Cluster-tilted algebras and slices

Ibrahim Assem^{1},

Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec), J1K 2R1, Canada

Thomas Brüstle^{2},

Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec),

J1K 2R1, Canada and Department of Mathematics, Bishop’s University, Lennoxville, (Québec), J1M 1Z7, Canada

Ralf Schiffler^{3},

Department of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst, MA 01003-9305, USA

Abstract

We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.

Key words: Local slice, cluster-tilted algebra, relation-extensions of tilted algebras 1991 MSC: 16S70, 16G20

Email addresses: ibrahim.assem@usherbrooke.ca (Ibrahim Assem), thomas.brustle@usherbrooke.ca (Thomas Brüstle), schiffler@math.umass.edu (Ralf Schiffler). 1 Partially supported by the NSERC of Canada and the University of Sherbrooke 2 Partially supported by the NSERC of Canada and the universities of Sherbrooke and Bishop’s 3 Partially supported by the University of Massachusetts

Preprint submitted to Elsevier Preprint 6 July 2007

0 Introduction

Cluster categories were introduced in [7] and, for type 𝔸, also in [11], as a categorical model allowing to understand better the cluster algebras of Fomin and Zelevinsky [13]. Cluster-tilted algebras were defined in [11] for type 𝔸, and in [8] for arbitrary hereditary algebras as follows: Let A be a hereditary algebra and T̃ be a tilting object in the associated cluster category C_{A}, that is, an object such that Ext_{C_{A}}^{1}(T̃, T̃) = 0 and the number of isomorphism classes of indecomposable summands of T̃ equals the rank of the Grothendieck group of A, then the algebra B = End_{C_{A}} T̃ is called cluster-tilted. These algebras have been studied by several authors, see, for instance, [1,8–10,12,17]. In particular, they were shown in [1] to be closely related to the tilted algebras introduced by Happel and Ringel in the early eighties [15]. Indeed, let C be a tilted algebra, then the trivial extension C̃ = C ⋉ Ext_{C}^{2}(DC, C) of C by the C-C-bimodule Ext_{C}^{2}(DC, C) is cluster-tilted, and every cluster-tilted algebra is of this form. Thus, we have a surjective map C ↦ C̃ from tilted to cluster-tilted algebras. However, easy examples show that this map is not injective. Our objective in this paper is to give a criterion allowing to verify whether for two tilted algebras C_{1} and C_{2}, the corresponding cluster-tilted algebras C̃_{1} and C̃_{2} are isomorphic or not.

Since tilted algebras are characterised by the existence of complete slices in their Auslander-Reiten quiver (see, for instance, [15,20,19,21] or [3]), it is natural to study the corresponding concept for cluster-tilted algebras. For this purpose, we introduce what we call a local slice, by weakening the axioms of complete slice (thus, in a tilted algebra, complete slices are local slices). We show that a complete slice in a tilted algebra C embeds as a local slice in C̃ (and, in fact, any local slice in C̃ is of this form).

Our main theorem is the following:

Theorem 1 Let B be a cluster-tilted algebra. Then a tilted algebra C is such that B = C ⋉ Ext_{C}^{2}(DC, C) if and only if there exists a local slice Σ in mod B such that

C = B/Ann_{B}Σ.

We also show that cluster-tilted algebras have many local slices. In fact, all but at most finitely many indecomposable modules lying in the transjective component of the Auslander-Reiten quiver belong to a local slice. If the cluster-tilted algebra is of tree type (which is the case, for instance, if it is of Dynkin type or of Euclidean type distinct from 𝔸̃), then this is the case for all indecomposable modules in this component.

We now describe the contents of our paper. In the first section, we introduce the notion of local section in a translation quiver and in the second section we study sections and local sections in the derived category. In the third section, we introduce the concept of a local slice and prove our main result, and in section four, we prove that cluster-tilted algebras of tree type have sufficiently many local slices.

We would like to thank David Smith for useful comments on the paper.

1 Preliminaries on translation quivers

1.1 Notation

Throughout this paper, all algebras are connected finite dimensional algebras over an algebraically closed field k. For an algebra C, we denote by mod C the category of finitely generated right C-modules and by ind C a full subcategory of mod C consisting of exactly one representative from each isomorphism class of indecomposable modules. When we speak about a C-module (or an indecomposable C-module), we always mean implicitly that it belongs to mod C (or to ind C, respectively). Also, all subcategories of mod C are full and so are identified with their object classes. Given a subcategory C of mod C, we sometimes write M ∈ C to express that M is an object in C. We denote by add C the full subcategory of mod C with objects the finite direct sums of modules in C and, if M is a module, we abbreviate add {M} as add M. We denote the projective (or injective) dimension of a module M as pd M (or id M, respectively). The global dimension of C is denoted by gl.dim.C and its Grothendieck group by K_{0}(C). Finally, we denote by Γ(mod C) the Auslander-Reiten quiver of an algebra C, and by τ_{C} = D T r, τ_{C}^{−1} = T r D its Auslander-Reiten translations. For further definitions and facts needed on mod C or Γ(mod C), we refer the reader to [3]. We also need facts on the bounded derived category D^{b}(mod C) of mod C, for which we refer to [14].

1.2 Sections

For translation quivers, we refer to [3,20]. Let (Γ, τ) be a connected translation quiver. We recall that a path x = x_{0} → x_{1} → … → x_{t} = y in Γ is called sectional if, for each i with 0 < i < t, we have τ x_{i+1} ≠ x_{i−1}. A full connected subquiver Σ of Γ is said to be convex in Γ if, for any path x = x_{0} → x_{1} → … → x_{t} = y in Γ with x, y ∈ Σ_{0}, we have x_{i} ∈ Σ_{0} for all i. It is called acyclic if there is no cycle x = x_{0} → x_{1} → … → x_{t} = x (with t > 0) which is entirely contained in Σ. The following definition is due to Liu and Skowroński (see [18,21] or else [3]).

Definition 2 Let (Γ, τ) be a connected translation quiver. A connected full subquiver Σ of Γ is a section in Γ if:

(S1) Σ is acyclic. (S2) For each x ∈ Γ_{0}, there exists a unique n ∈ Z such that τ^{n} x ∈ Σ_{0}. (S3) Σ is convex in Γ.

This definition is motivated by the study of tilted algebras. The well-known criterion of Liu and Skowroński asserts that, if C is an algebra, and Σ is a faithful section in a component of its Auslander-Reiten quiver such that Hom_{C}(X, τ_{C}Y) = 0 for all X, Y ∈ Σ_{0}, then C is tilted having Σ as complete slice (see [18,21,3]).

We note that, if a translation quiver Γ contains a section, then Γ is acyclic.

1.3 Presections

We need some weaker notions. The first one is the following.

Definition 3 Let (Γ, τ) be a connected translation quiver. A connected full subquiver Σ of Γ is called a presection in Γ if it satisfies the following two conditions:

(P1) If x ∈ Σ_{0} and x → y is an arrow, then either y ∈ Σ_{0} or τy ∈ Σ_{0}. (P2) If y ∈ Σ_{0} and x → y is an arrow, then either x ∈ Σ_{0} or τ^{−1}x ∈ Σ_{0}.

The following Lemma collects the elementary properties of presections. Recall that a translation quiver is called stable if there are neither projective, nor injective points in Γ.

Lemma 4 Let (Γ, τ) be a connected translation quiver.

(a) If Σ is a section in Γ, then Σ is a presection.

(b) Any path entirely contained in a presection Σ is a sectional path.

(c) If the translation quiver Γ is stable, then conditions (P1) and (P2) are equivalent.

(d) If Σ is a presection in a stable translation quiver Γ, then Σ intersects every τ-orbit of Γ at least once.

PROOF. (a) This is well-known (see, for instance, [3, VIII.1.4 p.304]).

(b) Assume that Σ is a presection in Γ, and that x = x_{0} → x_{1} → … → x_{t} = y is a path lying entirely in Σ. If this path is not sectional, then there exists a least i with 0 < i < t and τ x_{i+1} = x_{i−1}. But, in this case, we have arrows x_{i−1} → x_{i} and x_{i} → τ^{−1} x_{i−1} = x_{i+1} with x_{i}, x_{i−1}, x_{i+1} ∈ Σ_{0}, a contradiction.

(c) Assume (P1) holds and that x → y is an arrow with y ∈ Σ_{0}. Since x is not injective, there exists an arrow y → τ^{−1}x. Applying (P1) to the latter yields that τ^{−1}x ∈ Σ_{0} or x = τ(τ^{−1}x) ∈ Σ_{0}. Thus (P2) holds. The converse is shown in the same way.

(d) It suffices to prove that, if x ∈ Σ_{0} and y ∈ Γ_{0} lie in two neighbouring τorbits, then Σ intersects the τ-orbit of y. Since Γ is stable, there exists m ∈ Z such that there is an arrow x → τ^{m}y. But then τ^{m}y ∈ Σ_{0} or τ^{m+1}y ∈ Σ_{0}. □

For instance, let Γ be a stable tube. A ray in Γ is a presection but clearly not a section. On the other hand, if Γ is a component of type ZA_{∞}, then a ray is a section.

1.4 Local sections

We now define an intermediate notion between those of presection and section.

Definition 5 Let (Γ, τ) be a connected translation quiver. A presection Σ in Γ is called a local section if it satisfies moreover the following additional condition:

Σ is sectionally convex, that is, if x = x_{0} → x_{1} → … → x_{t} = y is a sectional path in Γ such that x, y ∈ Σ_{0}, then x_{i} ∈ Σ_{0} for all i.

The following Lemma is immediate.

Lemma 6 Any section is a local section.

PROOF. Indeed, any section is convex, and hence sectionally convex. We then apply Lemma 4 (b). □

The main result of this section is the following.

Proposition 7 Let Q be a finite acyclic quiver and Γ = ZQ. The following conditions are equivalent for a connected full subquiver Σ of Γ such that |Σ_{0}| = |Q_{0}|:

(a) Σ is a section.

(b) Σ is a local section.

(c) Σ is a presection.

PROOF. Because of Lemma 6 (and the definition of local section), it suffices to prove that (c) implies (a). Since Γ is a stable translation quiver, then, because of Lemma 4 (d), Σ intersects every τ-orbit of Γ at least once. Then, it follows from the hypothesis that |Σ_{0}| = |Q_{0}| that Σ intersects each τ-orbit of Γ exactly once. Since Σ is clearly acyclic (because Γ is), there only remains to prove convexity. Suppose that there exists a path x = x_{0} → x_{1} → … → x_{t} = y in Γ with x, y ∈ Σ_{0} but x_{1}, …, x_{t−1} ∉ Σ_{0} (t ≥ 2). Since Σ is a presection, we have τ x_{1} ∈ Σ_{0}. From the arrow τ x_{1} → τ x_{2}, we deduce that either τ x_{2} ∈ Σ_{0} or τ^{2}x_{2} ∈ Σ_{0}. Repeating this argument t times, we get that one of τ y, τ^{2}y, …, τ^{t}y lies in Σ. This contradicts the fact that Σ cuts each τ-orbit exactly once, because y ∈ Σ_{0}. □

2 Sections and local sections in the derived category.

Let A denote a hereditary algebra, and D = D^{b}(mod A) denote the bounded derived category of mod A. We denote by τ_{D}, τ_{D}^{−1} the Auslander-Reiten translations in D. We recall that the Auslander-Reiten quiver of D consists of two types of components: the regular (which correspond to the regular components of Γ(mod A) and their shifts) and the transjective (which are the form ZQ, where Q denotes the ordinary quiver of A), see [14].

We need the following result, known as Skowroński’s Lemma [22,3].

Lemma 8 Let C be an artin algebra. Assume that a C-module M is the direct sum of m pairwise non-isomorphic indecomposable modules and is such that Hom_{C}(M, τ_{C}M) = 0. Then m ≤ rk K_{0}(C).

The following Lemma is motivated by [21,19].

Lemma 9 Let Σ be a section in a connected component Γ of Γ(D) then the following conditions are equivalent:

(a) Σ is convex in ind D, that is, if X = X_{0} → X_{1} → … → Y is a sequence of non-zero morphisms between indecomposable objects in D such that X, Y ∈ Σ_{0}, then X_{i} ∈ Σ_{0} for all i.

(b) Hom (X, τ_{D}Y) = 0 for all X, Y ∈ Σ_{0}.

(c) Hom (τ_{D}^{−1}X, Y) = 0 for all X, Y ∈ Σ_{0}.

(d) Σ_{0} is finite.

(e) |Σ_{0}| = rk K_{0}(A).

(f) Γ is a transjective component.

PROOF. The equivalence of (b) and (c) follows trivially from the fact that τ is an automorphism of D.

(a) implies (b). A non-zero morphism X → τ_{D} Y induces a path X → τ_{D} Y → ∗ → Y in ind D. The hypothesis implies that both Y and τ_{D} Y lie in Σ, a contradiction.

(b) implies (f). If Γ is not a transjective component, then we may assume without loss of generality that Γ is concentrated in degree zero. By Skowroński’s Lemma 8, Σ is finite. Now this contradicts the fact that Σ intersects each τ_{D}-orbit exactly once.

(f) implies (e). The number of τ_{D}-orbits in a transjective component is exactly rk K_{0}(A).

(e) implies (d). This is trivial.

(d) implies (a). Suppose that X = X_{0} →^{f_{1}} X_{1} →^{f_{2}} … →^{f_{t}} X_{t} = Y is a path, where the f_{i} are non-zero morphisms, the X_{i} lie in Γ and X, Y ∈ Σ_{0}. Now, since Σ_{0} is finite, then Γ has finitely many τ_{D}-orbits. Therefore, Γ is transjective. This implies that the f_{i} lie in a finite power of the radical of D. Therefore the path above can be refined to a path of irreducible morphisms. Convexity of Σ in Γ then implies that X_{i} ∈ Σ_{0} for all i. □

Corollary 10 Let Σ be a connected full subquiver in a connected component Γ of Γ(D) such that |Σ_{0}| = rk K_{0}(A). The following are equivalent:

(a) Σ is a section.

(b) Σ is a local section.

(c) Σ is a presection.

PROOF. Again, it suffices to prove that (c) implies (a). Since Γ is a stable translation quiver, then the presection Σ intersects each τ_{D}-orbit of Γ at least once. Since, by hypothesis, |Σ_{0}| = rk K_{0}(A) < ∞, then Γ is a transjective component. The statement then follows from Proposition 7. □

3 Local slices

3.1 Definition and examples

We now define the main concept of this paper.

Definition 11 Let C be a finite dimensional algebra. A local slice Σ in mod C is a local section in a component Γ of Γ(mod C) such that |Σ| = rk K_{0}(C).

Remark 12 Let C be an algebra, and Σ be a local slice in mod C.

(a) Since local sections are presections, every path entirely contained in Σ is sectional (because of Lemma 4 (b)). This implies that Σ is acyclic.

(b) If Γ is a stable component of Γ(mod C) and Σ ⊂ Γ then, by Lemma 4 (d), Σ intersects any τ_{C}-orbit of Γ at least once. Since |Σ_{0}| < ∞, this implies that Γ has only finitely many τ_{C}-orbits.

Example 13 Let C be a tilted algebra, and Γ be a connecting component of Γ(mod C). Then any complete slice in Γ is also a local slice. We shall prove below that any cluster-tilted algebra has (many) local slices in its Auslander-Reiten quiver.

Example 14 The following is an example of an algebra which is neither tilted, nor cluster-tilted, but whose Auslander-Reiten quiver contains a local slice. Let B be given by the quiver

\xymatrix{1 &  & 4 \ar[dl]_{α} \\  & 3 \ar[ul]_{β} \ar[dl]_{δ} \\ 2 \ar[rr]_{ϵ} &  & 5 \ar[ul]_{γ}}

bound by α β = 0, δ ϵ = 0, ϵ γ = 0 and γ δ = 0. The Auslander-Reiten quiver Γ(mod B) of B is shown in Figure 1, where modules are represented by their Loewy series and we identify the two copies of the underlined module \underline{2}. Here

Σ = \{ \begin{matrix}3 \\ 2 1 \end{matrix} , 32 , 4 3 2 , 31 , \begin{matrix}5 \\ 3 \\ 1 \end{matrix} \} and Σ′ = \{ \begin{matrix}2 \\ 5 \end{matrix} , 5 , {4 5 \atop 3} , 53 , \begin{matrix}5 \\ 3 \\ 1 \end{matrix} \} are local slices.

Note that neither Σ nor Σ′ is a section, because both intersect twice the τ_{B}- orbit of \underline{2}. It is an interesting question to identify the algebras which have local slices.

\xymatrix{ &  &  & 4 3 2 \ar[dr] &  &  &  & 25 \ar[dr] \\ 1 \ar[dr] &  & 32 \ar[ur] \ar[dr] &  & 43 \ar[dr] &  & 5 \ar[ur] &  & \underline{2} \\  & {3 \atop 2 1} \ar@{-->}[ur] \ar@{-->}[dr] &  & 3 \ar[ur] \ar[dr] &  & {4 5 \atop 3} \ar@{-->}[ur] \ar[dr] \\ \underline{2} \ar[ur] &  & 31 \ar[ur] \ar[dr] &  & 53 \ar@{-->}[ur] &  & 4 \\  &  &  & 5 3 1 \ar[ur]}

Fig. 1. Auslander-Reiten quiver of Example 13

3.2 Cluster-tilted algebras

Let A be a hereditary algebra. The cluster category C_{A} of A is defined as follows. Let F be the automorphism of D = D^{b}(mod A) defined as the composition τ_{D}^{−1}[1], where τ_{D}^{−1} is the Auslander-Reiten translation in D and [1] is the shift functor. Then C_{A} is the orbit category D/F, that is, the objects of C_{A} are the F -orbits X̃ = (F^{i}X)_{i∈Z}, where X ∈ D, and the set of morphisms from X̃ = (F^{i}X)_{i∈Z} to Ỹ = (F^{i}Y)_{i∈Z} is

Hom_{C_{A}}(X̃, Ỹ) = ⨁_{i∈Z} Hom_{D}(X, F^{i}Y).

It is shown in [7,16] that C_{A} is a triangulated category with almost split triangles. Furthermore, the projection functor π_{0} : D → C_{A} is a functor of triangulated categories and commutes with the Auslander-Reiten translations in both categories. We refer to [7] for facts about the cluster category.

An object T̃ in C_{A} is called a tilting object provided Ext_{C_{A}}^{1}(T̃, T̃) = 0 and the number of isomorphism classes of indecomposable summands of T̃ equals rk K_{0}(A). The endomorphism algebra B = End_{C_{A}}(T̃) is then called a clustertilted algebra [8]. Of particular interest to us is the fact that the functor Hom_{C_{A}}(T̃, −) : C_{A} → mod B induces an equivalence

C_{A}/add (τT̃) ≅ mod B,

where τ denotes the Auslander-Reiten translation in C_{A}, see [8]. This result entails several interesting consequences. For instance, it is shown it [17] that any cluster-tilted algebra is 1-Gorenstein and hence of global dimension 1 or ∞. For the convenience of the reader, we give here a short proof of this fact.

Proposition 15 Let B be a cluster-tilted algebra.

(a) For every injective B-module I, we have pd I ≤ 1 (and for every projective B-module P, we have id P ≤ 1).

(b) gl.dim. B ∈ {1, ∞}.

PROOF. (a) By [3, (IV.2.7) p.115], we need to prove that Hom_{B}(DB, τ_{B}I) = 0. Now mod B ≅ C_{A}/add (τT̃) (where A and T are as above) and, under this equivalence, every injective C̃-module is the image of an object of the form τ^{2}T̃_{0} ∈ C_{A}, where T̃_{0} ∈ add T̃. It thus suffices to show that, for every T̃_{0} ∈ add T̃, we have Hom_{C_{A}}(τ^{2}T̃, τ^{3}T̃_{0}) = 0. But τ is an equivalence in C_{A}, hence the result follows from Hom_{C_{A}}(T̃, τT̃_{0}) ≅ Ext_{C_{A}}^{1}(T̃, T̃_{0}) = 0.

(b) This is the proof of [17], but we include it for completeness. It suffices to prove that, for every B-module M, id M = d < ∞ implies pd M ≤ 1. Thus, let

\xymatrix{0 \ar[r] & M \ar[r] & I^{0} \ar[r]^{f^{1}} & I^{1} \ar[r] & ⋯ \ar[r]^{f^{d}} & I^{d} \ar[r] & 0}

be a minimal injective coresolution. Let K^{i} = Im f^{i} for every i. Then the exact sequence 0 → K^{d−1} → I^{d−1} → I^{d} → 0 and (a) give pd K^{d−1} ≤ 1. An easy induction yields pd M ≤ 1. □

It is also shown in [8] that the equivalence C_{A}/add (τT̃) ≅ mod B commutes with the Auslander-Reiten translations in both categories. Let π denote the composition of the functors

\xymatrix{D \ar@{->>}[r]^{π_{0}} & C_{A} \ar@{->>}[r]^{Hom_{C_{A}}(T̃,−)} & mod B,}

where π_{0} is, as above, the canonical projection. We notice that π commutes with the Auslander-Reiten translations in both categories and also that, if X ∈ D, then π(X) = 0 if and only if X ∈ add (τ_{D}T̃).

3.3 Auslander-Reiten quivers of cluster-tilted algebras

With the above notations, we deduce the shape of the Auslander-Reiten quiver of C_{A} and mod B. Let Q be the ordinary quiver of A. If A is representationfinite, then the Auslander-Reiten quiver Γ(C_{A}) is of the form ZQ/ < φ >, where φ is the automorphism of ZQ induced by the functor F. Since Γ(C_{A}) is stable and has sections isomorphic to Q, we say that it is transjective. If, on the other hand, A is representation-infinite, then Γ(C_{A}) consists of a unique component of the form ZQ, which we call transjective because it is the image under π_{0} of the transjective component of Γ(D), and also of components which we call regular because they are the image under π_{0} of the regular components of Γ(D). In both cases, we deduce Γ(mod B) from Γ(C_{A}) by deleting the |Q_{0}| points corresponding to the summands of τT̃. In particular, Γ(mod C̃) always has a unique transjective component, deduced from that of Γ(C_{A}) upon applying the functor Hom_{C_{A}}(T̃, −).

Lemma 16 Let Γ be a component of the Auslander-Reiten quiver of a clustertilted algebra B. If Γ contains a local slice, then Γ is the transjective component.

PROOF. Assume that Γ is not transjective. Then Γ is either a stable tube or a component of type ℤ𝔸_{∞}, or is obtained from one of these by deleting finitely many points. Also, since the functor Hom_{C_{A}}(T̃, −) : C_{A} → mod B commutes with the Auslander-Reiten translations, deleting these points will not change the τ-orbits. Consequently, a local slice Σ in Γ lifts to a unique finite local section Σ̃ in a regular component Γ̃ of Γ(C_{A}). In particular, Γ̃ is stable. Let thus X̃ ∈ Σ̃ and X̃ = X̃_{0} → X̃_{1} → … → X̃_{i} → … be a sectional path of irreducible morphisms (a ray) starting at X. Then X̃_{1} or τX̃_{1} belongs to Σ̃. By induction, for each i ≥ 0, one of the objects τ^{i}X̃_{i}, τ^{i−1}X̃_{i}, …, X̃_{i} belongs to Σ̃. Therefore Σ̃ is infinite, a contradiction. □

3.4 Lifting to the derived category

Lemma 17 Let Σ be a connected full subquiver of the transjective component of the Auslander-Reiten quiver of a cluster-tilted algebra B arising from a hereditary algebra A and \overline{Σ} be a connected full subquiver of D = D^{b}(mod A) such that π|\overline{Σ} : \overline{Σ} → Σ is bijective. Then Σ is a local slice in mod B if and only if \overline{Σ} is a section in D such that |Σ_{0}| = rk K_{0}(A).

PROOF. Since both subquivers are full, then the bijection π|\overline{Σ} induces an isomorphism of quivers. Assume that Σ is a local slice in mod B. We claim that \overline{Σ} is a presection in D. Assume that \overline{X} → \overline{Y} is an irreducible morphism in D with \overline{X} ∈ \overline{Σ_{0}}. Then we have two cases to consider:

(1) If π(\overline{Y}) ≠ 0, then either π(\overline{Y}) ∈ Σ_{0} or τ_{B}π(\overline{Y}) = π(τ_{D}\overline{Y}) ∈ Σ_{0}. Therefore \overline{Y} ∈ \overline{Σ_{0}} or τ_{D}\overline{Y} ∈ \overline{Σ_{0}}.

(2) If π(\overline{Y}) = 0, then π(τ_{D}\overline{Y}) ≠ 0 because Hom_{D}(\overline{Y} , τ_{D}\overline{Y} [1]) ≠ 0. But we have an arrow \overline{X} → \overline{Y} which gives an arrow τ_{D}\overline{Y} → \overline{X}, hence an arrow π(τ_{D}\overline{Y}) → π(\overline{X}). Since π(\overline{X}) ∈ Σ_{0}, then π(τ_{D}\overline{Y}) ∈ Σ_{0} and so τ_{D}\overline{Y} ∈ \overline{Σ}_{0}.

Because of Lemma 4 (c), this shows that \overline{Σ} is a presection. Since π|\overline{Σ} is a bijection, we have |\overline{Σ}_{0}| = |Σ_{0}| = rk K_{0}(A). By Corollary 10, \overline{Σ} is a section.

Conversely, assume that \overline{Σ} is a section in D and that |\overline{Σ}_{0}| = rk K_{0}(A). By Lemma 9, \overline{Σ} lies in a transjective component of Γ(D). Note that |\overline{Σ}_{0}| = |Σ_{0}| and rk K_{0}(A) = rk K_{0}(B). Hence |Σ_{0}| = rk K_{0}(B).

We show that Σ is sectionally convex. Let X = X_{0} → X_{1} → … → X_{t} = Y be a sectional path with X, Y ∈ Σ_{0}. It lifts to a unique path \overline{X} = \overline{X}_{0} → \overline{X}_{1} → … → \overline{X}_{t} = \overline{Y} in D with \overline{X}, \overline{Y} ∈ \overline{Σ}_{0}. Since \overline{Σ} is a section, then this path is sectional and all \overline{X}_{i} ∈ \overline{Σ}_{0}. Applying π, we get that all X_{i} lie in Σ.

Finally, we show that Σ is a presection. Assume that X → Y is an irreducible morphism in mod B with X ∈ Σ_{0}. Let \overline{X} = π|−1 ; \overline{Σ}(X) and choose \overline{Y} in π^{−1}(Y) such that we have an irreducible morphism \overline{X} → \overline{Y} in D. Then \overline{Y} ∈ \overline{Σ}_{0} or τ_{D}\overline{Y} ∈ \overline{Σ}_{0}. Hence Y ∈ Σ_{0} or τ_{B}Y ∈ Σ_{0}. Note that if τ_{D}\overline{Y} ∈ \overline{Σ}_{0} then π(τ_{D}\overline{Y}) ≠ 0 because π|\overline{Σ} is a bijection. We treat similarly the case of an irreducible morphism X → Y with Y ∈ Σ_{0}. □

3.5 Construction of local slices

There is a close relation between tilted and cluster-tilted algebras. First, if B is a cluster-tilted algebra, then there exist a hereditary algebra A and a tilting A-module T such that B ≅ End_{C_{A}}(T̃), see [7, 3.3]. Also, if A is a hereditary algebra and T is a tilting A-module so that the algebra C = End_{A}(T) is tilted, the trivial extension C̃ = C⋉Ext_{C}^{2}(DC, C) (called the relation-extension of C) is cluster-tilted, and conversely, every cluster-tilted algebra is of this form, see [1]. Now, since C̃ = C⋉Ext_{C}^{2}(DC, C), then any C-module can be considered as a C̃-module under the standard embedding i : mod C → mod C̃. Note that, in general, i does not preserve irreducible morphisms. We consider the complete slice Σ = add Hom_{A}(T, DA) in mod C, where C = End T_{A} (see, for instance,

[3] or [20]) and denote its image as i(Σ) = Σ′ in mod C̃. The following Lemma collects the important properties of Σ′.

Lemma 18 Let Σ = Hom_{A}(T, DA) and Σ′ = i(Σ), then

(a) The image Σ′ is a local slice in mod C̃.

(b) i induces an isomorphism of quivers between Σ and Σ′.

(c) AnnC̃ Σ′ ≅ Ext_{C}^{2}(DC, C) as C-C-bimodules.

PROOF. (a) We have Hom_{C_{A}}(T̃, D̃A) = Hom_{A}(T, DA) because T is an Amodule. Thus the image i(Σ) = Σ′ of Σ is equal to the image of Σ under the composition

\xymatrix{mod C \ar[r]^{j} & D^{b}(mod C) \ar[r]^{ϕ}_{≅} & D^{b}(mod A) \ar[r]|{π} & mod C̃,}

where j is the inclusion in degree zero and ϕ the equivalence −⊗_{C}^{L} T. Let \overline{Σ} be a connected full subquiver of D^{b}(mod A) such that the restriction π|\overline{Σ} : \overline{Σ} → Σ′ is bijective, that is, π(\overline{Σ}) = Σ′ = i(Σ). Using the fact that ϕ is a quasi-inverse of the derived functor RHom_{A}(T, −) : D^{b}(mod A) → D^{b}(mod C), we see that \overline{Σ} = ϕ ∘ j(Σ) is equal to DA which is a connected section in a transjective component of D^{b}(mod A). Since add (τ T) ∩ add (DA) = ∅, then π(I) ≠ 0 for each I ∈ add DA, so that π|_{add DA} : DA → Σ′ is bijective. By Lemma 17, Σ′ is a local slice in mod C̃.

(b) This follows from the fact that Σ′ = πϕj(Σ) and each one of j|_{Σ}, ϕ and π|_{ϕj(Σ)} preserves irreducible morphisms (in the case of π, this is because add DA ∩ add τ T = ∅).

(c) By [1], we have an isomorphism Ext_{C}^{2}(DC, C) ≅ Hom_{D}(T, F T) of C-C-bimodules. Now

Hom_{A}(T, DA) · Hom_{D}(T, F T) = Hom_{D}(T, DA) · Hom_{D}(T, F T)
⊂ Hom_{D}(T, F (DA))
= Hom_{D}(T, A[2]) = 0,

because T is an A-module. Therefore Hom_{D}(T, F T) ⊂ AnnC̃ Σ′.

Now let f : T → T be a non-zero morphism. Since its image is an A-module, then Hom_{A}(Imf, DA) ≠ 0. Let g : Imf → DA be a non-zero morphism and denote by p : T → Imf the canonical epimorphism. Since DA is an injective module, there exists g′ : T → DA such that g′f = gp ≠ 0. Therefore AnnC̃ Σ′ ∩ Hom_{A}(T, T) = 0. Since, as a k-vector space, C̃ = Hom_{A}(T, T) ⊕ Hom_{D}(T, F T). We have dim C̃ = dim Hom_{D}(T, T) + dim Hom_{D}(T, F T) ≤ dim Hom_{D}(T, T) + dim AnnC̃Σ′, because dim Hom_{D}(T, F T) ⊂ AnnC̃Σ′. Since Hom_{D}(T, F T) and AnnC̃Σ′ are in direct sum, then we have dim Hom_{D}(T, T)+ dim AnnC̃Σ′ = dim(Hom_{D}(T, T) + AnnC̃Σ′) ≤ dim C̃. Hence dim AnnC̃Σ′ = dim Hom_{D}(T, F T). This shows that Hom_{D}(T, F T) and AnnC̃Σ′ are equal as subspaces of C̃, and the statement follows. □

3.6 Main result

We are now ready for the proof of our main theorem.

Theorem 19 Let B be a cluster-tilted algebra. Then a tilted algebra C is such that B = C ⋉ Ext_{C}^{2}(DC, C) if and only if there exists a local slice Σ in mod B such that C = B/Ann_{B} Σ.

PROOF. Necessity. It is well-known (see, for instance, [3]) that any complete slice Σ in mod C is of the form Σ = add Hom_{A}(T, DA) for some hereditary algebra A and some tilting A-module T. By Lemma 18 (a), Σ embeds as a local slice in mod B (because B = C̃). Moreover, by Lemma 18 (c), we have Ann_{B} Σ ≅ Ext_{C}^{2}(DC, C) as C-C-bimodules. Therefore C = B/Ext_{C}^{2}(DC, C) = B/Ann_{B} Σ.

Sufficiency. Let B be cluster-tilted, and Σ be a local slice in mod B. Set C_{1} = B/Ann_{B} Σ. Because of the definition of a cluster-tilted algebra, there exist a hereditary algebra A and a tilting object T̃ ∈ C_{A} such that B = End_{C_{A}}(T̃). Let \overline{Σ} be a connected component of the preimage π^{−1}(Σ) of Σ in D^{b}(mod A). Since the local slice Σ can only occur in the transjective component of Γ(mod B), because of Lemma 16, then \overline{Σ} lies in a transjective component of Γ(D^{b}(mod A)). Since Σ and \overline{Σ} have the same number of points, then π|\overline{Σ} : \overline{Σ} → Σ is bijective, whence \overline{Σ} is a section in Γ(D^{b}(mod A)) such that |\overline{Σ}_{0}| = rk K_{0}(A), by Lemma 17. We may suppose without loss of generality that \overline{Σ} = add DA.

The fact that π|_{add DA} : add DA → Σ is a bijection implies that the F - orbit τ_{D} π^{−1}(T̃) in D^{b}(mod A) does not intersect add DA. Therefore, we have add π^{−1}(T̃) ∩ add A[1] = ∅, because τ_{D}^{−1}DA = A[1] in D^{b}(mod A). Thus, we can choose a representative T in the F -orbit π^{−1}(T̃) such that T is an Amodule and π(T) = T̃. Then T is a tilting A-module. Let C_{2} = End_{A}(T) be the corresponding tilted algebra. By [1], we have B = C_{2}⋉Ext_{C_{2}}^{2}(DC_{2}, C_{2}). By Lemma 18 (c), we have Ext_{C_{2}}^{2}(DC_{2}, C_{2}) = Ann_{B} Σ. Thus C_{2} = B/Ann_{B} Σ = C_{1}. In particular, C_{1} is tilted and verifies B = C_{1} ⋉ Ext_{C_{1}}^{2}(DC_{1}, C_{1}). □

Corollary 20 Let C be a tilted algebra and C̃ be the corresponding clustertilted algebra. Then any complete slice in mod C embeds as a local slice in mod C̃ and any local slice in mod C̃ arises this way.

PROOF. This follows directly from Lemma 18 and Theorem 19. □

3.7 Computing the annihilator

Our Theorem 19 actually gives a concrete way to compute the tilted algebra C starting from C̃. Given a local slice Σ in mod C̃, one computes its annihilator using the following result.

Corollary 21 Let B be a cluster-tilted algebra and Σ be a local slice in mod B. Then Ann_{B}Σ is generated (as an ideal) by arrows in the quiver of B.

\xymatrix{ & 21 \ar[dr] &  & 3 \ar[dr] &  & 42 \ar[dr] &  & τT̃_{2} &  & 21 \\ \underline{1} \ar[dr] \ar@{-->}[ur] & τT̃_{4} & {2 3 \atop 1} \ar[dr] \ar@{-->}[ur] \ar[r] & 4 2 3 1 \ar[r] & {4 \atop 2 3} \ar[dr] \ar@{-->}[ur] & τT̃_{1} & 4 \ar[r] & 14 \ar[r] & \underline{1} \ar@{-->}[ur] \ar[dr] & τT̃_{4} \\  & 31 \ar[ur] &  & 2 \ar[ur] &  & 43 \ar@{-->}[ur] &  & τT̃_{3} &  & 31}

Fig. 2. Auslander-Reiten quiver of Example 3.8

PROOF. This follows from [4, 1.3] using that C̃ is a trivial extension (hence a split extension) of C by the C-C-bimodule Ext_{C}^{2}(DC, C) ≅ AnnC̃Σ. □

We can be a bit more precise. Let C̃ = kQ̃/Ĩ. Since C̃ = End_{C_{A}} T̃, there is a bijection between the points x ∈ Q̃_{0} and the indecomposable summands T̃_{x} of T̃ so that each arrow α : x → y in Q̃_{1} corresponds to a non-zero morphism f_{α} ∈ Hom_{C_{A}}(T̃_{y}, T̃_{x}) = Hom_{D}(T_{y}, T_{x}) ⊕ Hom_{D}(T_{y}, F T_{x}). With this notation, AnnC̃Σ is generated by all arrows α : x → y such that f_{α} ∈ Hom_{D}(T_{y}, F T_{x}). This indeed follows immediately from the isomorphisms

Hom_{D}(T, F T) ≅ Ext_{C}^{2}(DC, C) ≅ AnnC̃Σ.

Note that, as shown in [2], the arrows α which generate AnnC̃Σ have to satisfy certain conditions. Moreover, if C = kQ/I, then I = Ĩ∩ kQ.

3.8 Example

Let C̃ be the cluster-tilted algebra (of type 𝔻_{4}) given by the quiver

\xymatrix{ & 2 \ar[dl]_{β} \\ 1 \ar[rr]^{ϵ} &  & 4 \ar[ul]_{α} \ar[dl]^{γ} \\  & 3 \ar[ul]^{δ}}

bound by αβ = γδ, βϵ = 0, δϵ = 0, ϵα = 0, ϵγ = 0. The Auslander-Reiten quiver of C̃ is of the form shown in Figure 2 where indecomposable modules are represented by their Loewy series, and one identifies the two copies of the

underlined simple module 1 (and also the two copies of the modules 21 and \begin{matrix}3 \\ 1 \end{matrix} ).

The entries τT̃_{i} indicate the position of τT̃_{i} in the cluster category. We find

[figure: 2; β; C_{1}; 1; 4; α; γ; Σ_{1} = \{ \begin{matrix}4 \\ 2 3 \\ 1 \end{matrix} , {4 \atop 2 3} , 42 , \begin{matrix}4 \\ 3 \end{matrix} \}; 3; δ; 2; β; C_{2}; 1; ϵ; 4; Σ_{2} = \{ \begin{matrix}1 \\ 4 \end{matrix} , 1 , 21 , \begin{matrix}3 \\ 1 \end{matrix} \}; 3; δ; 2; C_{3}; 1; ϵ; 4; α; γ; Σ_{3} = \{ \begin{matrix}4 \\ 2 \end{matrix} , 43 , 4 , \begin{matrix}1 \\ 4 \end{matrix} \}; 3]

Fig. 3. Tilted algebras and corresponding local slices

easily all the local slices hence all the tilted subalgebras of C̃ which realise it as a relation-extension. There are only the three algebras C_{i} shown in Figure 3 each corresponding to a local slice Σ_{i}. with the inherited relations in each case.

4 Cluster-tilted algebras of tree type

In view of our main result, Theorem 19, it is reasonable to ask whether there exist sufficiently many local slices in the Auslander-Reiten quiver of a clustertilted algebra. Since the latter is deduced from the Auslander-Reiten quiver of the cluster category by dropping finitely many points, and since local slices can only occur in the transjective component, then all but at most finitely many indecomposables lying in the transjective component of the clustertilted algebra belong to a local slice. That not necessarily all indecomposables in the transjective component belong to a local slice is seen in the following example.

Example 22 Let A be the hereditary algebra given by the quiver

\xymatrix{ & 2 \ar[dl] \\ 1 &  & 3 \ar[ul] \ar[ll]}

and consider the tilting A-module T = 1 ⊕ 3 1 2 1 ⊕ 31 . Note that while T_{1} = 1

and T_{2} = 3 1 2 1 are projective A-modules, the module T_{3} = 31 is simple regular

non-homogeneous. The transjective component of the Auslander-Reiten quiver of C̃ = End_{C_{A}} T̃ is thus of the form

\xymatrix{ &  &  & · \ar[ddr] &  &  & M \ar[drr] &  &  & · \ar[drr] \ar[ddr] &  &  & · \ar[ddr] \\ … &  & · \ar[ur] \ar[drr] &  &  & • &  &  & P_{1} \ar@{-->}[ur] \ar[drr] &  &  & · \ar[ur] \ar[drr] &  &  & · & … \\  & · \ar[ur] \ar[uurr] &  &  & · \ar[uurr] &  &  & • &  &  & P_{2} \ar@{-->}[ur] \ar[uurr] &  &  & · \ar[ur]}

where P_{i} = Hom_{C_{A}}(T̃, T_{i}), for i ∈ {1, 2}, and the two • represent τT_{1} and τT_{2}. It is easily seen that the indecomposable C̃-module M = rad P_{1} lies on no local slice.

We say that a cluster-tilted algebra End_{C_{A}}(T̃) is of tree type if the ordinary quiver of A is a tree.

Theorem 23 Let C̃ be a cluster-tilted algebra of tree type. Then any indecomposable C̃-module lying in the transjective component lies on a local slice.

PROOF. Let M̃ be an indecomposable C̃-module in the transjective component and M be an indecomposable object in π^{−1}(M̃); here, as in section 3, π denotes the composition of the natural functors

\xymatrix{D^{b}(mod A) \ar@{->>}[r]^{π_{0}} & C_{A} \ar@{->>}[r]^{Hom_{C_{A}}(T̃,−)} & mod C̃} .

We claim that M lies on a section Σ in D^{b}(mod A) such that |Σ_{0}| = rk K_{0}(A) and Σ ∩ add τ T̃ = ∅.

Before proving this claim, we show that the theorem follows from it. Indeed, under these conditions, π|_{Σ} : Σ → π(Σ) is bijective and π(Σ) is a connected full subquiver of the transjective component of the Auslander-Reiten quiver of C̃. By Lemma 17, π(Σ) is a local slice. Obviously M̃ = π(M) ∈ π(Σ).

We now prove the claim. First, we fix some terminology. For any connected full subquiver T of Γ(D^{b}(mod A)) such that T is a tree and M ∈ T_{0}, there is a unique reduced walk in T from M to any other point N ∈ T_{0}. We define the distance d_{T} (M, N) between M and N in T to be the number of arrows of this walk. Moreover, we define

d_{T} = \{ \begin{matrix}rk K_{0}(A) & if T ∩ add τ T̃ = ∅ \\ min\{d_{T} (M, N) | N ∈ T ∩ add τ T̃\} & if T ∩ add τ T̃ ≠ ∅. \end{matrix}

Thus d_{T} measures the distance between M and the set add τ T̃ in the tree T . Note that d_{T} ≥ 1 because M ∉ add τ T̃ and d_{T} = rk K_{0}(A) if and only if T ∩ add τ T̃ = ∅.

Now, let Σ_{1} be any section in D^{b}(mod A) containing M. By hypothesis, Σ_{1} is a tree. If d_{Σ_{1}} = rk K_{0}(A) we are done. Suppose that d_{Σ_{1}} < rk K_{0}(A). Consider the set N = {N ∈ Σ_{1} ∩ add τ T̃ | d_{Σ_{1}} = d_{Σ_{1}}(M, N)} and let N ∈ N. Consider the unique reduced walk

M = X_{0} X_{1} … X_{(d_{Σ_{1}}−1)} = L X_{d_{Σ_{1}}} = N

in Σ_{1} from M to N. Deleting the edge L N cuts Σ_{1} into two subtrees. Let Σ_{1}^{M} be the subtree containing M (and L) and let Σ_{1}^{N} the subtree containing N. There are two cases to consider, according to the orientation of the arrow between L and N.

(1) If L → N, we define Σ_{2} to be the full subquiver of Γ(D^{b}(mod A)) having as points those of Σ_{1}^{M} ∪ τ Σ_{1}^{N}.

(2) If L ← N, we define Σ_{2} to be the full subquiver of Γ(D^{b}(mod A)) having as points those of Σ_{1}^{M} ∪ τ^{−1} Σ_{1}^{N}.

By construction, Σ_{2} is a connected tree, it lies in the transjective component and it intersects every τ-orbit exactly once.

We now show that Σ_{2} is also convex. Assume that L → N (the proof in case L ← N is entirely similar). Then Σ_{2} has two subtrees Σ_{1}^{M} and τ Σ_{1}^{N}, connected by the arrow τ N → L. We show first that these two subtrees are convex. Suppose that X = X_{0} → X_{1} → … → X_{s} = Y is a path in Γ(D^{b}(mod A) with X, Y ∈ Σ_{1}^{M}. By convexity of Σ_{1}, we have X_{i} ∈ Σ_{1} for all i. Since there is exactly one walk from X to Y in Σ_{1} we actually have X_{i} in Σ_{1}^{M} for all i. Thus Σ_{1}^{M} is convex. Similarly, τ Σ_{1}^{N} is also convex. Now suppose that X = X_{0} → X_{1} → … → X_{s} = Y is a path with X, Y ∈ Σ_{2}. If X ∈ Σ_{1}^{M}, then Y ∈ Σ_{1}^{M} because of the structure of the transjective component, and then the convexity of Σ_{1}^{M} implies the result. If X, Y ∈ τ Σ_{1}^{N} we are done by convexity of τ Σ_{1}^{N}. Hence the only remaining case is when X ∈ τ Σ_{1}^{N} and Y ∈ Σ_{1}^{M}. Suppose that there is an i ∈ {1, …, s − 1} such that X_{i} ∉ Σ_{1}. We may suppose, without loss of generality, that i = 1. Now τ Σ_{1} is a section such that |(τ Σ_{1})_{0}| = rk K_{0}(A), hence, in particular, it is a presection. Therefore X ∈ τ Σ_{1}, X → X_{1} and X_{1} ∉ τ Σ_{1} imply τ X_{1} ∈ τ Σ_{1}, hence X_{1} ∈ Σ_{1}. Then all X_{i} with 1 ≤ i ≤ s lie in Σ_{1} because Σ_{1} is convex. Since X_{1} ∉ Σ_{2}, we have X_{1} ∈ Σ_{1}^{N}. Then, since Y ∈ Σ_{1}^{M} and the subtrees Σ_{1}^{N}, Σ_{1}^{M} are only joined by the arrow L → N, there exists j such that X_{j} = N and X_{j+1} = L. But then there is an arrow L ← N, a contradiction. This shows that Σ_{2} is convex and hence is a section in D^{b}(mod A) satisfying |(Σ_{2})_{0}| = rk K_{0}(A).

We repeat this construction for every element N of N. Note that, since N ∈ add τ T̃, neither τ N nor τ^{−1}N are in add τ T̃. In this way, we obtain, after k = |N | steps, a section Σ_{k+1} in Γ(D^{b}(mod A)) that contains M, such that |(Σ_{k+1})_{0}| = rk K_{0}(A) and

d_{Σ_{k+1}} > d_{Σ_{1}}.

If d_{Σ_{k+1}} = rk K_{0}(A) then Σ_{k+1} ∩ add (τ T̃) = ∅ and we are done. Otherwise, we repeat the construction with the set

N_{k+1} = {N ∈ Σ_{k+1} | d_{Σ_{k+1}}(M, N) = d_{Σ_{k+1}}}.

This algorithm will find the required section Σ in a finite number of steps. □

Corollary 24 Let C̃ be a representation-finite cluster-tilted algebra. Then any indecomposable C̃-module lies on a local slice.

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