebra Λ, having a finitely generated generator M , such that there exists a tilting EndΛ (M )op -module T , the module T can be extended to a tilting category of modΛ . We study next locally finite infinite quivers Q, and consider the quiver algebra -category. We prove that for locally finite infinite quivers, a section on the preprojective component produces a tilting category. To apply the tilting functor is analogous to apply an infinite sequence of partial Coxeter functors, to change the

26


orientation of the quiver. We next use these results to computate the AuslanderReiten components of the locally finite infinite Dynkin quivers. To describe the shape of all Auslander-Reiten components, it is enough to compute them for a fixed orientation, and then to apply tilting. We choose quivers with only sinks and sources. A simple modification of the arguments given by [ABPRS], proves that regular Auslander-Reiten components of locally finite infinite quivers are of type A∞ .

3.1. Extending tilting. In this subsection we will prove that there exist examples of tilting subcategories. The first source of examples is produced in the following way: Let C be a K-category, K a field, which is Hom-finite, with an additive generator Λ, such that there is a tilting End(Λ)op -module T , C ′ = {addΛ} is a subcategory of C containing the generator of C, Mod(C ′ ) is equivalent to Mod(End(Λ)op ) and it has tilting subcategory which corresponds to T under the equivalence. We will extend first the tilting subcategory of Mod(C ′ ) to a partial tilting subcategory of Mod(C) and then using Bongartz’ argument [B], [CF] we will extend it to a tilting subcategory of Mod(C). Lemma 7. Let C ′ be a subcategory of C. If T is a self orthogonal subcategory of Mod(C ′ ), consisting of finitely presented objects, then C ⊗C ′ T is a self orthogonal subcategory of Mod(C). Proof. Let T1 , T2 be objects of T , with presentations C ′ ( , C1′ ) → C ′ ( , C0′ ) → T1 → 0, C ′ ( , C1′′ ) → C ′ ( , C0′′ ) → T2 → 0. Let’s consider the functors: res : Mod(C) → Mod(C ′ ), C⊗C ′ : Mod(C ′ ) → Mod(C), as in [A]. We claim Ext1C (C ⊗C ′ T1 , C ⊗C ′ T2 ) = 0. Indeed, let e be an element of Ext1C (C ⊗C ′ T1 , C ⊗C ′ T2 ), e : 0 → C ⊗C ′ T2 → F → C ⊗C ′ T1 → 0. We apply C⊗C ′ to the projective presentations of T1 and T2 , to get projective presentations of the corresponding extension. By the Horse Shoe Lemma, we obtain the following commutative exact diagram:

0 0 / C( , C ′ ) C( , C1′ ) 0 a a C( , C1′ C1′′ ) / C( , C0′ C0′′ ) / C( , C ′′ ) C( , C1′′ ) 0 0 0

0 / C ⊗C ′ T1 / F / C ⊗C ′ T2 0

/ 0 / 0 / 0

By [A Prop. 3.2], there exists an isomorphism C ⊗C ′ res(F ) ∼ = IdMod(C) (F ) = F .

CLASSICAL TILTING

27

Since the functor res is exact, we have the following commutative exact diagram: 0 0 / C ′( , C ′ ) C ′ ( , C1′ ) 0 a a C0′′ ) C1′′ ) / C ′ ( , C0′ C ′ ( , C1′ / C ′ ( , C ′′ ) C ′ ( , C1′′ ) 0 0

0

0 / T1

/ 0

/ resF / T2

/ 0 / 0

0

By assumption, the column at the right of the diagram splits, hence tensoring with C⊗C ′ and using C ⊗C ′ res(F ) ∼ = F , the columns in the first diagram split, hence e splits. Definition 12. A full subcategory T of Mod(C) is a partial tilting, if its objects satisfy the following two conditions: (i) For each object T in T there is an exact sequence 0 → P1 → P0 → T → 0, with Pi finitely generated projective. (ii) For every pair of objects Ti and Tj in T , Ext1C (Ti , Tj ) = 0. Proposition 23. Let C ′ be a subcategory of C containing an additive generator Λ of C. If T is a partial tilting subcategory of Mod(C ′ ), then C ⊗C ′ T is a partial tilting subcategory of Mod(C). Proof. By Lemma 7, C ⊗C ′ T is self orthogonal. (ii) Let T be an object in T , and C ′ ( ,f )

0 → C ′ ( , C1′ ) −−−−→ C( , C0′ ) → T → 0, a projective resolution with C0′ and C1′ in C( ,f )

C ′ . Applying C ⊗ − , we get the projective presentation C( , C1′ ) −−−−→ C( , C0′ ) → C ⊗C ′ T → 0. Since Λ is in C ′ , then C ′ (Λ, C0′ ) = C(Λ, C0′ ) and C ′ (Λ, C1′ ) = C(Λ, C1′ ). In consequence, C⊗C ′ T (Λ) = T (Λ). We need to see C( , f ) is monomorphism. Let C be an object in C. Since Λ is an additive generator of C, then there is an epimorphism g : Λn → C → 0, and hence a monomorphisms 0 → C(C, Ci′ ) → C(Λ, Ci′ ), for i = 0, 1. In this way, we obtain a commutative exact diagram: 0

0

C(C, C1′ )

/ C(C, C ′ ) 0

C(C,f )

C(g,C1′ )

n

C(g,C0′ )

C(Λ ,f ) / C(Λn , C ′ ) C(Λn , C1′ ) 0

0

/ C ′ (Λn , C ′ )C 1

′

(Λn ,f ) ′ /

/ C ⊗C ′ T (C)

C (Λn , C0′ )

C⊗

/ C ⊗C ′

/ 0

′ T (g)

C T (Λn )

/ T (Λn )

and for any object C in C the map C(C, f ) is a monomorphism.

/ 0

/ 0

28


Theorem 7. Let C be an Hom-finite, K-category with additive generator Λ and let RΛ be the endomorphism ring RΛ = End(Λ)op . Assume Mod(RΛ ) has a partial tilting module. Then Mod(C) has a tilting subcategory. Proof. From the equivalence Mod(RΛ ) and Mod(add{Λ}), it follows the category Mod(add{Λ}) has a partial tilting object. The category C ′ = add{Λ}, is a subcategory of C containing Λ. By Proposition 23, there exists an object T in Mod(C) which is a partial tilting in Mod(C). Since T is finitely presented, we have an exact sequence 0 → L → C( , C0 ) → T → 0,

(3.1)

with L finitely generated. Let C be an object in C. Applying HomC ( , C( , C)) to (3.1), and using the long homology sequence, we obtain the following exact sequence: HomC (C( , C0 ), C( , C)) → HomC (L, C( , C)) → ExtC (T, C( , C)) → 0. Since L is finitely generated, there is an epimorphism C( , C1 ) → L → 0, and hence a monomorphism 0 → HomC (L, C( , C)) → HomC (C( , C1 ), C( , C)) ∼ = HomC (C1 , C). Since C is Hom-finite, then HomC (C1 , C) is a finite dimensional K-vector space. It follows HomC (L, C( , C)) and Ext1C (T, C( , C)) are finite dimensional K-vector spaces. We proceed now as in [B]: Let e1 , · · · , ed be a K-base of Ext1C (T, C( , C)). Represent each ei as a short exact sequence gi

fi

0 → C( , C) −→ Ei −→ T → 0. Consider the following diagram with exact raws: / C( , C)d

0

f

d a

/

Ei

g

/ Td

/ 0

i=1 ∇

0

1

u

/ C( , C)

v

/ EC

w

/ Td

/ 0

where f and g are sums of morphisms fi , and gi respectively and ∇ = [1, . . . , 1]. Let’s denote by eC the element of Ext1C (T d , C( , C)) represented by the exact sequence w v → T d → 0. eC : 0 → C( , C) − → EC − Let ui : T → T d be the i-th inclusions. We claim for each i = 1, . . . , d, ei = Ext1C (ui , C( , C))eC . Indeed, let’s consider the following commutative diagram with exact raws: fi / 0 / Ei gi / T 0 / C( , C) u′i

′′

ui

0

/ C( , C)d f/

ui

d a

Ei

g

/ Td

/ 0

i=1 ∇

0

/ C( , C)

1

u v

/ EC

w

/ Td

/ 0

CLASSICAL TILTING

29

where u′i , u′′i denote the corresponding inclusions. Since ∇u′′i = 1C( following commutative diagram with exact raws: 0 0

/ C( , C) fi / Ei uu′i 1 v / C( , C) / EC

gi

w

/ T ui / Td

,C) ,

we get the

/ 0 / 0

hence, the claim follows. Applying HomC (T, ) to the sequence eC , we obtain by he long homology sequence, an exact sequence δ

HomC (T, T d ) − → Ext1C (T, C( , C)) → Ext1C (T, EC ) → Ext1C (T, T d) = 0. Since ei = Ext1C (ui , C( , C))eC = δ(ui ), it follows each basic element of the vector space Ext1C (T, C( , C)) is in the image of the connecting morphism δ, which is then surjective. Therefore: Ext1C (T, EC ) = 0. ` We will see that the full subcategory T of Mod(C), with T = {T EC }C∈C , is a tilting category. Since EC is an extension of finitely presented ` functors, it is finitely presented. Hence, for any object C in Mod(C), the sum T EC is finitely presented. ` (i) pdimT EC ≤ 1. Since T is a partial tilting object object in Mod(C), it has pdimT ` ≤ 1, and from the exact sequence eC it follows pdimEC ≤ 1. Therefore: pdimT EC ≤ 1. ` ` (ii) For any pair of objects C and C ′ in C, Ext1C (T EC , T EC ′ ) = 0. Applying HomC ( , T ) and HomC ( , EC ′ ) to eC , we obtain by the long homology sequence, exact sequences: 0

= Ext1C (T d , T ) → Ext1C (EC , T ) → Ext1C (C( , C), T ) = 0,

0

= Ext1C (T d , EC ′ ) → Ext1C (EC , EC ′ ) → Ext1C (C( , C), EC ′ ) = 0.

Hence, for any pair of objects C and C ′ in Mod(C), we have: Ext1C (T, EC ′ ) = Ext1C (EC , T ) = Ext1C (EC , EC ′ ) = 0. Using the fact Ext1C (T, T ) = 0 and the above equalities, the condition (ii) follows. (iii) is immediate. 3.2. The Hereditary Case. In this subsection Q = (Q0 , Q1 ) will be a locally finite, infinite quiver. We will consider the quiver algebra KQ as a subadditive K-category C, and the finite dimensional representations of the quiver, will be identified with the category of finitely presented contravariant functors mod(KQ) from KQ to he category of finite dimensional K-vector spaces. The category C is Homfinite, dualizing and Krull-Schmidt. By [AR] mod(C) has almost split sequences. We will describe the Auslander-Reiten components, beginning with he preprojective components. The preprojective component K of the Auslander-Reiten Γ(KQ), of KQ, can be computed as in the finite quiver situation. It is easy to prove that it is a translation quiver of the form (−N∆, τ ), ∆ = Qop . The category K is the quiver algebra K(−N∆) module the mesh relations K =K(−N∆)/ < mx > [R, Lemma 3, Section 2.3]. We will need the following combinatorial lemma, which can be proved as in [BGP].

30


Lemma 8. Let Q be a locally finite infinite quiver that is not of type A∞ , A∞ ∞, D∞ . Given a finite subquiver Q′′ of Q, there exists a finite full connected subquiver Q′ ⊂ Q such that Q′′ is a subquiver of Q′ and Q′ is not Dynkin. Definition 13. [ASS] Let (Γ, τ ) be a connected translation quiver. A connected full subquiver Σ is a section of Γ if the following conditions are satisfied: (S1) If X0 → X1 → · · · → Xt is a path in Σ of lenght l ≥ 1, then X0 6= Xt . (S2) For each X ∈ Γ0 , there exist a unique n ∈ Z such that τ n X ∈ Σ0 . (S3) If X0 → X1 → · · · → Xt is a path in Γ with X0 , Xt ∈ Σ, then Xi ∈ Σ0 for all i such that 0 ≤ i ≤ t. Definition 14. [ARS] Let I be the integers in one of the intervals (−∞, n], [n, ∞), fi

[m, n] for m < n or {1, . . . , n} modulo n. Let · · · → Xi −→ Xi+1 → · · · be a path in the Auslander-Reiten quiver (Γ, τ ) with each index in I. This path is said to be sectional if τ Xi+2 ≇ Xi . f1

f2

fn−1

Lemma 9. Let C be a dualizing variety, and X1 → X2 → · · · Xn−1 → Xn sectional path in mod(C). Then the composition fn−1 fn−2 ...f1 is not zero. Proof. The proof is as in [ARS Theo. 2.4].

Remark 3. Let Q be a locally finite infinite quiver, K the preprojective component of the Auslander-Reiten quiver of KQ, that we identify with the quiver algebra with quiver (−NQ, τ ) and the mesh relations. Then given an object X in K and a positive integer n, there exists a finite number of directed paths in (−NQ, τ ) with starting vertex X and length ≤ n. Theorem 8. Let Q be a locally finite infinite quiver and Σ a section of (−NQ, τ ) with no infinite directed paths, then the following is true (a) For any vertex X of (−NQ, τ ) the number of directed paths from X to the section is finite and there is a non zero path from X to the section. (b) For any vertex X of (−NQ, τ ) the number of directed paths from the section to X is finite and there is a non zero path from the section to X. Proof. (a) The proof will be by induction on the least n ≥ 0 such that τ −n X is in Σ. In the case there is not such an n, then X is a successor of the section and the number of paths to the section is zero. If n = 0, then X is on the section and the claim is true, since the number of directed paths on the section is finite. Assume n = 1 and consider X an element of the preprojective component with almost split sequence: (σαi )

0→X →

n a

(αi )

Ei → τ −1 X → 0

i=1

If all Ei are in the section, then there is nothing to prove. Let αi : Ei → τ −1 X be a map that it is not in the section. Then σ −1 αi : −1 τ X → τ −1 Ei is in the section, assume there is a irreducible map β : Ei → Y , different from αi . Then there is an irreducible map σ −1 β : Y → τ −1 Ei . Let n be the maximum of the lengths of the paths in Σ starting at τ −1 X. By the above remark, there exists only a finite number of paths of length ≤ n starting at Y and

CLASSICAL TILTING β

βn−1

β2

β1

31

assume Y → Y1 → Y2 → ... → Yn is a path that does not meet any of the paths starting at τ −1 X and ending at Σ. Then for k ≤ n we have the following diagram of irreducible maps: −2

σ τ −1 Yk−1 o O

(3.2)

σ 1 βk

Yk o

βk−1 −1

τ

−2

Yk−2 O

σ−1 βk−1 βk

Yk−1

σ τ −1 Yk o O

βk−1

σ τ −1 E1 o O

σ−1 β1

Y1 o

−1

Y o

τ −1O X o

α2

β

E1 o

E2 O σα2

α

σ−1 β β1

α

σα

X

with τ −1 Yk−1 in Σ, and there is no irreducible map τ −1 Yk−1 →Z with Z on Σ. By the definition of section, σ −1 βk :Yk → τ −1 Yk−1 is in Σ. Assume there is only a finite number of paths for any X in (−NQ, τ ) with τ −m+1 X ∈ Σ0 . As before, we consider an almost split sequence starting at X and a map αi : Ei → τ −1 X that is not in the section. Then, by induction hypothesis, there is a finite number of paths from τ −1 X to the section. Let n be the length of the largest such path. Assume there is a irreducible map β : Ei → Y , different from β

β1

β2

βn−1

αi . Arguing as above, we consider a path Y → Y1 → Y2 → ... → Yn that does not meet any of the paths starting at τ −1 X. Then we obtain a diagram of irreducible maps as in (3.2) and for some integer k ≤ n, τ −1 Yk−1 in Σ. Then either σ −1 βk :Yk → τ −1 Yk−1 is in Σ, or τ −1 Yk is in Σ0 . In any case, it follows from the case n = 1, there is only a finite number of paths from X to Σ. Moreover, since sectional paths have non zero composition, there is a non zero path from X to the section. (b) Is proved using dual arguments or going to the opposite category. Proposition 24. Let K be a preprojective component Γ(KQ) and Σ a section of K without infinite directed paths. Let P be an indecomposable projective. Then there exist a short exact sequence, 0 → P → T 0 → T 1 → 0, with T 0 , T 1 ∈ addΣ0 . Proof. (1) We will separate the proof in two cases, assuming first Q is not of type ∼ A∞ , A∞ ∞ or D∞ . We use the isomorphism K = K(−N∆)/ < mX >, where mX denotes the set of mesh relations. The projective P is identified with a vertex v0 in ∆0 . Let V0− = {vi− ∈ Q0 |there is a path vi− → v0 }, let V0 = {v1 , v2 ..., vn } be the vertices of Σ that are the ending vertex of a path starting at some vi− . Since Σ has no infinite directed paths, the set V1 = {v0 , v1 ..., vm } of all vertices of Σ that are connected with a length zero or directed path to vertices of V0 , is also finite and contains V0 , Σ′′ is the full subquiver of Σ with vertices V1 . Denote by VP the collection of all paths starting at some vertex of V0− and ending at Σ. Let {T0 , T1 ..., Tm } be the objects of K corresponding to the vertices V1 and denote by P denote the finite set of indecomposable projective that appear as summands in the minimal projective presentations of any of the Ti for 1 ≤ i ≤ m. The objects in P correspond to vertices of a finite subquiver Q′′ of Q, which by Lemma 8 is contained in a finite, full, non Dynkin subquiver Q′ of Q. Let ∆′ = Q′op , since ∆′ is not Dynkin, (−N∆′ ) is a connected, full subquiver of (−N∆) and VP is completely contained in (−N∆′ ). We identify the preprojective component K′ of the Auslander-Reiten quiver Γ(KQ′ ), with K(−N∆′ )/ < m′X >, where m′X is the set of mesh relations in K(−N∆′ ). Let’s consider the ideal I of


32

K(−N∆) defined by: f ∈ I(X, Y ), if and only if, f factors through τ −i Z, for some Z ∈ ∆0 /∆′0 , i ∈ N By the universal properly of quiver algebras, there exists a functor: −

F : K(−N∆′ ) →

K(−N∆) I+ < mX >

The set of meshes can be separated in three kinds: (i) Meshes m′Xj which are also meshes in K(−N∆), (ii) The mesh mXk of K(−N∆) that is of the form Σ{α∈Γ(Λ)|F (α)=Xk } ασ(α), where ασ(α) factors through an object in (−N∆)r(−N∆′ ) and (iii) The meshes mXl of K(−N∆) of the form mXl = ρ1j + ρ2l , where ρ1l is a sum of morphisms factoring through some object in (−N∆) r (−N∆′ ) and ρ2l consists of morphims which do not factor through (−N∆) r (−N∆′ ) . Then it is clear that −

the kernel of F is < m′X > and there exists a full, faithful and dense functor: F :

K(−N∆′ ) K(−N∆) → < m′X > I+ < mX >

Let Σ′ be the subquiver of Σ consisting of all vertices which correspond to orbits under the inverse Auslander-Reiten translation of projective corresponding to the vertices of Q′ , since Q′ is non Dynkin, Σ′ is a section of K′ . (2) According to [HR Theo. 7.2], there exists a short exact sequence of KQ′ modules (gji )ji a (fi )i a Tj → 0 Ti −−−−→ 0 → P −−−→ j

i

Σ′0 ,

with Ti and Tj corresponding to vertices in and fi , gji are K-linear combinations of paths in K(−N∆′ ). P By the equivalence in part (1), (fi )i is a monomorphism and i gji fi = 0 in K. We have the exact sequence of KQ-modules (hki )ki a (fi )i a Ti −−−−→ Ck → 0 0 → P −−−→ i

k

` where C = Coker(gji )ji , is the cokernel of (gji )ji in K and C = k Ck its decomposition in sum of indecomposable KQ-modules. property of the cokernel, there exists a morphism (ljk )jk : P ` ` By the universal k ljk hki : Ti → Tj . By j Tj such that the map gij is equal to k Ck → condition (S3) of the definition of section, each Ck ∈ Σ0 . ′ ′op If Q is of type A∞ , A∞ large enough ∞ or D∞ , then we can choose ∆ = Q ′ in order to the injective KQ -modules of the Auslander-Reiten quiver of Γ(KQ′ ) appear as successor of Σ′′ and then apply an argument similar to the first case. We next prove he main result in this subsection. Theorem 9. Let Q be a locally finite infinite quiver, K a field and KQ the quiver algebra consider as a preadditive category. Let K be a preprojective component of the Auslander-Reiten quiver of KQ and Σ a section of K without infinite oriented paths. Then, addΣ0 is a tilting subcategory of mod(KQ).

CLASSICAL TILTING

33

Proof. Since Q is locally finite, KQ is hereditary and (iii) was proved in Proposition 24, we only need to prove condition (ii). Let T1 and T2 be non projective objects in Σ0 , since KQ is a dualizing variety, Auslander-Reiten formula Ext1KQ (T1 , T2 )=D(HomKQ (T2 , τ T1 )) holds. Let ϕ : T2 → τ T1 be a non zero morphism, then using the Ausalnder-Reiten sequence we have morphisms between indecomposable objects: T2 → τ T1 → Ei → T1 From condition (S3), τ T1 ∈ Σ, a contradiction. Corollary 7. If T = addΣ0 is a tilting subcategory, then Mod(T ) has global dimension one. Proof. Let P be an indecomposable projective in mod(T ). Since T is Krull-Schmidt P ∼ = ( , T ) with T in Σ0 . By Brenner-Butler’s theorem, the subcategory Y = {N |Tor1T (N, T ) = 0} ⊂ mod(T ) is a torsion free class containing the projective, and it is equivalent to T = {M |Ext1C (Ti , M ) = 0, Ti ∈ Σ0 } ⊂ mod(C), via the functor φ. Let Y be a non zero sub-object of P . Then there is a monomorphism α : Y → P . Since P is contained in the subcategory Y ⊂ mod(T ), then Y ∈ Y , since it is closed under sub-objects. By the equivalence φ : T → Y , there exists an indecomposable object M in T such that φ(M ) = ( M )T ∼ = Y . Moreover, the inclusion Y ֒→ P is of the form ( , f ) : ( , M )T → ( , T ), with f : M → T a non zero morphism. Since Y is a non zero functor, there exists an object T ′ ∈ Σ0 such that 0 6= Y (T ′ ) = Hom(T ′ , M ). Let g ∈ Hom(T ′ , M ) be a non zero morphism, then there is a chain of morphisms g

f

→M − →T T′ − Then by Property (S3) of a section, M is in Σ0 and we conclude Y is projective. 3.2.1. Representations of Infinite Dynkin Diagrams. In this sub section we will apply the results of the previous subsection to compute the Auslander-Reiten quivers of the infinite Dynkin quivers A∞ , A∞ ∞ or D∞ without infinite paths. The Auslander Reiten quivers of the infinite Dynkin quivers were computed in [ReVan III. 3] for a fixed orientation. We will apply here the tilting theory so far developed to compute the Auslander Reiten quivers for arbitrary orientations. We give first the computation of the Auslander Reiten for an infinite Dynkin quiver with only sinks and sources, then we change the orientation by tilting with the objects in a section of the preprojective component with no infinite paths, and we prove that tilting with objects in a section behaves as in the finite dimensional case, it removes a portion from the preprojecive component and glues it on the preinjective component leaving the other components invariant. Proposition 25. Let Q be an infinite Dynkin quiver with only sinks and sources, Γ(Q) the Auslander-Reiten quiver of Q then: (a) Γ(A∞ ) have only two components: the preprojective and the preinjective components. (b) Γ(D∞ ) have three components: the preprojective, the preinjective and a regular component, the regular component are of type A∞ . (c) Γ(A∞ ∞ ) has 4 components: the preprojective, the preinjective and two regular components, the regular components are of type A∞ .

34


Proof. In the case of locally finite infinite quivers, we have almost split sequences and we can compute them, and the preprojective component, as in the finite dimensional case; proceeding by induction starting with the indecomposable projective. Every indecomposable representation has finite support, hence it can be considered as a representation of a finite Dynkin quiver. (a) Consider the quiver A∞ : 0 ← 1 → 2 ← 3 → · · · , for each pair of integers b ≥ a ≥ 0, let Mab ∈ rep(A∞ ), be the representation defined as (Mab )i = K if a ≤ i ≤ b and 0 in the remaining vertices. The simple projective objects are 2(n+1) 2n P (2n) = M2n , and the non simple projective are P (2n + 1) = M2n , with b 2m n ≥ 0. The non projective representation Ma with b even, can be written as M2k , 2m 2m M2k+1 , with m > k + 1, k ≥ 0 and M1 , with m ≥ 1. A computation shows 2(m−1)

2(m−1)

2(m−1)

2m 2m ) = M2k−1 , and DTr(M12m ) = M0 , hence DTrM2k = M2(k+1) , DTr(M2k+1 b each Ma with b even, is in the preprojective component. By a similar computation Mab with b odd is in the preinjective component. We look now to the quiver D∞ 0 o 2 / 3 o 4 / ··· 1

We use the fact that every indecomposable representation has support in a finite Dynkin quiver whose representations we know, (See [R]). For each pair of integers m,n, with m ≥ n ≥ 2 we have indecomposable representations, Mnm , defined as follows: (Mnm )i = K, if n ≤ i ≤ n, and 0 in any other vertex. For m ≥ 2 we define the indecomposable representations ( ( K if 1 ≤ i ≤ m, K if i = 0 or 1 < i ≤ m, (N0m )i = , (N1m )i = 0 if i = 0 or i > m. 0 in any other vertex. For integers m and l, with m > l ≥ 2, let’s define  2  if 2 ≤ i ≤ l, K m (Ll )i = K if i ∈ {0, 1} or l + 1 ≤ i ≤ m,   0 in the other vertices.

For each integer m ≥ 0, let’s define ( K m (L )i = 0

if 0 ≤ i ≤ m, in the other vertices.

To compute the preprojective components of the Auslander-Reiten quiver K(D∞ ) we compute the orbits of P (1) and P (0) under TrD and to compute the preinjective component we take the orbits of I(1) and I(0) under DTr, to obtain {P (1), N13 , N05 , N17 , N09 , . . .}, {P (0), N03 , N15 , N07 , N19 , . . .} {I(1), N14 , N06 , N18 , N010 , . . .}, {I(0), N04 , N16 , N08 , N110 , . . .} We see that the representations that lies in the preproyective component are: N1m , N0m , Lm and Mnm with n and m odd or 0, and Lm l with l and m odd. The representations that lies in the preinyective component are: N1m , N0m , Mnm with n and m odd or 0, and Nlm with l and m even. m m Finally, the representations Mnm , Lm n with m + n and l + m odd, and L , M0 with m even, lies in a component of type ZA∞ .

CLASSICAL TILTING

··· N6 ? 5 ??? · · · N36 o N5 ? 4 ? ??? 4 o 6 ··· N N3 o ?? ? ? ? 6 o o 4 · · · M3 N23 ? N ?? ??? ? ? ? 6 o 4 o 2 M M N o

35

···

5

3

··· ??? ? o ???

N25

··· ???

? M3 o

M25

2

··· ??? M45

(b) We consider next the quiver A∞ ∞ : · · · → −2 ← −1 → 0 ← 1 → 2 ← 3 → · · · As above, for each pair of integers b ≥ a, let Mab ∈ rep(A∞ ∞ ), be the representation defined as (Mab )i = K if a ≤ i ≤ b and 0 in the remaining vertices. The projective 2(n+1) 2n simple are P (2n) = M2n , and the non simple projective are P (2n + 1) = M2n , b with n ∈ Z. The non projective representations Ma with a and b even can be 2(m−1) 2m 2m written as M2k with m > k + 1, and k ≥ 0, and we have TrD(M2k ) = M2(k+1) , and by induction Mab with a and b even are in the preprojective component. Using the same argument, we can see that in case a and b are odd, then Mab is in the preinjective component. b+2 If a is even and b odd, then DTr(Mab ) = Ma+2 , and all these representations are b−2 in a regular component, if a is odd and b is even, then DTr(Mab ) = Ma−2 and we obtain the elements of the second regular component. If a is even and b odd. The almost split sequences are of the form b+2 0 → Ma+2 → Mab+2 → Mab → 0 a b+2 b 0 → Ma+2 → Ma+2 Mab+2 → Mab → 0

the first appears in on the border of the regular component If a is odd and b even. The almost split sequences are of the form b−2 b 0 → Ma−2 → Mab → 0 → Ma−2 a b−2 b → Mab → 0 0 → Ma−2 → Mab−2 Ma−2

the first appears in on the border of the regular component Proceeding as in the finite dimensional case, we see that in the three cases the preprojective components are of the form (−NQ, τ ) and the preinjective components of the form (NQ, τ ). If Q is a locally finite quiver then KQ is a dualizing variety, if Σ is a section without infinite paths, then T = addΣ0 of mod(KQ), the functor φ : Mod(C) → Mod(T ) restricts to the category of finitely presented functors φmod(C) : mod(C) → mod(T ). We also proved T is a dualizing hereditary category, therefore both KQ and T splits. Choosing a section without infinite paths corresponds with changes in the orientation of the quiver Q. We observe next that tilting with the objects of a section without infinite paths in the preprojective component, behave as in the finite quiver situation; The Auslander Reiten components of the tilted category are as follows:


36

we cut all the predecessors of the section in the preprojective component to get the preprojective component of the tilted category. We glue what we cut as successors of the injective to build the preinjective component. The regular components remain without changes. Let P(C) be the preprojective component of the Auslander-Reiten quiver, Γ(KQ) and Q(T ) the preinjective component of the Auslander-Reiten quiver de Γ(T ). By Auslander-Reiten formula we have T (T ) = =

{X ∈ mod(KQ)|Ext1 (T, X) = 0, T ∈ Σ0 } {X ∈ mod(KQ)|Hom(X, τ T ) = 0, T ∈ Σ0 }.

We also know F (T ) = {X ∈ mod(KQ)|Hom(T, X) = 0, T ∈ Σ0 }. Definition 15. The set of predeccessors of Σ (successors, succΣ), preΣ, is the set of indecomposables X such that there is a T ∈ Σ0 and an integer n > 0 with T = τ −n X (T = τ n X). Lemma 10. Let X be an indecomposable object in mod(KQ). Then, X is in T (T ), if and only if, X is not a predecessor of Σ. Moreover, F (T ) = preΣ. Proof. Assume X is in T (T ). If X is a predeccesssor of Σ, then, by Theorem 8, there is a non zero path from X to τ Σ, a contradiction. Assume now X is not a predeccessor of Σ. If there exists, T ∈ Σ0 and a non zero map f : X → τ T , then for some positive integer k, τ k T is projective, hence X is preprojective and it has to be a succesor of Σ. Then there is a T0 ∈ Σ0 and a non zero map g : T0 → X. Hence we have maps: T0 → X → τ T → E → T , contradicting the fact, T0 , T ∈ Σ0 . The last claim follows from the fact that the tilting category T separates. We have the following: Proposition 26. The follwing statements hold: (a) For each T ∈ Σ0 , the T -module Ext1 ( , τ T )T is injective. (b) For any positive integer k, and any object T in Σ, there is an isomorphism τ k Ext1 ( , τ T )T ∼ = Ext1 ( , τ k+1 T )T . (c) Given an indecomposable projective (−, C) there is a natural isomorphism in mod(T op ) : D(Ext1 (−, (−, C))T ) ∼ = τ (((−, C), −)T ). Proof. (a) For each X ∈ mod(T ), and each non projective T ∈ T there is an isomorphism (X, Ext1 ( , τ T )T ) ∼ = DX(T ). Indeed, let 0 → ( , T1 ) → ( , T0 ) → X → 0 be a projective resolution of X, and T in T non projective. Applying Auslander-Reiten formula, there is an isomorphism η : (X, Ext1 ( , τ T )T ) → DX(T ), such that the following diagram commutes 0 → (X, Ext1 ( , τ T )) / (( , T0 ), Ext1 ( , τ T )) / (( , T1 ), Ext1 ( , τ T )) ∼ ∼ η = = / D(T, T1 ) / D(T, T0 ) 0 → D(X(T ))

CLASSICAL TILTING

37

(b) Consider an almost split sequence in F (T ): 0 → τ 2T → E → τ T → 0 By Lemma 5, it induces an almost split sequence in X (T ): 0 → Ext1 (−, τ 2 T )T → Ext1 (−, E)T → Ext1 (−, τ T )T → 0 ∼ Ext1 ( , τ 2 T )T . from which it follows τ Ext1 ( , τ T )T = The rest is by induction. (c) Let (−, C) be an indecomposable projective in mod(KQ), then there is an exact sequence: 0 → (−, C) → T1 → T0 → 0 which induces by the long homology sequence exact sequences: 0 → (−, (−, C)) → (−, T1 ) → (−, T0 ) → Ext1 (−, (−, C))T → 0 0 → (T0 −) → (T1 , −) → ((−, C), −)T → 0 The second exact sequence gives a projective presentation of ((−, C), −)T , taking the transpose and dualizing we obtain the exact sequence: 0 → τ (((−, C), −)T ) → D((−, T0 )) → D((−, T1 )) → 0 Dualizing the first exact sequence we obtain an exact sequence: 0 → D(Ext1 (−, (−, C))T ) → D((−, T0 )) → D((−, T1 )) → 0 which implies: D(Ext1 (−, (−, C))T ) ∼ = τ (((−, C), −)T ) The results in the proposition can be interpreted as the construction of the Auslander-Reiten components of mod(T ) by gluing the predecessors of Σ as successors of the injectives and leaving the remaining components unchanged, as illustrated in the following diagram: ❅ ❅ ❅ ❅

❅ ❅ F (T )

φ

❄

Y (T )❅ ❅ ❅ ❅

DtrΣ

Σ

T (T )

P(T )

Ext1 ( , −)T ❄ X (T )

Ext1C ( , DTrΣ)T

Q(T )

❅ ❅

As a corollary we obtain the main theorem of the subsection. Theorem 10. Let Q be a locally finite infinite Dynkin quiver, Γ(Q) the AuslanderReiten quiver of Q then:

38


(a) Γ(A∞ ) have only two components: the preprojective and the preinjective components. (b) Γ(D∞ ) have three components: the preprojective, the preinjective and a regular component, the regular component are of type A∞ . (c) Γ(A∞ ∞ ) has 4 components: the preprojective, the preinjective and two regular components, the regular components are of type A∞ . 3.2.2. The regular components. In the previous subsection we were concerned with the preprojective components and with all the components of a locally finite infinite quiver of type: A∞ , A∞ ∞ or D∞ . In this subsection we study the regular components of an arbitrary locally finite infinite quiver and prove that the regular components are of type A∞ . To prove this, we will follow very closely the proof given by ABPRS, [see ARS]. Since KQ is hereditary and the radical of mod(KQ) has properties very similar to the finite dimensional case, we can also use length arguments. We follow the first part of ABPRS’s proof to conclude that the number of indecomposable summands, α(M ), of a KQ-module M in a regular component, is at most three, and in case α(M ) = 3, there exist chains of irreducible monomorphisms Bi,ti → Bi,ti−1 → · · · → Bi,1 = Bi → M , with α(Bi,ti ) = 1 and α(Bi,j ) = 2 for j < ti , where f1 f2 a a g1 g2 g3 f3 B2 0 → DTrM −−−→ B1 B3 −−−−−−−→ M → 0 is an almost split sequence (see [ARS Prop. 4.11]). To exclude the case α(M ) = 3, we will reduce to a finite quiver situation to obtain a contradiction. We will make use of the following lemma:

Lemma 11. Let C be a Hom-finite dualizing locally finite Krull-Schmidt K-category. Let B = {Bi }i∈I be a finite family of objects in mod(C). For each i ∈ I, consider almost split sequences in mod(C) fi

gi

0 → DTr(Bi ) −→ Ei −→ Bi → 0. Then, there is a finite subcategory C ′ ⊂ C, such that the restriction gi |C ′

fi |C ′

0 → DTr(Bi )|C ′ −−−→ Ei |C ′ −−−→ Bi |C ′ → 0 is an almost split sequence in mod(C ′ ) Proof. We leave the proof to the reader.

Theorem 11. Let Q = (Q0 , Q1 ) be a connected locally finite infinite quiver and M an indecomposable module in the regular component of the Auslander-Reiten quiver Γ(KQ). Then α(M ) ≤ 2. Proof. Since we already know the shape of the Auslander-Reiten components of the infinite Dynkin quivers, we may assume Q is non Dynkin. `3 Let’s suppose α(M ) = 3, and let 0 → DTr(M ) → i=1 Bi → M → 0 be an almost split sequence and chains of irreducible monomorphisms Bi,ti → · · · → Bi,1 = Bi → M, i = 1, 2, 3, as above. (a) We proceed as in the lemma 11, defining B = {M } ∪ {Bi,j } ∪ {TrD(Bi,j )}, i = 1, 2, 3, 1 ≤ j ≤ ti , to find a subcategory C ′ ⊂ C, such that the almost split

CLASSICAL TILTING

39

sequences of the objects in B, restrict to almost split sequences in mod(C ′ ) 0

→ DTr(Bi,j )|C ′ → Ei |C ′ → Bi,j |C ′ → 0, 1 ≤ j ≤ ti

0

→ Bi,j |C ′ → Ei′ |C ′ → TrDBi,j |C ′ → 0, 1 ≤ j ≤ ti

0

→ DTrM |C ′ →

3 a

Bi |C ′ → M |C ′ → 0,

i=1

Adding a finite number of objects, if needed, we can identify C ′ with KQ′ , where Q′ is a finite connected non Dynkin full subquiver of Q. Observe that these almost split sequences will be almost split sequences for any quiver algebra of a finite subquiver Q′′ of Q′ containing Q′′ , since it will contain the support of the objects in the sequences. Since α(M ) = 3, the module M |C ′ is in a preprojective or in a preinjective component. We assume it is in the preprojective component, the other case will follow by duality. τ −1 B1,3 τ −1 B2,3 : . JJ : : . tt tt JJ JJ : :t t t J tt J t t J J t t JJ t t JJ t $ tt $ tt JJ $ tt $ $ . tt − −1 B1,2 B2,2 τ B1,2 τ B2,2 . : : J J t : JJ : JJ JJ : tt t: tt JJ J t t J t J t t J t JJ JJ t JJ tt tt J$ JJ tt $ $ $ tt $ . ttt $ −1 −1 TTTT B B τ B τ B T 1 2 : 1 2 T T T : T T O TTTT O TTT: T TTTT TTTT TTTT TTTT TTTT TTTT TTTT TTTT$ TTTT$ TTT*$ * * . M JJ :t JJ : JJ JJ JJ t: t tt t J t t J t JJ t J t JJ t JJ JJ tt tt JJ tt J$ JJ tt tt tt $ $ . tt τ − B3 : JJJ : B3 JJ : JJ tt t J t t J t JJJ t JJ JJ t tt JJ tt JJ tt t J t t J t $ t JJ t $ t t t $ B3,2 τ −1 B3,2

B1,3 .

.O .

.

B2,3

JJJ : JJJ J$

.

.

.O .

.

Since we are assuming Q′ is non Dynkin, the preprojective component is of the from (−N∆′ , τ ), with Q′ = ∆′ , hence, the section consisting of the three paths Bi,ti → · · · → Bi,1 = Bi → M, i = 1, 2, 3 is isomorphic to Q′ after a change of orientation. But for any larger subquiver Q′′ the algebra KQ′′ will have the same section in its preprojective component as KQ′ , which implies Q is a finite quiver, contradicting our hypothesis. These results are used in the next section. The shape of the Auslander-Reiten components of infinite quivers was found in an independent way by [BSP]. 4. The Auslander Reiten Components of a regular Auslander Reiten Component In the last section we use the results of the previous section to describe the Auslander Reiten components of a regular Auslander-Reiten component of a finite dimensional algebra, to do this, we need the concepts and results of [MVS1], [MVS2], [MVS3], which we briefly recall here.

40


Let Λ be a finite dimensional algebra over an algebraically closed field K. We denote by modΛ the category of finitely generated left Λ-modules and by Agr (modΛ ) the category with the same objects as modΛ and whose morphisms between A and B in Agr (modΛ ) are given by: ` HomAgr (modΛ ) (A, B) = i≥0 radiΛ (A, B)/radi+1 Λ (A, B).

In [MVS1] the definition of Koszul and weakly Koszul categories was given and it was proved that modΛ is weakly Koszul and Agr (modΛ ) is Koszul. Koszul categories are a natural generalization of Koszul algebras and the main results of Koszul theory extend to Koszul categories in particular we have Koszul duality. Next, recall the construction of the Ext-category of a full subcategory of an abelian category. For an abelian category D we consider the Ext-category E(D′ ) of the full subcategory D′ of D. The graded category E(D′ ) has the same objects as D′ and maps given by: HomE(D′ ) (A, B) =

a

ExtiD (A, B),

i≥0

for all objects A and B in E(D′ ) . We recall the application of this construction to mod(modΛ ). Let’s denote by C the category mod(modΛ ) and by S(C) the full additive subcategory of C generated by the simple functors SC =HomΛ (-, C)/rad(-,C):(modΛ )op , for all indecomposable objects C in modΛ . The categoryAgr (modΛ ) is graded and according to [IT] the category of graded functors Gr(Agr (modΛ )) has global dimension two. Then the Ext-category E(S(C)) has the same object as S(C) and maps: HomE(Agr (modΛ )) (SA , SB ) =

a

ExtiC (SA , SB )

i≥0

The category E(S(C)) is locally finite of Loewy length 3. Moreover, the following theorem was proved in [MVS3]: Theorem 12. Let C be an indecomposable Λ-module, and consider the indecomposable projective functor PC = HomE(Agr (modΛ )) (−, SC ) in Gr(E(Agr (modΛ ))). Then one of the following statements is true: (i) PC is a simple projective, when C is a simple injective Λ-module. (ii) PC is a projective of Loewy length 2, when C is a non simple injective module. (iii) PC is a projective injective functor of Loewy length 3, when C is non injective. Let K be a fixed component of the Auslander-Reiten quiver of Λ, Agr(K) is the full subcategory of Agr (modΛ ) generated by objects in K the corresponding Extop category E(Agr(K)) is equivalent to E(K). The full subcategory grPI (E(K) ) of op op gr(E(K) ) consisting of objects in gr(E(K) ) without projective-injective summands it is equivalent to a radical square zero category. Radical square zero categories are stably equivalent to hereditary K-categories. The stable equivalence is obtained in a way similar to the radical square zero artin algebras.

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Proposition 27. [MVS3]Let K be the additive closure of a connected component of the Auslander-Reiten quiver of a finite dimensional algebra Λ over an algebraically closed field K . (a) The category gr(E(K)op /rad2 ) ∼ =gr(Agr(K)/rad2 ) is stably equivalent to gr(H), where H is a disjoint union of hereditary full subcategories corresponding to the sections of the Auslander-Reiten component of K given by HC for an indecomposable module C in K. (b) Each subcategory gr(HC ) is equivalent to the category of representations of the separated quiver of the component K of the Ausalnder-Reiten quiver at C. For a regular component we have he following: Theorem 13. [MVS3] Let K be a connected regular component of the AuslanderReiten quiver of a finite dimensional algebra Λ over an algebraically closed field K. Then op (a) The category gr(E(K) /rad2 ) ∼ =gr(Agr(K)/rad2 ) is stably equivalent to gr(H), where H is a disjoint union ∪ Hi of hereditary full subcategories i∈Z

corresponding to the sections of the Auslander-Reiten component of K (b) The categories Hi , Hi+2 are equivalent and the categories Hi , Hi+1 are opposite categories for all i. (c) If Hi is finite, then it is non-Dynkin. It was also proved in [MVS3] (See also [MVS2] ) that a regular component K the category Agr(K) is Artin-Schelter regular, its Ext-category E(K) is locally finite, Frobenius of radical cube zero. We will have a situation very similar to the preprojective algebra and we can use the same line of ideas as in [MV] to prove the following: Theorem 14. The following statements hold: (1) E(K) is selfinjective of radical cube zero, for any indecomposable object M op in grPI (E(K) ) generated in degree zero Ω(M ) is either simple generated in degree 2 or it is generated in degree one. op (2) The indecomposable non Koszul objects in gr(E(K) ) are the objects M n such that for some integer n, the n-th syzygy Ω (M ) is simple. (3) For any indecomposable non projective object M the Ausalnder-Reiten translation is Ω2 (N (M )), with N the Nakayama equivalence. Hence for n = 2k the n-th syzygy Ωn (M ) is simple, if and only if, Dtrk M is simple and for n = 2k +1, the syzygy Ωn (M ) is simple, if and only if , Dtrk−1 M is P/socP, with P an indecomposable projective. (4) grPI (E(K)op ) is radical square zero. It is stably equivalent to the disjoint union ∪ Hi of hereditary full subcategories corresponding to the sections i∈Z

of the Auslander-Reiten component of K and the categories Hi , Hi+2 are equivalent and the categories Hi , Hi+1 are opposite categories for all i, the functor producing the stable equivalence sends all the almost split sequences 0 → DtrM → E → M → 0 with DtrM non simple, to almost split sequences and nodes (in this case all simple are nodes) correspond to two simple objects, one projective and one injective. (5) The Auslander-Reiten components of E(K) containing the projective objects are built by gluing for each i, a preprojective component of Hi with a


42

(6) (7) (8) (9)

preinjective component of Hi by identifying the simple projective with the corresponding simple injective objects and adding projective injective objects to recover the almost split sequences with a projective injective in the middle term. All the regular components are of type A∞ . The objects in the preinjective and regular components are Koszul. The objects in the preprojective and regular components are co Koszul. The category of indecomposable non projective Koszul object in E(K) has almost split sequences to the left and they are almost split in the whole op category gr(E(K) ).

Corollary 8. The following statements hold: (a) If K is of type A∞ it is a tube, then E(K) has a finite number of preprojective components and it has no regular component. (b) If K is of type A∞ and it is not a tube, then E(K) has a countable number of preprojective components and it has no regular component. (c) If K is of type D∞ , then E(K) has a countable number of preprojective components and it has no regular component. (b) If K is of type A∞ ∞ , then it has a countable number of preprojective components and a countable number of regular components of type A∞ If K is a regular component, then E(K) is Koszul and E(E(K)) = Agr(K). Koszul duality Ψ : KE(K) → KAgr(K)op sends almost split sequences to almost split sequences, from these facts, we have the following analogous of the preprojective algebra: ([MV] Theorem 2.8 and [MVS3]). Theorem 15. Let K be a regular Auslander-Reiten component of a finite dimensional algebra Λ. The Koszul functors of Agr(K) have the following properties: (i) Every indecomposable, non simple, non projective Koszul functor M has projective dimension one. (ii) For every indecomposable, non simple, Koszul functor M, there exists a non splittable short exact sequence graded objects and maps: 0 → M → E → r2 σM [2] → 0 where σ is an auto equivalence of Agr(K) and r denotes the radical. The objects E and r2 σM are Koszul and the sequence is almost split in KAgr(K)op . (iii) The indecomposable Koszul functors of Agr(K) are distributed in components, preprojective components, corresponding to the preprojective components of the sections of K and regular components, corresponding to the regular components of the sections of K. (i’) Every indecomposable, non simple, non injective co Koszul functor M has injective dimension one. (ii’) For every indecomposable, non simple, co Koszul functor M, there exists a non splittable short exact sequence graded objects and maps: 0 → σM/soc2 σM → E → M → 0 where σ is an auto equivalence of Agr(K) and soc2 M denotes the second socle. The objects E and σM/soc2 σM are co Koszul and the sequence is almost split in the category of co Koszul functors coKAgr(K)op .

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(iii’) The indecomposable co Koszul functors of Agr(K) are distributed in components, preinjective components, corresponding to the preinjective components of the sections of K and regular components, corresponding to the regular components of the sections of K. This theorem has a nicer interpretation in the quotient category module the functors of finite length [MVS3]. We recall this construction. We will obtain results similar to the case considered in [MZ]. Let K be a regular Auslander-Reiten component of a finite dimensional algebra Λ. To simplify the notation we will denote by C the category Agr(K) and by gr(C)0 the finitely presented graded functors and degree zero maps, by [MVS1] gr(C)0 is abelian. Denote by tors(C) the full subcategory of gr(C)0 of all functors of finite length. This is a Serre category, we can take the quotient category Qgr(C)= gr(C)0 /tors(C). The category Qgr(C) has the same objects as gr(C)0 and maps: ′ ′ ′ HomQgr(C) (πM, πN ) = lim −→(M ,N )∈L Homgr(C)0 (M´, N/N ), where L={(F,G)| F⊂M, G⊆N, and M/F, G in tors(C)}. Let π :gr(C)0 → Qgr(C) be the canonical projection. P It is known [P], Qgr(C) is abelian and π is exact. If we denote by t(M ) = L and L a subfunctor L∈tors(C)

of M , then t is an idempotent radical and we say that a functor M is torsion if t(M ) = M and torsion free if t(M ) = 0. It was proved im [MVS2] that for a finetely presented functor M the torsion part t(M ) is of finite lenght, in particular finetely presented. Denote by M≥k the truncation subfunctor of the graded functor M , this is (M≥k )j =0, if j < k and (M≥k )j = Mj , if j ≥ k. By definition, M/M≥k is of finite length and the maps in Qgr(C) can be written as follows: HomQgr(C) (πM, πN ) = limk Homgr(C)0 (M≥k , N/t(N )). −→ Since π(M ) ∼ = π(M/t(M )) we may always assume N is torsion free and in such a case the exact sequence: 0 → M≥k → M≥k−1 → Mk−1 /M≥k → 0 induces an exact sequence: 0 → Homgr(C)0 (Mk−1 /M≥k , N ) → Homgr(C)0 (M≥k−1 , N ) → Homgr(C)0 (M≥k , N ) whose first therm is zero and lim −→k Homgr(C)0 (M≥k , N ) = ∪Homgr(C)0 (M≥k , N ). k

Since C has global dimension two, any finitely presented functor M has a truncation M≥k such that M≥k [k] is Koszul (see [MVS3] or [AE]) and in Qgr(C) to objects πM and πN are isomorphic, if and only if there are truncations M≥k , N≥k such that M≥k ∼ = N≥k in gr(C)0 . ∼

∼

Hence Qgr(C) = ∪ K C [i] where K C [i] is the image under π of the Koszul Ci∈Z

functors, shifted by i. The situation is analogous to the sheaves on projective space studied in [MZ]. From this it follows: Theorem 16. If we denote by C the associated graded category of a regular component of a finite dimensional algebra, then the category Qgr(C) has almost split sequences, they are of the form: 0 → πM [k] → πE[k] → πσM [k + 2] → 0 with σan auto equivalence of C. The category Qgr(C) is the union of connected components of the Auslander-Reiten quiver and these components are of the following kind: Proj[k] and Reg[k], this means the k-th shift of π(Pr ojC ) or π(RegC ), where Pr ojC

44


and Reg C denote a preprojective component and a regular component of C, respectively. π(Pr ojC ) and π(Pr ojC ), respectively, π(RegC ) and RegC , have isomorphic translation quivers. Proof. By the above observation, any indecomposable object in Qgr(C) is of the form π(M )[ℓ] with M an indecomposable Koszul functor. The endomorphism ring of π(M )[ℓ] is EndQgr(C) (π(M )[ℓ] )=∪Homgr(C)0 (M≥k , M ). k

The truncations M≥k [k] are isomorphic to rk M [k],hence they correspond under Koszul duality to Ωk M [k] , since E(C) is selfinjective, they are indecomposable with local endomorphism ring. A map f ∈EndQgr(C) (π(M )[ℓ] ) is represented by a map f : M≥k → M and restricting to the image a map M≥k → M≥k , hence it is either an iso or nilpotent and it follows EndQgr(C) (π(M )[ℓ] ) is local. We have proved Qgr(C) is Krull-Schmidt. j

p

Consider the almost split sequence: 0 → M → E → r2 σM [2] → 0 apply π and π(j)[k]

π(p)[k]

shift it, to obtain to obtain the exact sequence: 0 → πM [k] → πE[k] → πσM [k + 2] → 0 if the sequence splits. there exists a map h:πσM [k + 2] → πE[k] such that π(p)[k]h = 1. The map h is of the form π(t) with t : rℓ σM [k + 2] → rℓ E[k] . It follows the j

p

exact sequence 0 → rℓ M [ℓ] → rℓ E[ℓ] → rℓ+2 σM [ℓ + 2] → 0 splits. Using Koszul duality Φ : KC → KE(C) it follows 0 → Ω2+ℓ σΦ(M ) → Ωℓ Φ(E)) → Ωℓ Φ(M ) → 0 splits. Therefore 0 → Ω2 σΦ(M ) → Φ(E)) → Φ(M ) → 0 splits, a contradiction. Let πN [j] be an indecomposable object with N Koszul and f : πN [j] → πσM [k+ 2] a non isomorphism. As above, there exists a map t : N≥ℓ → σM [k − j + 2] such that πt = f , which induces a non isomorphism of Koszul objects t : N≥ℓ [ℓ] → σM [k+ℓ-j+2]≥k+ℓ−j+2 . j

p

The sequence: 0 → M≥k+ℓ-j [k+ℓ-j] → E≥k+ℓ-j [k+ℓ-j] → r2 σM≥k+ℓ-j [k+ℓ-j+2] → 0 is almost split. Then there exists a map s : N≥ℓ [ℓ] → E≥k+ℓ-j [k+ℓ-j] with ps = t. It follows f lifts to πE[k]. π(j)[k]

In a similar way we prove the map πM [k] → πE[k] is left almost split. The remaining claims are clear.

We will end the paper with the following remark: Remark 4. Consider a locally finite infinite quiver Q and construct the translation quiver ZQ. For each arrow α of ZQ there exits a unique arrow σα such that the end of σα coincides with the starting of α, and a unique arrow σ −1 α starting at the end of α. We define the following set ρ of relations in ZQ: (i) If α and β are arrows with the same end, then ασα − βσβ is a relation. (ii) If γ is an arrow ending at the start of α and different from σα, then αγ is a relation. (iii) If γ is an arrow starting at the end of α and different from σ −1 α, then γα is a relation. Consider the category of representations rep(ZQ, ρ) over a field K of the quiver with relations (ZQ, ρ).The category rep(ZQ, ρ) is by construction selfinjective of radical cube zero. It follows from the arguments in [M] it is also Koszul . The theory developed in [MVS1], [MVS2], [MVS3] [MVS4] applies to this situation and we obtain as Ext-category of E(ZQ, ρ),The category of representations of the quiver ZQ with mesh relations η. It follows rep(ZQ,η) is an Artin-Shelter regular Koszul

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category of global dimension two. We do not know whether or not it corresponds to an Auslander-Reiten quiver of a finite dimensional algebra, however the above theorems for regular Auslander-Reiten components hold, in particular they also hold for a connected component of the stable non regular Ausalnder-Reiten quiver, as considered in [MVS3]. References [A]

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[MVS1] [MVS2] [MVS3] [MVS4] [MZ] [P] [ReVan]

[R] [Ri] [S] [SM]

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´ ticas UNAM, Unidad Morelia, Mexico Instituto de Matema E-mail address: [email protected] ´ ticas UNAM, Unidad Morelia, Mexico Instituto de Matema E-mail address: [email protected] URL: http://www.matmor.unam.mx