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Representation Dimension of Cluster Concealed Algebras

We are going to show that the representation dimension of a cluster-concealed algebra B is 3. We compute its representation dimension by showing an explicit Auslander generator for the cluster-tilted algebra.


Introduction
Auslander [3] introduced the concept of representation dimension for artin algebras, motivated by the connection of arbitrary artin algebras with representation finite artin algebras. He expected this notion to give a reasonable way of measuring how far an artin algebra is from being of representation finite type. The representation dimension is a Morita-invariant of artin algebras and characterizes the artin algebras of finite representation type. It was shown by Auslander in [3] that an algebra A is representation-finite if and only if rep.dimA ≤ 2. Later, Iyama proved in [14] that the representation dimension of an artin algebra is always finite, using a relationship with quasihereditary algebras. The interest in representation dimension revived when Igusa and Todorov showed that the representation dimension is related to the finitistic dimension conjecture. They proved that if an artin algebra has representation dimension at most three, then its finitistic dimension is finite [13]. Recently, Rouquier showed in [18] an exterior algebra with representation dimension 4. In fact, he has constructed examples of algebras with arbitrarily large representation dimensions.
On the other hand, Cluster algebras were introduced by Fomin-Zelevinsky [19]. Later, Marsh-Reineke-Zelevinsky [15] found that there is a deep connection between cluster algebras and quiver representations. Buan-Marsh-Reineke-Reiten-Todorov [6] defined the cluster category and developed a tilting theory using a especial class of objects, namely the cluster tilting objects. In [8], Buan-Marsh-Reiten introduced the cluster-tilted algebras as the endomorphism algebras End C (T ) op of a cluster-tilting object T in a cluster category C. These algebras are connected to tilted algebras, which are the algebras of the form End H (T ) op for a tilting module T over a hereditary algebra H. This motivates us to investigate the relationship between the module theory of cluster-tilted algebras and the module theory of hereditary algebras.
A cluster concealed algebra is given by B = End C (T ) op where T is a cluster tilting object induced by a postprojective tilting H-module. The objective of this paper is to compute the representation dimension of cluster-concealed algebras by showing an explicit Auslander generator. In order to do this, tilting and torsion theory of hereditary algebras became very useful tools. Also the concept of slices and local slices, the last ones defined in [1], became a key tool to find Auslander generators for cluster-concealed algebras. The notions of covariantly and contravariantly finite categories [5] are very useful for the proof, together with their relationship with torsion pairs in tilted algebras [4].
In section 2, we give some notations and preliminary concepts needed for proving our main result. In section 3, we present our main theorem and the previous results required for proving it.

preliminaries
Through this paper we are going to use the following notation. A denotes a finite dimensional algebra over an algebraically closed field and mod A represents the category of all finitely generated right A-modules. H denotes a finite dimensional hereditary algebra and we denote the bounded derived category of H by D b (H) and by [ ] the shift functor. We will often identify the objects concentrated in degree zero with the corresponding H-module.
2.1. Tilting theory. We start this section by giving the definition of tilting module. For more details on tilting modules see [12]. Let A be an algebra and T an A-module. T is said to be a tilting module in modA if it satisfies the following conditions (a) pdT ≤ 1.
If a module satisfies condition (b), we say that the module is exceptional, or equivalently, we say that T is a tilting module if pdT ≤ 1, is rigid and has n non-isomorphic indecomposable direct summands, where n is the number of non-isomorphic simple modules.
Remark 2.1. We recall that for a hereditary algebra H, an H-module T is said to be a tilting module if T is rigid and has n non-isomorphic indecomposable direct summands.
A tilting module is said to be basic if all of its direct summands are nonisomorphic. The endomorphism ring of a tilting module over a hereditary algebra is said to be a tilted algebra. In particular, hereditary algebras are tilted algebras.
We recall that a path from X to Y is a sequence of indecommposable modules and non-zero morphisms X = X 0 → X 1 → ... → X t = Y . Given X, Y ∈ indA, we say that X is a predecessor of Y or that Y is a successor of X, provided that there exists a path from X to Y . A tilting module T is convex if, for a given pair of indecomposable summands of T , X Y in add T , any path from X to Y contains only indecomposable modules in add T . Following [2], we say that a set Σ T in mod A is a complete slice if T = M ∈Σ T M is a convex tilting module with End A T hereditary. For the original definition of complete slices, we refer to [16], [17]. Tilted algebras are characterized by the existence of a complete slice in its module category.
For a given tilting module in modH there exist two full disjoint subcategories of modH, namely The free torsion class and torsion class respectively. Furthermore, if T is a convex tilting, then modH = F(T ) T (T ). We have that, in this case, F(T ) is closed under predecessors and T (T ) is closed under successors.

2.2.
Cluster categories and cluster tilted Algebras. For the convenience of the reader, we start this section recalling some definitions and results of cluster categories from [6]. Let C be the cluster category associated to H and given by D b (H)/F , where F is the composition functor τ −1 D [1]. We represent byX the class of an object X of D b (H) in the cluster category. We recall that Hom C (X,Ỹ ) = i∈Z Hom D b (H) (X, F i Y ). We recall that S = indH H [1] is the fundamental domain of C. If X and Y are objects in the fundamental domain, then we have that Hom D b (H) (X, F i Y ) = 0 for all i = 0, 1. Moreover, any object in C is of the formX with X ∈ S.
For C, we say thatT in C is a tilting object if Ext 1 C (T ,T ) = 0 andT has a maximal number of non-isomorphic direct summands. A tilting object in C has finite summands.
There exists the following nice correspondence between tilting modules and basic tilting objects. In [8], Buan, Marsh and Reiten introduced the cluster-tilted algebra. LetT be a tilting object over the cluster category C, we recall that B is the cluster tilted algebra if B = End C (T ). It is also shown in [8] that, ifT is a tilting object in C, the functor Hom C (T , ) induces an equivalence of categories between C/add(τT ) and modB.
Thus using the equivalence above, we can compute the Hom B (X ′ , Y ′ ) in terms of the cluster category C.
First we begin by recalling the notion of approximations. Let X be a full subcategory of modA closed under direct summands and isomorphisms. We say that an A-module Y is contravariantly finite over X if there exists a mor- Moreover, by [5,Proposition 3.9], an A-module Y is contravariantly finite over X if and only if there exists a right X -approximation of Y and an A-module Y is covariantly finite over X if and only if there exists a left X -approximation of Y .
The category X is called covariantly (contravariantly) finite in modA if every A-module Y has a left (right) minimal X -approximation.

Representation dimension.
We recall that an A-module M is a generator for modA if for each X ∈ modA, there exists an epimorphism M ′ → X with M ′ ∈add(M ). Observe that A is a generator for modA. Dually, we say that an A-module M is a cogenerator if for each Y ∈ modA there exists a monomorphism Y → M ′ with M ′ ∈add(M ). Note that DA is a cogenerator for modA. In particular, any module M containing every indecomposable projective and every indecomposable injective module as a summand is a generator-cogenerator module for modA.
The original definition of representation dimension (we will note it by rep.dim) of an artin algebra A is due to Auslander. For more facts on this topic, we refer the reader to [3]. The following is a nice characterization of representation dimension, in the case that A is a non semisimple algebra, also due to Auslander. This characterization is given as follows

is a generator-cogenerator for modA}
A module M that reaches the minimum is called an Auslander generator and gl.dim End A (M ) = rep.dimA if M is an Auslander generator.
The representation dimension can also be defined in a functorial way, which will result us more convenient. The next definition (see [2], [11], [18]) will be very useful for the rest of this work. Definition 2.3. The representation dimension rep.dimA is the smallest integer i ≥ 2 such that there is a module M ∈ modA with the property that, given any A-module X (a) , there is an exact sequence of following [11], we say the module M has the i-resolution property and that We have rep.dimA =rep.dimA op .

Representation dimension for cluster concealed algebras
In this section we present our main result. LetT be a tilting object in a cluster category C and let be B =End C (T ), the associated cluster-tilted algebra. To simplify some proofs we choose without lose of generality T and τ T without projective summands. As we are interested in compute the representation dimension of cluster concealed algebras, through this section T will be a postprojective tilting module on modH andT denotes the class of T on the cluster category C. Recall thatT is a tilting object in C. We consider H, and thus B, of infinite representation type.
We consider a complete slice Σ on the postprojective component of the hereditary algebra H, such that Σ ∈ T (T ). Note that this condition implies Σ does not have projective summands. We can construct a convex tilting module of the form U = E∈Σ E, and we denote T (Σ) to the category T (U ) and F(Σ) to F(U ). Since Σ ∈ T (T ) also U ∈ T (T ) and thus we have T (Σ) ⊂ T (T ). Then we have Σ ′ = Hom C (T ,Σ), the image ofΣ by the BMR-equivalence [8], which is a local slice in modB [1].
Using the BMR-equivalence, we can describe the indecomposable modules in modB as follows. Let Y be an indecomposable in modB, then we have one of the following cases Observe that if Y is indecomposable also Q, X or G is indecomposable. Since we choose Σ in the postprojective component of modH, we have that F(Σ) has only a finite number of non isomorphic indecomposable H-modules, then we only have a finite number of isomorphism classes of indecomposable B-modules Y satisfying the second case. As well as in the first case, because we only have a finite number of indecomposable projective H-modules. This implies we can consider the following modules, Q ′ = Hom C (T , H [1]) and G = G Hom C (T , G) with G indecomposable in F(Σ), in modB.
We want to determine an Auslander generator for modB, this is a module M with the properties of Definition 2.3, so we want a module which provides add(M )-approximations for every one of the modules in the previous cases. Since by the Remark 2.4, a module M satisfies Definition 2.3 if satisfies condition (a) and contains a cogenerator as a direct summand, we will focuses on find modules that satisfies condition (a). If we let M ′ to be the module G ⊕ Q ′ , clearly, M ′ approximates trivially the modules of the first and second case.
Hence we will concentrate from now on, in the modules of the third case. For a given complete slice Γ we consider the set D Γ defined as that is the modules in D Σ are the ones of the third case. Our objective now is to approximate the modules of D Σ by Σ ′ .
Let Y ∈ D Σ , then we have Y = Hom C (T ,X) with X ∈ T (Σ) ⊂ modH. Now, X ∈ T (Σ) = Gen(Σ) then there exist an epimorphism f f : E → X → 0 a right add(Σ)-approximation of X. Recall that a complete slice on a tilted algebra induces a convex tilting module, then since H is hereditary and X ∈ Gen(Σ) by [2, Proposition :)], we have that K = Kerf ∈ add(Σ) and hence that the exact short sequence is an add(Σ)-resolution of length two for X. We want to construct a similar exact sequence in modB for Y . The idea is to prove that the image of this sequence in C induces such a sequence in modB.
This short exact sequence induces a triangle in D b (H) which induces a triangle in the cluster category C by taking the respective quotient classesK Our objective is to study the image of this triangle in mod B by the functor Hom C (T , ) Proof. Recall that Σ do not have projective summands, hence, since E is in add(Σ), E is not projective and thus τ E is in modH.
since X and τ E are H-modules, thusX and τẼ are in the fundamental domain of C therefore where all the others summands are zero.
We are going to see that also Hom D b (H) (X, τ E) is zero. Now Hom D b (H) (X, τ E) = Hom H (X, τ E) , since X and τ E are H modules. Moreover, since E ∈ add(Σ) we have that τ E ∈ F(Σ) and X ∈ T (Σ).
we are in conditions to show that the modules in D Σ are generated by modules in add(Σ ′ ). Proposition 3.2. Let T be a postprojective tilting module over modH. Let Σ be a complete slice on the postprojective component of modH such that Σ is contained on T (T ). Then (i) D Σ ⊂ Gen(Σ ′ ) in modB.
(ii) Moreover, consider τ −1 Σ, wich is also a complete slice, there exist a short exact sequence of the form Proof. Let Y ∈ D Σ , recall we have a triangleK →Ẽ →X → τK in C and recall the functor Hom C (T , ) induces an equivalence between C/add(τT ) and modB. Now, applying this functor to the last triangle, we obtain a long exact sequence in modB We want to construct an epimorphism from add(Σ ′ ) to Y ∈ D Σ , we know that Σ ′ = Hom C (T ,Σ) and E is an add(Σ)-approximation. Consider E ′ = Hom C (T ,Ẽ) and f * = Hom C (T ,f ). We are going to prove that f * : E ′ → Y is an epimorphism. In order to do this, it suffices to see that Hom B (Hom C (T ,X), Hom C (T , τK)) = 0.
By lemma 3.1, since K ∈ add(Σ) we can assume that h ∈ Hom D b (H) (X, K [1]). Since we need to prove that h factorizes by add(τT ) we are going to consider F τ T = T [1], then is enough to show that h factorizes by We can assume without lose of generality that there exist H ′ hereditary, derived equivalent to H such that X, T [1] and K [1] are H ′ modules, in fact, we can choose H ′ =End H (τ −1 Σ).
Then T [1] results a tilting module over H ′ and we may consider the category T (T [1]) which is a full subcategory of modH ′ .
Under the hypothesis taken, we know that K ∈ add(Σ) ⊂ T (Σ) ⊂ T (T ). Then we have K[1] ∈ T (T [1]) and since X is in modH we also have 0 = and X are H ′ -modules. Thus we have shown that X ∈ F(T [1]).
We have the following commutative diagram: it is, h should factorizes by add (T [1]), thus h :X → τK factorizes by add(τT ).
Therefore we have that the long exact sequence in ( * ) factors through zero This finishes the proof of (i).
Only remains to see that Hom C (T , X[−1]) → Hom C (T ,K) is zero also. Then, as we did before, we want to see that Hom B (Hom C (T , X[−1]), Hom C (T ,K)) = 0, then we are going to prove that any h ∈ Hom C ( X[−1],K) factors through add(τT ). Observe that X[−1] = τ −1X then we have that where Hom D b (H) (τ −1 X, K) ≃ Hom D b (H) (X, τ K) = Hom H (X, τ K) = 0 because X ∈ T (Σ) and τ K ∈ F(Σ). Therefore ), then we know by (i) that any h in Hom C (τ −1X ,K) will factor through add(τT ) in C and we have 0 → Hom C (T ,K) → Hom C (T ,Ẽ) → Hom C (T ,X) → 0 Therefore, if we let M be the B-module M ′ ⊕ Σ ′ our last proposition shows that we have M generates all modules in modB because add(Σ ′ ) is contained in add(M ). More over, our next lemma will prove that the epimorphism constructed before is an add(Σ ′ )-approximation of the modules in D Σ .
Proof. Let U ′ be an indecomposable module in add(Σ ′ ). We want to show that, Using the [BMR]-equivalence we can assume there exists X ∈ T (T ), U ∈ add(Σ) such that U ′ = Hom C (T ,Ũ ) and Y = Hom C (T ,X). Hence we can write, h ∈ Hom B (U ′ , Y ) = Hom B (Hom C (T ,Ũ ), Hom C (T ,X)) ≃ Hom C (Ũ ,X)/add(τT ), then there exist h ′ ∈ Hom C (Ũ ,X) such that h = Hom C (T , h ′ ). Since h = 0 we can assume that h ′ don't factorize by the add(τT ) in C. Let's compute Hom C (Ũ ,X) Since U and X are taken in the fundamental domain, even more U is postprojective(U is not contained in any oriented cycle), we know that at most one of the last terms is different from zero, [8].
Then we have only one case, h ′ ∈ Hom D b (H) (U, X).
Suppose h ′ ∈ Hom D b (H) (U, X) then h ′ ∈ Hom H (U, X) because U and X are H-modules, and then, since f : E → X is an add(Σ)-approximation of X ∈ mod(H), there exist k ′ : U → E such that the following diagram commutes it is, f k ′ = h ′ . This commutative diagram induces a commutative diagram on D b (H), which induces the following commutative diagram in C U ւ ↓ E →X Again, applying Hom C (T , ) we obtain We are going to see now that this approximation is in fact, an add(M )approximation of the modules in D Σ ; To achieve this objective we are going to prove that the morphism Hom Before we establish our next result, we are going to state the next technical lemma.

Lemma 3.4. For any indecomposable G in F(Σ) and any indecomposable X in T (Σ)
Hom C (G,X) = Hom H (G, X).
Proof. We know that G and X are indecomposable H-modules. We can assume thatG andX are in the fundamental domain of C. Therefore we can compute Hom C (G,X) = Hom D b (H) (G, X) ⊕ Hom D b (H) (G, F (X)), because since X and G are in the fundamental domain, we know by [6], that only this summands can be different from zero. We want to see that H (G, P ) = 0, because H is hereditary. If X is not injective, then we have that τ −1 X ∈ modH and then where the last isomorphism is given by the Auslander-Reiten formula, but we have, if G is not projective, τ G ∈ F(Σ) and τ −1 X ∈ T (Σ) because T (Σ) is closed under successors and F(Σ) is closed under predecessors, so Hom H (τ −1 X, τ G) = 0 and therefore and Ext 2 H (G, P ) = 0. Then we have proof that Hom D b (H) (G, F (X)) = 0, and so Hom C (G,X) = Hom D b (H) (G, X) = Hom H (G, X) because G and X are H-módules.
We are now in conditions, of proving that f * : E ′ → Y is an add(Σ ⊕ G)approximation of Y . Proof. Let C G = Hom C (T ,G) with G ∈ F(Σ) indecomposable. Suppose that there is a map 0 = h : C G → Y , we want to show that the following diagram: We can write, as we did before, using the [BMR]-equivalence h ∈ Hom B (C G , Y ) ≃ Hom C (G,X)/add(τT ), then there exist h ′ ∈ Hom C (G,X) such that h = Hom C (T , h ′ ). We can assume that h ′ don't factorize by the add(τT ) in C, otherwise h will be zero in modB.
Hence, we have that h = Hom C (T , h ′ ) with h ′ ∈ Hom H (G, X). Then, since f : E → X is an add(Σ)-approximation of X ∈ modH and Σ is a complete slice in modH, any morphism from G to X, will factorize by E because T (Σ) is covariantly finite. This is, there exist k ′ : G → E such that the following diagram commutes G ւ ↓ E → X We may now proceed as in lemma 3.3, and show that this diagram induces a commutative diagram on modB We have proved that there exist k = Hom C (T ,k ′ ) such that f * k = Hom C (T ,fk ′ ) = Hom C (T ,h ′ ) = h. Then, h : C G → Y factors through E ′ ; observe that k = 0 because k ′ don't factorize by add(τT ), otherwise h ′ would factorize by add(τT ). This finishes proof of the lemma.
It only rest to prove that, if we have a map from a module Q in add(Q ′ ), in any module Y ∈ D Σ , this map also factors through E ′ . Our next lemma proves the desired result. Let h be, h = Hom C (T , h ′ ) with h ′ ∈ Hom C ( P [1],X). Then, we have where, Hom D b (H) (P [1], X) = 0 because X ∈ modH and P [1] ∈ modH[1] therefore there is no non zero map between X and P [1] in D b (H).
Recall f : E → X was constructed as an add(Σ)-approximation of X, for X ∈ T (Σ). Now we can take We now can construct the following commutative diagram on modH Since P is projective, then the morphism h ′ [−1] factors through τ −1 E, it is, there exists k ′ : P → τ −1 E such that the following diagram Applying the shift functor [ ] to this diagram, we obtain a new commutative diagram in D b (H).
And, as we did before we apply the functor Hom C (T , ) to obtain a commutative diagram on modB Hom C (T , P [1]) ւ ↓ Hom C (T ,Ẽ) → Hom C (T ,X) Then, if we choose k ≃ Hom C (T , k ′ [1]), we get that f * k = h, where the commutativity is given by the above diagram. Proof. Let N be an indecomposable direct summand of M . Then we can consider the following cases: (a) N ∈ add(Σ ′ ) (b) N ∈ add(Q ′ ) (c) N ∈ add(G) Hence we know the results holds for N in all the above cases. Case (a) follows from Lemma 3.3, case (b) from Lemma 3.5 and case (c) from Lemma 3.6 Then the proposition follows for M by the additivity of the functor Hom B (N, ). This proves that there exist an add(M )-resolution of length at most two, for every module in modB. Note that this proposition proves that M satisfies the condition (a) of definition 2.3 for i = 2 and therefore that gl.dim End B (M ) ≤ 3. We are going to proceed now to state and prove our main result. Proof. Let M be the module in the above proposition. Observe that if I is injective in modB, then I = Hom C (T , τ 2T ), but τ 2 T ∈ F(Σ), recall τ T does not have projective summands and F(T ) ⊂ F(Σ) due to the hypothesis taken on Σ on H , therefore any indecomposable injective module in modB is a direct summand of M The same occurs for any indecomposable P ∈ modB, then P = Hom B (T ,T ) and againT lies in F(Σ), thus the add(G) contains every indecomposable projective and every indecomposable injective module. Then G is generator-cogenerator for modB and hence M is a generator-cogenerator for modB. Moreover, proposition proves that gl.dim End B (M ) ≤ 3 and thus rep.dimB ≤ 3.
We are going to see some applications of our main theorem. Let A be basic connected finite dimensional algebra, A is said to be of minimal infinite type if it is of infinite representation type and for each vertex e in the quiver of A we have that A/AeA is of finite type. Recall that for any cluster-tilted algebra B with a vertex e in the quiver of B we have that B/BeB is again a cluster-tilted algebra(see [9] section 2).
So, as a consequence of the theorem, we have the following corollary.