MUTATION OF CLUSTER-TILTING OBJECTS AND POTENTIALS

ASLAK BAKKE BUAN, OSAMU IYAMA, IDUN REITEN, AND DAVID SMITH

Abstract. We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials.

We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.

Contents

Introduction 1 1. Preliminaries on mutation 3 1.1. Mutation of quivers 3 1.2. Mutation of quivers with potentials 4 1.3. Mutation of cluster-tilting objects 5 1.4. Cluster-tilted algebras 6 1.5. Mutation of tilting modules over 3-Calabi-Yau algebras 6 2. Cluster-tilted algebras are determined by their quivers 7 3. Preliminaries on presentation of algebras 9 4. Jacobian algebras and weak 2-almost split sequences 13 5. Mutation of quivers with potentials and cluster-tilting objects 16 6. 2-CY-tilted algebras associated with elements in Coxeter groups 24 7. Nearly Morita equivalence for neighboring Jacobian algebras 29 References 36

Introduction

The Fomin-Zelevinsky mutation of quivers plays an important role in the theory of cluster algebras initiated in [FZ]. There is, motivated by this theory via [MRZ], a mutation of cluster-tilting objects in cluster categories, and more generally Homfinite triangulated 2-Calabi-Yau (2-CY for short) categories over an algebraically closed field K [BMRRT, IY]. This has turned out to give a categorical model for the quiver mutation in certain cases [BMR2, BIRS]. On the other hand there is the recent theory of mutation of quivers with potentials (Q, W), initiated in [DWZ1].

2000 Mathematics Subject Classification. Primary 16G20, Secondary 18E30, 16S38.

All authors were supported by the Storforsk-grant 167130 from the Norwegian Research Council, the second author also by JSPS Grant-in-Aid for Scientific Research 18740007, the third author also by a Humboldt Research Prize, and the fourth author also by NSERC of Canada.

Associated with cluster-tilting objects T in 2-CY categories C are the endomorphism algebras End_{C}(T), called 2-CY-tilted algebras. And associated with a quiver with potential (Q, W) are algebras called Jacobian algebras in [DWZ1]. The mutation of cluster-tilting objects induces an operation on the associated 2-CY-tilted algebras and the mutation of quivers with potentials induces an operation on the associated Jacobian algebras. The theme of this paper is to investigate these classes of algebras and their relationships, in particular with respect to mutation. In addition to cluster-tilting objects in triangulated 2-CY categories, we also deal with mutation of tilting modules over 3-CY algebras, and their relationship to mutation of quivers with potentials. We now state the main results of this paper, referring to section 1 for definitions and background material.

(A) Starting with a cluster-tilting object T in a 2-CY category C such that End_{C}(T) is Jacobian, our main theorem states that the two mutations “coincide” (Theorem 5.1). The same type of result is proved for tilting modules over 3-CY algebras (Theorem 5.2). Our basic idea for the proof is to use the relationship between Jacobian algebras and 2-almost split sequences introduced in [I2, IY], which is a higher analogue of the classical relationship between mesh categories of translation quivers and almost split sequences [Rie, BG, IT1, IT2, I1].

(B) The results in (A) can be used to show that large classes of 2-CY-tilted algebras are Jacobian. In particular, a main result is that cluster-tilted algebras, which are an important subclass of 2-CY-tilted algebras, are Jacobian (Corollary 5.12). This is an easy consequence of Theorem 5.1.

More generally we show that a large class of 2-CY-tilted algebras coming from triangulated 2-CY categories associated with elements in Coxeter groups (see [BIRS]) are also Jacobian (Theorem 6.4).

(C) It is an open question whether there exists a (mutation-) operation on algebras with the following property: End_{C}(T) should be sent to End_{C}(T^{∗}), when T is a cluster-tilting object in a triangulated 2-CY category and T^{∗} is a cluster-tilting object obtained from T by mutation.

Similarly, in [DWZ1] a question was posed, asking if there is a (mutation-) operation defined on algebras with the property: the Jacobian algebra P(Q, W) should be sent to the Jacobian algebra P(Q′, W′), when (Q, W) is a quiver with potential and (Q′, W′) is a quiver with potential obtained from (Q, W) by mutation.

It is an easy consequence of (A) that both questions have an affirmative answer for 2-CY-tilted algebras which are Jacobian. By recent results in [A], Jacobian algebras of Jacobi-finite QP’s have this property (Theorem 5.13, Corollary 5.14).

(D) Another main result in this paper is that cluster-tilted algebras are determined by their quivers. This was shown for finite representation type in [BMR3]. We here give a short proof of this fact (Theorem 2.3). Alternatively, it is also a direct consequence of (A).

(E) One of the starting points of tilting theory is the reflection functors [BGP, APR], giving a nearly Morita equivalence, in the terminology of [Rin], meaning that \frac{mod Λ}{[add S_{k}]} ≃ \frac{mod Λ′}{[add S′_{k}]}, where S_{k} and S′_{k} are the simple modules associated with the vertex k, where k is a sink or a source in the quiver of the path algebra Λ, and Λ′ is the path algebra obtained by changing direction of all arrows adjacent to k. An important property of cluster-tilted algebras (and more generally 2-CY-tilted algebras), is that this nearly Morita equivalence generalizes in the sense that one replaces reflection at a source/sink with mutation at any vertex [BMR1, KR1]. A consequence of this is that the cluster-tilted algebras (in one mutation class) have the same “number” of indecomposable modules. In view of the close connection between 2-CY-tilted algebras and Jacobian algebras it is natural to investigate if a property known to hold for one of the classes also holds for the other one. In this spirit we show that nearly Morita equivalence holds also for two Jacobian algebras where one is obtained from the other by mutation of potentials (Theorem 7.1). We show this also for algebras which are not finite dimensional, dealing with the category of finite length modules. In view of our previous results, this gives a generalization of the above result for cluster-tilted algebras. For the proof we use a functorial approach to results and techniques of [DWZ1].

The organisation of the paper is as follows. In section 1 we collect some known results on the different kinds of mutation and on cluster-tilted algebras. In section 2 we prove that cluster-tilted algebras are determined by their quiver. The connection between cluster-tilting mutation and mutation of quivers with potential is given in sections 3,4 and 5. In section 6 we show that a class of 2-CY-tilted algebras associated with reduced expressions of elements of the Coxeter groups are given by QP’s, which are rigid in the sense of [DWZ1]. The result on nearly Morita equivalence is given in section 7.

The results in this paper have been presented at conferences in Oxford and Oberwolfach [Re, I3], That cluster-tilted algebras and some of the 2-CY-tilted algebras associated with elements in Coxeter groups are Jacobian is proved independently in [K2] using completely different methods (see also [A]). We have been informed by Zelevinsky that the result on nearly Morita equivalence is independently proved in [DWZ2].

Conventions All modules are left modules, and a composition ab of morphisms (respectively, arrows) a and b means first a and then b. We denote by s(a) the start vertex of an arrow or path a and e(a) denotes the end vertex. For an algebra Λ we denote by mod Λ the category of finitely generated Λ-modules, and by f. l. Λ the category of finite length Λ-modules. We denote by J_{Λ} the Jacobson radical of an algebra Λ, and by J_{C} the Jacobson radical of an additive category C. For an object T in an additive category C we denote by Q_{T} the quiver of End_{C}(T).

1. Preliminaries on mutation

In this section we discuss different kinds of mutations. We first recall the Fomin-Zelevinsky mutation of quivers and the recent mutation of quivers with potentials from [DWZ1]. Then we consider the mutation of cluster-tilting objects in 2-Calabi-Yau (2-CY) categories, which often gives a categorical modelling of quiver mutation. A main theme of this paper is the comparison of the last two mutations. Since cluster-tilted algebras play a central role in this paper, we recall their definition and some of their basic properties. Finally, we consider mutation of tilting modules over 3-Calabi-Yau algebras, which will also be compared to mutation of quivers with potentials.

1.1. Mutation of quivers. Let Q be a finite quiver with vertices 1, …, n, and having no loops or 2-cycles. For any vertex k, Fomin-Zelevinsky [FZ] defined a new quiver μ_{k}(Q) as follows. Let b_{ij} and b′_{ij} denote the number of arrows from i to j minus the number of arrows from j to i in Q and μ_{k}(Q), respectively. Then we have

b′_{ij} = \begin{cases}−b_{ij} & if i = k or j = k, \\ b_{ij} + \frac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2} & else. \end{cases}

Clearly we have μ_{k}(μ_{k}(Q)) ≃ Q. This mutation is an essential ingredient in the theory of cluster algebras initiated in [FZ]. Note that if the vertex k is a sink or a source, then this mutation coincides with the Bernstein-Gelfand-Ponomarev reflection of quivers [BGP].

1.2. Mutation of quivers with potentials. Let Q be a finite connected quiver without loops, and with vertices 1, …, n, and set of arrows Q_{1}. We denote by KQ_{i} the K-vector space with basis Q_{i} consisting of paths of length i in Q, and by KQ_{i,cyc} the subspace of KQ_{i} spanned by all cycles. Consider the complete path algebra

K̂Q = ∏_{i≥0} KQ_{i}

over an algebraically closed field K. A quiver with potential (QP for short) is a pair (Q, W) consisting of a quiver Q without loops and an element W ∈ ∏_{i≥2} KQ_{i,cyc} (called a potential). It is called reduced if W ∈ ∏_{i≥3} KQ_{i,cyc}. The cyclic derivative ∂_{a}W is defined by ∂_{a}(a_{1} ⋯ a_{ℓ}) = ∑_{a_{i}=a} a_{i+1} ⋯ a_{ℓ}a_{1} ⋯ a_{i−1} and extended linearly and continuously. A QP gives rise to what has been called the associated Jacobian algebra [DWZ1]

P(Q, W) = K̂Q/ J (W),

where J (W) = \overline{⟨∂_{a}W | a ∈ Q_{1}⟩} is the closure of the ideal generated by ∂_{a}W with respect to the JK̂Q-adic topology.

Two potentials W and W′ are called cyclically equivalent if W −W′ ∈ \overline{[KQ, KQ]}, where [−, −] denote the vector space spanned by commutators. Two QP’s (Q, W) and (Q′, W′) are called right-equivalent if Q_{0} = Q′_{0} and there exists a K-algebra isomorphism ϕ: K̂Q → K̂Q′ such that ϕ|_{Q_{0}} = id and ϕ(W) and W′ are cyclically equivalent. In this case ϕ induces an isomorphism P(Q, W) ≃ P(Q′, W′). It is shown in [DWZ1] that for any QP (Q, W) there exists a reduced QP (Q_{red}, W_{red}) such that P(Q, W) ≃ P(Q_{red}, W_{red}), which is uniquely determined up to rightequivalence. We call (Q_{red}, W_{red}) a reduced part of (Q, W). For example, a reduced part of the QP (Q, W) below is given by the QP (Q_{red}, W_{red}) below.

(Q, W) = ( [figure: 1; c; a; 2; b; 3; d], cd + abd) (Q_{red}, W_{red}) = ( [figure: 1; a; 2; b; 3], 0)

A QP (Q, W) is called rigid [DWZ1] if the deformation space P(Q, W)/(KQ_{0} + \overline{[P(Q, W), P(Q, W)]}) is zero, or equivalently JK̂Q = J (W) + \overline{[KQ, KQ]} holds.

A mutation μ_{k}(Q, W) of a QP (Q, W) is introduced in [DWZ1] for each vertex k in Q not lying on a 2-cycle. It is defined as a reduced part of (Q′, W′) = μ̃_{k}(Q, W), the latter one being given as follows. Replacing W by a cyclically equivalent potential, we assume that no cycles in W start at k.

(a) Q′ is a quiver obtained from Q by the following changes.

(i) Replace the fixed vertex k in Q by a new vertex k^{∗}. (Although k and k^{∗} are identified in [DWZ1], we distinguish them to avoid any confusion.)

(ii) Add a new arrow [ab]: i → j for each pair of arrows a: i → k and

b: k → j in Q. (iii) Replace each arrow a: i → k in Q by a new arrow a^{∗}: k^{∗} → i. (iv) Replace each arrow b: k → j in Q by a new arrow b^{∗}: j → k^{∗}.

(b) W′ = [W] + Δ where [W] and Δ are the following:

(i) [W] is obtained by substituting [ab] for each factor ab in W with a: i → k and b: j → k.

(ii) Δ = ∑_{{a,b∈Q_{1} \atop e(a)=k=s(b)}} a^{∗}[ab]b^{∗}.

Then k^{∗} is not contained in any 2-cycle in μ_{k}(Q, W), and μ_{k^{∗}}(μ_{k}(Q, W)) is rightequivalent to (Q, W) [DWZ1].

For example, we calculate μ_{2}(Q, W) and μ_{2^{∗}}(μ_{2}(Q, W)) for the QP (Q, W) below. (For simplicity we denote a^{∗∗} and b^{∗∗} by a and b respectively.)

\begin{aligned}& (Q, W) = ( [figure: 1; a; 2; b; 3], 0) & ⟶^{μ_{2}} & ( [figure: 1; [ab]; 2^{∗}; a^{∗}; 3; b^{∗}], a^{∗}[ab]b^{∗}) \\ \underrightarrow{μ_{2^{∗}}} & ( [figure: 1; [ab]; a; 2; b; 3; [b^{∗}a^{∗}]], [ab][b^{∗}a^{∗}] + b[b^{∗}a^{∗}]a) & ⟶^{reduced} & ( [figure: 1; a; 2; b; 3], 0) \end{aligned}

1.3. Mutation of cluster-tilting objects. Let C be a Hom-finite triangulated K-category, where K is an algebraically closed field. We denote by [1] the shift functor in C, and Ext_{C}^{i}(A, B) = Hom_{C}(A, B[i]). Then C is said to be n-Calabi-Yau (n-CY for short) if there is a functorial isomorphism

D Hom_{C}(A, B) ≃ Ext_{C}^{n}(B, A)

for A, B in C and D = Hom_{K}( , K).

Let C be 2-CY. An object T in C is cluster-tilting if

add T = {X ∈ C | Ext_{C}^{1}(T, X) = 0}

(see [BMRRT, I2, KR1]). In this case the algebra End_{C}(T) is called a 2-CY-tilted algebra.

Let T = T_{1} ⊕ T_{2} ⊕ ⋯ ⊕ T_{n}, where the T_{i} are nonisomorphic indecomposable objects. For each k = 1, …, n there is a unique indecomposable object T_{k}^{∗} in C with T_{k}^{∗} ≄ T_{k} such that (T/T_{k}) ⊕ T_{k}^{∗} is a cluster-tilting object in C, and we write μ_{k}(T) = (T/T_{k}) ⊕ T_{k}^{∗}. Clearly we have μ_{k}(μ_{k}(T)) ≃ T. There are associated exchange triangles

T_{k}^{∗} →^{g} U_{k} →^{f} T_{k} → T_{k}^{∗}[1] and T_{k} →^{g′} U′_{k} →^{f′} T_{k}^{∗} → T_{k}[1]

where f and f′ are minimal right add(T/T_{k})-approximations, and g and g′ are minimal left add(T/T_{k})-approximations [BMRRT, IY].

We have an equivalence Hom_{C}(T, ): C /[add T [1]] ≃ mod End_{C}(T) by [BMR1, KR1]. Using this, we have an equivalence

\frac{mod End_{C}(T)}{[add S_{k}]} ≃ \frac{mod End_{C}(μ_{k}(T))}{[add S_{k}^{∗}]},

where S_{k} and S_{k}^{∗} are the simple modules associated with T_{k} and T_{k}^{∗}. Thus End_{C}(T) and End_{C}(μ_{k}(T)) are by definition nearly Morita equivalent.

We say that a 2-CY category C has no loops or 2-cycles if the quiver Q_{T} of End_{C}(T) has no loops or 2-cycles for any cluster-tilting object T in C (see [BIRS]). Under this assumption we have the following important connection with the Fomin-Zelevinsky quiver mutation. For a cluster-tilting object T we have

Q_{μ_{k}(T)} ≃ μ_{k}(Q_{T})

by [BMR2, BIRS]. Hence C has a cluster structure in the sense of [BIRS].

1.4. Cluster-tilted algebras. Cluster categories are by definition the orbit categories C_{Q} = D^{b}(KQ)/τ^{−1}[1], where Q is a finite connected acyclic quiver, D^{b}(KQ) is the bounded derived category of the finite dimensional (left) KQ-modules, and τ is the AR-translation in D^{b}(KQ), see [BMRRT]. These orbit categories are known to be triangulated by [K1], and are Hom-finite 2-CY [BMRRT]. The cluster-tilted algebras are by definition the 2-CY-tilted algebras coming from cluster categories [BMR1].

For cluster categories C_{Q}, the cluster-tilting objects are induced by tilting KQ′- modules over path algebras KQ′ derived equivalent to KQ. It is known [BMRRT] that given any two cluster-tilting objects in C_{Q}, then one can be obtained from the other by a finite sequence of mutations of cluster-tilting objects, and hence there is only one mutation class of cluster-tilted algebras by [BMRRT], based on [HU].

By [ABS] ([BR] in the Dynkin case) the quiver of End_{C_{Q}}(T) is obtained from the quiver of the tilted algebra End_{KQ′}(T) by adding an arrow in the opposite direction for each relation in a minimal set of relations. In particular, we have the following, which is also easily seen directly.

Lemma 1.1. Let T be a tilting module in mod KQ. Then End_{C_{Q}}(T) ≃ KQ′ for a quiver Q′ with no oriented cycles, if and only if End_{KQ}(T) ≃ KQ′.

If the quiver Q_{T} of a 2-CY-tilted algebra coming from a 2-CY category C is acyclic, then it follows from [KR1] that End_{C}(T) ≃ KQ_{T}. Moreover if C is a connected algebraic triangulated category (that is, the stable category of a Frobenius category), then C is equivalent to the cluster category C_{Q_{T}} [KR2].

1.5. Mutation of tilting modules over 3-Calabi-Yau algebras. Let R be a formal power series ring K[[x, y, z]] of three variables over an algebraically closed field K, and let Λ be an R-algebra which is a finitely generated R-module. We call Λ 3-Calabi-Yau (3-CY for short) if the bounded derived category D^{b}(f. l. Λ) is a 3-CY category [Boc, CR, G, IR]. It was shown in [IR] that Λ is 3-CY if and only if Λ is a Cohen-Macaulay R-module which is a symmetric R-algebra with gl. dim Λ = 3. Rickard proved that 3-CY algebras are closed under derived equivalences (see [IR]).

Let Λ be a 3-CY algebra. For a basic tilting Λ-module T of projective dimension at most one, we have another 3-CY algebra Γ = End_{Λ}(T). We take an indecomposable decomposition T = T_{1} ⊕ ⋯ ⊕ T_{n}. Let e_{k} ∈ Γ be the idempotent corresponding to T_{k} for k = 1, …, n. We let I_{k} = Γ(1 − e_{k})Γ. Let us recall the following result [IR, Th. 5.4, Th. 7.1].

Proposition 1.2. If Γ/I_{k} ∈ f. l. Γ, then there exists a minimal projective resolution

0 → Γe_{k} ⟶^{f_{2}} P_{1} ⟶^{f_{1}} P_{0} ⟶^{f_{0}} Γe_{k} → Γ/I_{k} → 0  (1)

with P_{0}, P_{1} ∈ add Γ(1 − e_{k}) satisfying the following conditions.

(a) Applying Hom_{Γ}(−, Γ) to (1), we have an exact sequence

0 → e_{k}Γ → Hom_{Γ}(P_{0}, Γ) → Hom_{Γ}(P_{1}, Γ) → e_{k}Γ → Γ/I_{k} → 0.  (2)

(b) Let T_{k}^{∗} = Ker(T ⊗_{Γ} f_{0}) and μ_{k}(T) = (T/T_{k}) ⊕ T_{k}^{∗}. Then μ_{k}(T) is a basic tilting Λ-module of projective dimension at most one and T_{k}^{∗} ≄ T_{k}.

(c) If Q_{T} and Q_{μ_{k}(T)} have no loops or 2-cycles, then Q_{μ_{k}(T)} ≃ μ_{k}(Q_{T}) holds.

Notice that we can not drop the assumption Γ/I_{k} ∈ f. l. Γ, which is automatically satisfied if Q_{T} has no loops at k. We need the following additional information for later application.

Proposition 1.3. If Γ/I_{k} ∈ f. l. Γ, then the following assertions hold.

(a) There exist exact sequences (called exchange sequences)

0 → T_{k}^{∗} ⟶^{g} U_{k} ⟶^{f} T_{k} and 0 → T_{k} ⟶^{g′} U′_{k} ⟶^{f′} T_{k}^{∗}

such that f and f′ are right add(T/T_{k})-approximations, and g and g′ are left add(T/T_{k})-approximations.

(b) In the category of finitely generated Λ-modules with projective dimension at most one, g and g′ are kernels of f and f′ respectively, and f and f′ are cokernels of g and g′ respectively.

(c) The complex T_{k} ⟶^{g′} U′_{k} ⟶^{f′g} U_{k} ⟶^{f} T_{k} induces exact sequences

0 → (T, T_{k}) ⟶^{g′} (T, U′_{k}) ⟶^{f′g} (T, U_{k}) ⟶^{f} J_{mod Λ}(T, T_{k}),

0 → (T_{k}, T) ⟶^{f} (U_{k}, T) ⟶^{f′g} (U′_{k}, T) ⟶^{g′} J_{mod Λ}(T_{k}, T),

where the right maps are surjective if Q_{T} has no loops at k.

Proof. (a)(c) We have Tor_{i}^{Γ}(T, Γ/I_{k}) = 0 for any i > 1. Applying T ⊗_{Γ} − to the sequence (1) and putting U_{k} = T ⊗_{Γ} P_{0} and U′_{k} = T ⊗_{Γ} P_{1}, we have the exact sequences in (a).

Since the functor T ⊗_{Γ} : add_{Γ}Γ → add_{Λ}T is an equivalence and the sequence (1) is exact, we have that f and f′ are right add(T/T_{k})-approximations, and that the upper sequence in (c) is exact if Q_{T} has no loops at k. Since the functor Hom_{Γ^{op}}( , T): add Γ_{Γ} → add_{Λ}T is a duality and the sequence (2) is exact, we have that g and g′ are left add(T/T_{k})-approximations, and that the lower sequence in (c) is exact if Q_{T} has no loops at k.

(b) Clearly g and g′ are kernels of f and f′ respectively. Since Λ is 3-CY, we have Ext_{Λ}^{i}(f. l. Λ, Λ) = 0 for 0 ≤ i ≤ 2 [IR]. Thus any Γ-module X with projective dimension at most one satisfies Ext_{Λ}^{i}(f. l. Λ, X) = 0 for i = 0, 1. On the other hand, since Γ/I_{k} ∈ f. l. Γ by our assumption, we have that Cok f = T ⊗_{Γ} (Γ/I_{k}) and Cok f′ = Tor_{1}^{Γ}(T, Γ/I_{k}) are in f. l. Λ. Thus we have Ext_{Λ}^{i}(Cok f, X) = 0 = Ext_{Λ}^{i}(Cok f′, X) for i = 0, 1. This implies that f and f′ are cokernels of g and g′ respectively. □

2. Cluster-tilted algebras are determined by their quivers

In this section we prove that cluster-tilted algebras are determined by their quivers. Alternative proofs using potentials will be given as a consequence of the main result in section 5.

First we prove the following.

Lemma 2.1. Let T be a cluster-tilting object in some cluster category C_{Q}, with Q_{T} an acyclic quiver. Then C_{Q_{T}} is equivalent to the cluster category C_{Q}.

Proof. Although the assertion follows from the main result in [KR2], we give an elementary proof here.

We know that T is induced by some tilting KQ′-module U, where KQ′ is derived equivalent to KQ. Since End_{C_{Q}}(T) ≃ KQ_{T}, we have by Lemma 1.1 that End_{KQ′}(U) ≃ End_{C_{Q′}} (T) ≃ End_{C_{Q}}(T) ≃ KQ_{T}, so that KQ′ and KQ_{T} are also derived equivalent. Hence KQ and KQ_{T} are derived equivalent, so that the cluster categories C_{Q} and C_{Q_{T}} are equivalent. □

Lemma 2.2. Let Q be an acyclic quiver and C_{Q} the associated cluster category. Let T be a cluster-tilting object in C_{Q} such that the quiver Q_{T} of End_{C}(T) is isomorphic to Q. Then there is an autoequivalence of triangulated categories F: C_{Q} → C_{Q} with F (T) ≃ KQ.

Proof. We have that T is induced by a tilting KQ′-module U, satisfying End_{KQ′}(U) ≃ KQ by Lemma 1.1. Then we have a commutative diagram

\xymatrix{U \ar@{|->}[d] & ∈ & D^{b}(KQ′) \ar[d] \ar[r]^{R Hom_{KQ′} (U, )} & D^{b}(KQ) \ar[d] & ∋ & KQ \ar@{|->}[d] \\ T & ∈ & _{C_{Q}} \ar@{-->}[r]^{F} & _{C_{Q}} & ∋ & KQ}

where R Hom_{KQ′}(U, ): D^{b}(KQ′) → D^{b}(KQ) is an equivalence of triangulated categories since U is a tilting module [H]. Hence R Hom_{KQ′}(U, ) commutes with τ and [1], so that there is an equivalence F: C_{Q} → C_{Q} of triangulated categories, with F (T) ≃ KQ. □

Using this we get the following

Theorem 2.3. Let T_{1} and T_{2} be cluster-tilting objects in the cluster categories C_{Q_{1}} and C_{Q_{2}} respectively, and assume that the quivers Q_{T_{1}} and Q_{T_{2}} are isomorphic. Then there is an equivalence of triangulated categories F: C_{Q_{1}} → C_{Q_{2}} such that F (T_{1}) ≃ T_{2}. In particular End_{C_{Q_{1}}} (T_{1}) ≃ End_{C_{Q_{2}}} (T_{2}) holds.

Proof. We have a sequence of mutations μ = μ_{k_{ℓ}} ∘ ⋯ ∘ μ_{k_{1}} such that μ(KQ_{1}) ≃ T_{1} in C_{Q_{1}}. Let T be a cluster-tilting object in C_{Q_{2}} such that μ(T) ≃ T_{2}. Since Q_{T_{1}} ≃ Q_{T_{2}}, we have Q_{1} ≃ Q_{T}. Hence by Lemmas 2.1 and 2.2 there is an equivalence of triangulated categories F: C_{Q_{1}} → C_{Q_{2}} such that F (KQ_{1}) ≃ T. Since clustertilting mutations commute with equivalences of triangulated categoires, we have

F (T_{1}) ≃ F (μ(KQ_{1})) ≃ μ(F (KQ_{1})) ≃ μ(T) ≃ T_{2},

and consequently End_{C_{Q_{1}}} (T_{1}) ≃ End_{C_{Q_{2}}} (T_{2}). □

We can strengthen the above theorem as follows.

Corollary 2.4. Let T_{1} be a cluster-tilting object in a cluster category and T_{2} a cluster-tilting object in an algebraic 2-CY triangulated category C without loops or 2-cycles. If Q_{T_{1}} ≃ Q_{T_{2}}, then End_{C}(T_{1}) ≃ End_{C}(T_{2}).

Proof. By the same argument as in the proof of Theorem 2.3, there exists a sequence μ of mutations such that μ(T) ≃ T_{2} and Q_{T} is acyclic. Since C is assumed to be algebraic, it follows from [KR2] that C is equivalent to C_{Q_{T}}, so the result follows from Theorem 2.3. □

3. Preliminaries on presentation of algebras

In this section we consider presentations of algebras, which are important for the rest of this paper.

Let K as before be an algebraically closed field and Γ a K-algebra with Jacobson radical J_{Γ}. We regard Γ as a topological algebra via the J_{Γ}-adic topology with a basic system {J_{Γ}^{i}}_{i≥0} of open neighborhoods of 0. Thus the closure of a subset S of Γ is given by

\overline{S} = ⋂_{ℓ≥0}(S + J_{Γ}^{ℓ}).  (3)

We assume that Γ satisfies the following conditions:

(A1) Γ/J_{Γ} ≃ K^{n} for some n > 0 and dim_{K}(J_{Γ}/J_{Γ}^{2}) < ∞,

(A2) the J_{Γ}-adic topology on Γ is complete and separated, i.e. Γ ≃ \underleftarrow{lim}_{ℓ≥0} Γ/J_{Γ}^{ℓ}.

For example, finite dimensional K-algebras, complete path algebras K̂Q of finite quivers Q, and algebras over R = K[[x_{1}, …, x_{d}]] which are finitely generated Rmodules satisfy (A1) and (A2). If ϕ : Λ → Γ is a homomorphism of K-algebras satisfying (A1), then J_{Λ} ⊂ ϕ^{−1}(J_{Γ}) holds. In particular ϕ is continuous.

Let Q be a finite quiver. It is convenient to introduce the following concepts. For a ∈ Q_{1}, define a right derivative ∂_{a}^{r}: JK̂Q → K̂Q and a left derivative ∂_{a}^{l}: JK̂Q →

K̂Q by

∂_{a}^{r}(a_{1}a_{2} ⋯ a_{m−1}a_{m}) = \{ \begin{matrix}a_{1}a_{2} ⋯ a_{m−1} & if a_{m} = a, \\ 0 & otherwise, \end{matrix}

∂_{a}^{l}(a_{1}a_{2} ⋯ a_{m−1}a_{m}) = \{ \begin{matrix}a_{2} ⋯ a_{m−1}a_{m} & if a_{1} = a, \\ 0 & otherwise, \end{matrix}

and extend to JK̂Q linearly and continuously. Then ∂_{a}^{r} is a homomorphism of K̂Q-

modules and ∂_{a}^{l} is a homomorphism of K̂Q^{op}-modules. Clearly for any p ∈ JK̂Q we have

p = ∑_{a∈Q_{1}}(∂_{a}^{r}p)a = ∑_{a∈Q_{1}} a(∂_{a}^{l}p).

For simplicity we call p ∈ K̂Q basic if p is a formal linear sum of paths in Q with a common start i and a common end j. In this case we write s(p) = i and e(p) = j. Clearly any ideal of K̂Q is generated by a set of basic elements.

We start with the following basic fact.

Proposition 3.1. Let Γ be a K-algebra satisfying (A1) and (A2), and let Q be a finite quiver. Assume that we have a K-algebra homomorphism ϕ_{0} : KQ_{0} → Γ and a homomorphism ϕ_{1} : KQ_{1} → J_{Γ} of (KQ_{0}, KQ_{0})-modules. Then the following assertions hold.

(a) ϕ_{0} and ϕ_{1} extend uniquely to a K-algebra homomorphism ϕ: K̂Q → Γ. Moreover Ker ϕ is a closed ideal of K̂Q.

(b) The following conditions are equivalent.

(i) ϕ: K̂Q → Γ is surjective.

(ii) ϕ_{0} and ϕ_{1} induce surjections KQ_{0} → Γ/J_{Γ} and KQ_{1} → J_{Γ}/J_{Γ}^{2}. (iii) ϕ_{0} induces a surjection KQ_{0} → Γ/J_{Γ}, and for any i ∈ Q_{0} we have an

exact sequence

⨁_{a∈Q_{1}, e(a)=i} Γ(ϕs(a)) ⟶^{(ϕa)_{a}} J_{Γ}(ϕi) → 0.

(iv) ϕ_{0} induces a surjection KQ_{0} → Γ/J_{Γ}, and for any i ∈ Q_{0} we have an exact sequence

⨁_{a∈Q_{1}, s(a)=i}(ϕs(a))Γ ⟶^{(ϕa)_{a}} (ϕi)J_{Γ} → 0.

Proof. (a) The first assertion is clear. We observed that ϕ is continuous. Thus {0} is a closed subset of Γ and its inverse image Ker ϕ is a closed subset of K̂Q.

(b) (ii)⇒(i) ϕ induces a surjection ϕ_{ℓ}: KQ_{ℓ} → J_{Γ}^{ℓ}/J_{Γ}^{ℓ+1} for any ℓ. For any x ∈ Γ, there exists x_{ℓ} ∈ J_{Γ}^{ℓ} and p_{ℓ} ∈ KQ_{ℓ} such that x_{ℓ+1} = x_{ℓ} − ϕ_{ℓ}(p_{ℓ}). Then we have ϕ(∑_{ℓ≥0} p_{ℓ}) = x. (i)⇒(iii) It is enough to show that ϕ(p) is in the image of (ϕa)_{a} for any basic element p ∈ JK̂Q with e(p) = i. This follows from the equality p = ∑_{a∈Q_{1}}(∂_{a}^{r}p)a since this implies ϕp = (ϕ∂_{a}^{r}p)_{a} · (ϕa)_{a}.

(iii)⇒(ii) The sequence in (iii) induces a surjection ⨁_{a∈Q_{1}} Γ/J_{Γ} ⟶^{(ϕa)_{a}} J_{Γ}/J_{Γ}^{2}. The rest follows similarly. □

We shall need the following easy fact.

Lemma 3.2. Let Q be a finite quiver and S a finite subset of JK̂Q. For the ideal

I = ⟨S⟩ of K̂Q, we have I = ∑_{v∈S}(K̂Q)v + I · Q_{1} and \overline{I} = ∑_{v∈S}(K̂Q)v + \overline{I} · Q_{1}.

Proof. Since I is generated by S, we have the first equality.

Let p ∈ \overline{I}. For any ℓ > 0 we can write p = r_{ℓ} + r′_{ℓ} for some r_{ℓ} ∈ I and r′_{ℓ} ∈ J_{Γ}^{ℓ}. Then p_{ℓ} = r_{ℓ+1}−r_{ℓ} = r′_{ℓ}−r′_{ℓ+1} ∈ I ∩J_{Γ}^{ℓ} for ℓ > 0. Letting p_{0} = r_{1}, we have p_{0}+p_{1}+ ⋯ + p_{ℓ} = p − r′_{ℓ+1}. Thus p = ∑_{ℓ≥0} p_{ℓ}. We write p_{ℓ} = ∑_{v∈S} p′_{ℓ,v}v + ∑_{a∈Q_{1}} p′′_{ℓ,a}a for p′_{ℓ,v} ∈ K̂Q and p′′_{ℓ,a} ∈ I. Then p = ∑_{v∈S}(∑_{ℓ≥0} p′_{ℓ,v})v + ∑_{a∈Q_{1}}(∑_{ℓ≥0} p′′_{ℓ,a})a holds. Thus we have the second equality. □

We now give a result on description of relations.

Proposition 3.3. Assume that the conditions in Proposition 3.1(b) are satisfied. For a finite set S of basic elements in JK̂Q, the following conditions are equivalent.

(a) Ker ϕ = \overline{I} holds for the ideal I = ⟨S⟩ of K̂Q.

(b) The following sequence is exact for any i ∈ Q_{0}.

⨁_{v∈S, e(v)=i} Γ(ϕs(v)) ⟶^{(ϕ∂_{a}^{r}v)_{v,a}} ⨁_{a∈Q_{1}, e(a)=i} Γ(ϕs(a)) ⟶^{(ϕa)_{a}} J_{Γ}(ϕi) → 0.  (4)

(c) The following sequence is exact for any i ∈ Q_{0}.

⨁_{v∈S, s(v)=i}(ϕe(v))Γ ⟶^{(ϕ∂_{a}^{l}v)_{v,a}} ⨁_{a∈Q_{1}, s(a)=i}(ϕe(a))Γ ⟶^{(ϕa)_{a}} (ϕi)J_{Γ} → 0.

Proof. (a)⇒(b) Since

(ϕ∂_{a}^{r}v)_{a} · (ϕa)_{a} = ϕ( ∑_{a∈Q_{1}, e(a)=i}(∂_{a}^{r}v)a) = ϕv = 0,

the sequence is a complex. Now we assume that (p_{a})_{a} ∈ ⨁_{{a∈Q_{1} \atop e(a)=i}} K̂Q(s(a)) satisfies

(ϕp_{a})_{a} · (ϕa)_{a} = 0. Since

∑_{a∈Q_{1}, e(a)=i} p_{a}a ∈ Ker ϕ = \overline{I},

there exists q_{v} ∈ K̂Q by Lemma 3.2 such that

∑_{a∈Q_{1}, e(a)=i} p_{a}a − ∑_{v∈S} q_{v}v ∈ \overline{I} · Q_{1}.

Applying ∂_{a}^{r} on both sides, we have

p_{a} − ∑_{v∈S} q_{v}∂_{a}^{r}v ∈ \overline{I} = Ker ϕ.

Thus (ϕq_{v})_{v} ∈ ⨁_{v∈S, e(v)=i} Γ(s(v)) satisfies

(ϕp_{a})_{a} = (ϕq_{v})_{v} · (ϕ∂_{a}^{r}v)_{v,a}.

(b)⇒(a) We shall show that Ker ϕ = \overline{I}. Take any v ∈ S with e(v) = i. Since (4) is exact, we have

ϕv = ϕ( ∑_{a∈Q_{1}, e(a)=i}(∂_{a}^{r}v)a) = (ϕ∂_{a}^{r}v)_{a} · (ϕa)_{a} = 0.

This implies v ∈ Ker ϕ and I ⊂ Ker ϕ. Since Ker ϕ is a closed ideal by Proposition 3.1, we have \overline{I} ⊂ Ker ϕ.

To prove Ker ϕ ⊂ \overline{I}, we will first show that, for any p ∈ Ker ϕ, there exists p′ ∈ I such that p − p′ ∈ (Ker ϕ) · Q_{1}. Without loss of generality, we can assume that p is basic with e(p) = i. Since

(ϕ∂_{a}^{r}p)_{a} · (ϕa)_{a} = ϕp = 0

holds, we have that (ϕ∂_{a}^{r}p)_{a} factors through the left map (ϕ∂_{a}^{r}v)_{v,a} in (4). Thus there exists q_{v} ∈ K̂Q such that (ϕ∂_{a}^{r}p)_{a} = (ϕq_{v})_{v} · (ϕ∂_{a}^{r}v)_{v,a}. Then

∂_{a}^{r}p − ∑_{v∈S, e(v)=i} q_{v}∂_{a}^{r}v ∈ Ker ϕ.

Hence

p − ∑_{v∈S, e(v)=i} q_{v}v = ∑_{a∈Q_{1}, e(a)=i}(∂_{a}^{r}p − ∑_{v∈S, e(v)=i} q_{v}∂_{a}^{r}v)a ∈ (Ker ϕ) · Q_{1}.

It follows that p′ = ∑_{v∈S, e(v)=i} q_{v}v ∈ I satisfies the desired condition. Consequently we have Ker ϕ ⊂ I + (Ker ϕ) · Q_{1}. This implies

Ker ϕ ⊂ I + (I + (Ker ϕ) · Q_{1}) · Q_{1} = I + (Ker ϕ) · Q_{1}^{2} ⊂ ⋯ ⊂ I + (Ker ϕ) · Q_{1}^{ℓ}

for any ℓ. By (3), we have Ker ϕ ⊂ \overline{I}.

(a)⇔(c) can be shown dually. □

We give a relationship between the dimension of Ext^{2}-spaces and minimal sets of generators for relation ideals in complete path algebras (cf. [Bon, Y]). Note that the corresponding result is not true in general if we deal with ordinary path algebras KQ instead of K̂Q. See [BIKR, section 7] for an example. Note however that if KQ/⟨S⟩ is finite dimensional, then the corresponding result is true.

Proposition 3.4. Let Q be a finite quiver, J = JK̂Q, I a closed ideal of K̂Q

contained in J^{2}, and Γ = K̂Q/I. Let S be a finite set of basic elements in I.

(a) S spans the K-vector space I/(IJ + JI) if and only if I = \overline{⟨S⟩}. Moreover S is a basis of the K-vector space I/(IJ + JI) if and only if S is a minimal set satisfying I = \overline{⟨S⟩}.

(b) If S is a minimal set satisfying I = \overline{⟨S⟩}, then we have

dim_{K} Ext_{Γ}^{2}(S_{i}, S_{j}) = #(iK̂Qj) ∩ S = dim_{K} i(I/(IJ + JI))j

for any simple Γ-modules S_{i} and S_{j} associated with i, j ∈ Q_{0}.

Proof. (a) Assume first that S spans I/(IJ + JI), and let I′ = \overline{⟨S⟩}. Then we have I = I′ + IJ + JI. This implies

I = I′+(I′+IJ+JI)J+J(I′+IJ+JI) = I′+IJ^{2}+JIJ+J^{2}I = ⋯ = I′+ ∑_{0≤i≤ℓ} J^{i}IJ^{ℓ−i}

for any ℓ > 0. Thus we have I ⊂ ⋂_{ℓ>0}(I′ + J^{ℓ}) = I′ and hence I = I′ = \overline{⟨S⟩}.

Next we assume that I = \overline{⟨S⟩}. By Lemma 3.2, we have I = ⟨S⟩ + IJ. Thus I = ⟨S⟩ + IJ + JI = KS + IJ + JI holds.

The second equivalence is an immediate consequence of the first assertion.

(b) The minimality of S implies that the projective resolution in Proposition 3.3(b) is minimal. Thus we have the left equality. The right equality follows from (a) since (iK̂Qj) ∩ S is a basis of the K-vector space i(I/(IJ + JI))j. □

Now let C be a K-linear category satisfying the following conditions:

(C1) C is Krull-Schmidt, i.e. any object in C is isomorphic to a finite direct sum

of objects whose endomorphism rings are local. (C2) End_{C}(X) satisfies (A1) and (A2) for any basic object X ∈ C.

For a finite quiver Q and a field K, we denote by K̂Q also the category of finitely generated projective K̂Q-modules. We have the following observation from Proposition 3.1(a).

Lemma 3.5. For a category C satisfying (C1) and (C2) and a quiver Q, assume that an object Φ_{0}i ∈ C for any i ∈ Q_{0} and a morphism Φ_{1}a ∈ C(Φs(a), Φe(a)) for any a ∈ Q_{1} are given. Then Φ_{0} and Φ_{1} extend uniquely to a K-linear functor Φ: K̂Q → C.

Restating Proposition 3.3, we have the following result.

Proposition 3.6. Let C be a category satisfying (C1) and (C2), and let Q be a finite quiver. Let Φ: K̂Q → C be a K-linear functor, and let T = ⨁_{i∈Q_{0}} Φi. For a finite set S of basic elements in JK̂Q, the following conditions are equivalent.

(a) Φ induces a surjection ϕ: K̂Q → End_{C}(T) with Ker ϕ = \overline{⟨S⟩}.

(b) For any i ∈ Q_{0}, we have a complex

⨁_{v∈S, e(v)=i} Φs(v) ⟶^{(Φ∂_{a}^{r}v)_{v,a}} ⨁_{a∈Q_{1}, e(a)=i} Φs(a) ⟶^{(Φa)_{a}} Φi

in add T which induces an exact sequence

C(T, ⨁_{v∈S, e(v)=i} Φs(v)) ⟶^{(Φ∂_{a}^{r}v)_{v,a}} C(T, ⨁_{a∈Q_{1}, e(a)=i} Φs(a)) ⟶^{(Φa)_{a}} J_{C}(T, Φi) → 0.

(c) For any i ∈ Q_{0}, we have a complex

Φi ⟶^{(Φa)_{a}} ⨁_{a∈Q_{1}, s(a)=i} Φe(a) ⟶^{(Φ∂_{a}^{l}v)_{a,v}} ⨁_{v∈S, s(v)=i} Φe(v)

in add T which induces an exact sequence

C( ⨁_{v∈S, s(v)=i} Φe(v), T) ⟶^{(Φ∂_{a}^{l}v)_{a,v}} C( ⨁_{a∈Q_{1}, s(a)=i} Φe(a), T) ⟶^{(Φa)_{a}} J_{C}(Φi, T) → 0.

4. Jacobian algebras and weak 2-almost split sequences

In this section we study properties of Jacobian algebras of QP’s. We also investigate a variation of 2-almost split sequences and AR 4-angles discussed in [I2, IY], which we call weak 2-almost split sequences. We show that there is a close relationship between Jacobian algebras and weak 2-almost split sequences, which will be used to prove the connection between mutation of QP’s and of cluster-tilting objects in the next section.

Let (Q, W) be a QP. For a pair of arrows a and b, we define ∂_{(a,b)}W by

∂_{(a,b)}(a_{1}a_{2} ⋯ a_{m}) = ∑_{a_{i}=a, a_{i+1}=b} a_{i+2} ⋯ a_{m}a_{1} ⋯ a_{i−1}

for any cycle a_{1} ⋯ a_{m} in W and extend linearly and continuously. Clearly e(a) ≠ s(b) implies ∂_{(a,b)}W = 0.

The following easy observation is useful.

Lemma 4.1. (a) ∑_{a∈Q_{1}}(∂_{(a,b)}W)a = ∂_{b}W = ∑_{c∈Q_{1}} c(∂_{(b,c)}W) for any b ∈

Q_{1}.

(b) ∂_{(a,b)}W = ∂_{a}^{r}(∂_{b}W) = ∂_{b}^{l}(∂_{a}W).

We have the following property of Jacobian algebras of QP’s.

Proposition 4.2. Let (Q, W) be a QP and Γ = P(Q, W). Let ϕ : K̂Q → Γ be a natural surjection. Then there exist complexes

Γ(ϕi) ⟶^{(ϕb)_{b}} ⨁_{\begin{matrix}b∈Q_{1} \\ s(b)=i \end{matrix}} Γ(ϕe(b)) ⟶^{(ϕ∂_{(a,b)}W)_{b,a}} ⨁_{\begin{matrix}a∈Q_{1} \\ e(a)=i \end{matrix}} Γ(ϕs(a)) ⟶^{(ϕa)_{a}} J_{Γ}(ϕi) → 0,

(ϕi)Γ ⟶^{(ϕa)_{a}} ⨁_{\begin{matrix}a∈Q_{1} \\ e(a)=i \end{matrix}} (ϕs(a))Γ ⟶^{(ϕ∂_{(a,b)}W)_{b,a}} ⨁_{\begin{matrix}b∈Q_{1} \\ s(b)=i \end{matrix}} (ϕe(b))Γ ⟶^{(ϕb)_{b}} (ϕi)J_{Γ} → 0

which are exact except at the second left terms.

Proof. Since ∂_{(a,b)}W = ∂_{a}^{r}(∂_{b}W) = ∂_{b}^{l}(∂_{a}W) holds by Lemma 4.1, the assertion follows immediately from Proposition 3.3(a)⇒(b)(c). □

As an immediate application, we have the following consequence.

Proposition 4.3. Let (Q, W) be a QP and Γ = P(Q, W).

(a) Ext_{Γ}^{2}(S, Γ) = 0 = Ext_{Γ^{op}}^{2}(S′, Γ) holds for any simple Γ-module S and any simple Γ^{op}-module S′.

(b) If Γ is a finite dimensional K-algebra, then id_{Γ}Γ ≤ 1 and idΓ_{Γ} ≤ 1.

Proof. (b) follows immediately from (a).

(a) Let S be a simple Γ-module. By Proposition 4.2, there exists a complex

P ⟶^{f_{2}} P_{1} ⟶^{f_{1}} P_{0} ⟶^{f_{0}} P

of projective Γ-modules, which is exact at P_{0}, such that Cok f_{0} = S and

Hom_{Γ}(P_{0}, Γ) ⟶^{f_{1}} Hom_{Γ}(P_{1}, Γ) ⟶^{f_{2}} Hom_{Γ}(P, Γ)  (5)

is exact. We can take a morphism f′_{2} : P′ → P_{1} such that

P ⊕ P′ \underrightarrow{\binom{f_{2}}{f′_{2}}} P_{1} ⟶^{f_{1}} P_{0} ⟶^{f_{0}} P → S → 0

gives a projective resolution of S. Since (5) is exact, we have that

Hom_{Γ}(P_{0}, Γ) ⟶^{f_{1}} Hom_{Γ}(P_{1}, Γ) \underrightarrow{\binom{f_{2}}{f′_{2}}} Hom_{Γ}(P ⊕ P′, Γ)

is exact. Consequently we have the desired properties. □

To study the property of Jacobian algebras of QP’s categorically, we introduce the following concept.

Definition 4.4. Let C be a category satisfying (C1) and (C2), and T ∈ C an object. We call a complex

X ⟶^{f_{2}} U_{1} ⟶^{f_{1}} U_{0} ⟶^{f_{0}} X

in add T a weak 2-almost split sequence if the following sequences are exact

C(T, U_{1}) ⟶^{f_{1}} C(T, U_{0}) ⟶^{f_{0}} J_{C}(T, X) → 0,

C(U_{0}, T) ⟶^{f_{1}} C(U_{1}, T) ⟶^{f_{2}} J_{C}(X, T) → 0.

In other words, f_{0} and f_{2} are right almost split and left almost split in add T respectively, and f_{1} is a pseudo-kernel of f_{0} and a pseudo-cokernel of f_{2} in add T. Note that we do not assume that f_{2} (respectively, f_{0}) is a pseudo-kernel (respectively, pseudo-cokernel) of f_{1}.

Remark The definition given here is more general than 2-almost split sequences and AR 4-angles in [I2, IY]. One difference is that we assume neither right minimality of f_{0} nor left minimality of f_{2}. This point is necessary to deal with Jacobian algebras of non-reduced QP’s. Another difference is that our complex is not assumed to be a gluing of two exact sequences (respectively, triangles). This point simplifies our argument below. But note that we do not have uniqueness of weak 2-almost split sequences.

The following observation shows that weak 2-almost split sequences appear naturally in cluster-tilting theory.

Proposition 4.5. Let C and T = T_{1} ⊕ ⋯ ⊕ T_{n} be one of the following. (a) C is a 2-CY triangulated category and T is a basic cluster-tilting object.

(b) C = mod Λ for a 3-CY algebra Λ and T is a basic tilting Λ-module of

projective dimension at most one (section 1.5). Assume that the quiver of End_{C}(T) has no loops at k. Gluing exchange trian-

gles/sequences T_{k}^{∗} ⟶^{g} U_{k} ⟶^{f} T_{k} and T_{k} ⟶^{g′} U′_{k} ⟶^{f′} T_{k}^{∗} given in sections 1.3 and 1.5,

we have a weak 2-almost split sequence T_{k} ⟶^{g′} U′_{k} ⟶^{f′g} U_{k} ⟶^{f} T_{k} in add T.

Proof. The case (a) is shown in [IY], and the case (b) is shown in Proposition 1.3(c). □

Now we give a key observation which gives a relationship between weak 2-almost split sequences and Jacobian algebras of QP’s.

Theorem 4.6. Let C be a category satisfying (C1) and (C2), and let (Q, W) be a QP. Let Φ: K̂Q → C be a K-linear functor, and T = ⨁_{i∈Q_{0}} Φi. Then the following conditions are equivalent.

(a) Φ induces an isomorphism P(Q, W) ≃ End_{C}(T).

(b) Any vertex i in Q gives rise to the following weak 2-almost split sequence in C

Φi ⟶^{(Φb)_{b}} ⨁_{b∈Q_{1}, s(b)=i} Φe(b) ⟶^{(Φ∂_{(a,b)}W)_{b,a}} ⨁_{a∈Q_{1}, e(a)=i} Φs(a) ⟶^{(Φa)_{a}} Φi.

Proof. We shall apply Proposition 3.6 to S = {∂_{a}W | a ∈ Q_{1}}.

(b)⇒(a) By Lemma 4.1(b), we have ∂_{(a,b)}W = ∂_{a}^{r}(∂_{b}W). Looking at the complex

⨁_{b∈Q_{1}, s(b)=i} Φe(b) ⟶^{(Φ∂_{(a,b)}W)_{b,a}} ⨁_{a∈Q_{1}, e(a)=i} Φs(a) ⟶^{(Φa)_{a}} Φi,

we have the assertion by Proposition 3.6(b)⇒(a).

(a)⇒(b) By Proposition 3.6(a)⇒(b)(c), we have exact sequences

C(T, ⨁_{b∈Q_{1}, s(b)=i} Φe(b)) ⟶^{(Φ∂_{a}^{r}(∂_{b}W))_{b,a}} C(T, ⨁_{a∈Q_{1}, e(a)=i} Φs(a)) ⟶^{(a)_{a}} J_{C}(T, Φi) → 0,

C( ⨁_{a∈Q_{1}, e(a)=i} Φs(a), T) ⟶^{(Φ∂_{b}^{l}(∂_{a}W))_{b,a}} C( ⨁_{b∈Q_{1}, s(b)=i} Φe(b), T) ⟶^{(b)_{b}} J_{C}(Φi, T) → 0.

Since ∂_{(a,b)}W = ∂_{a}^{r}(∂_{b}W) = ∂_{b}^{l}(∂_{a}W) holds by Lemma 4.1, we have the assertion. □

We end this section with the following sufficient condition for an algebra to be a Jacobian algebra of a QP, which we will apply in section 6. We call a cycle a_{1}a_{2} ⋯ a_{m} in a quiver Q full if the vertices s(a_{1}), ⋯, s(a_{m}) are distinct, and if there is an arrow b in Q with s(b) and e(b) in the cycle, then there is some arrow a_{i} with s(b) = s(a_{i}) and e(b) = e(a_{i}).

Proposition 4.7. Let Γ be an algebra satisfying (A1) and (A2), and let (Q, W) be a QP satisfying the following conditions.

(i) There exists a surjective K-algebra homomorphism ϕ: P(Q, W) → Γ such that Ker ϕ is the closure of a finitely generated ideal.

(ii) Every cycle in W is full.

(iii) The elements ∂_{a}W for all arrows a ∈ Q_{1} contained in cycles of W are linearly independent over K.

(iv) dim_{K} Ext_{Γ}^{2}(S, S′) ≤ dim_{K} Ext_{Γ}^{1}(S′, S) for any simple Γ-modules S, S′. Then ϕ: P(Q, W) → Γ is an isomorphism.

Proof. Let I be the kernel of the ring homomorphism K̂Q → Γ induced by ϕ in (i). Then ∂_{a}W ∈ I for any a ∈ Q_{1}.

Let J = JK̂Q. We show that the image of ∂_{a}W for all arrows a ∈ Q_{1} contained in cycles of W in I/(IJ + JI) are linearly independent over K. We assume z = ∑_{i=1}^{m} c_{i}∂_{a_{i}}W is in IJ +JI for some c_{1}, …, c_{m} in K and different arrows a_{1}, …, a_{m} contained in cycles of W. Fix a minimal set S′ of basic elements satisfying I = \overline{⟨S′⟩}, which is finite by the assumption (i). We have an equality z = ∑_{j} p_{j}q_{j}r_{j} (possibly an infinite sum) with q_{j} ∈ S′ and p_{j}, r_{j} ∈ K̂Q such that for each j one of p_{j} and r_{j} belongs to J. Assume z ≠ 0. Then there exists a cycle C in W such that ∂_{a_{i}}C appears in z for some i = 1, …, m. We have ∂_{a_{i}}C = pqr for paths p, q and r appearing in p_{j}, q_{j} and r_{j} respectively for some j, and one of p and r is a nontrivial path. Let S and S′ be simple Γ-modules corresponding to s(q) and e(q) respectively. Then we have Ext_{Γ}^{2}(S, S′) ≠ 0 by applying Proposition 3.4(b) to S′. By (iv), we have Ext_{Γ}^{1}(S′, S) ≠ 0, so there is an arrow in Q from e(q) to s(q). Since p or r is a nontrivial path, it follows that the cycle C is not full, a contradiction to (ii). Consequently we have z = 0. By (iii), we have that all c_{i} are 0.

For each arrow i →^{a} j lying on a cycle in W, we have a relation of the form ∂_{a}W, hence dim_{K} Ext_{Γ}^{1}(S_{i}, S_{j}) such relations, where S_{i} and S_{j} denote the simple modules at the vertices i and j. We then have

dim_{K} Ext_{Γ}^{1}(S_{i}, S_{j}) ≤ dim_{K} i(I/(IJ + JI))j

= dim_{K} Ext_{Γ}^{2}(S_{j}, S_{i}) ≤ dim_{K} Ext_{Γ}^{1}(S_{i}, S_{j}),

where the first inequality follows by the above, the equality follows by Proposition 3.4(b), and the last inequality holds by (iv). Hence we have I = \overline{⟨∂_{a}W | a ∈ Q_{1}⟩} by Proposition 3.4(a). It follows that ϕ is an isomorphism. □

5. Mutation of quivers with potentials and cluster-tilting objects

In this section we use the results from section 4 to show that there is a close connection between mutation of quivers with potentials on one hand and mutation of cluster-tilting objects in 2-CY triangulated categories in section 1.3, or mutation of tilting modules over 3-CY algebras in section 1.5, on the other hand. More specifically we will prove the following main results in this section.

Theorem 5.1. Let C be a 2-CY triangulated category with a basic cluster-tilting object T. If End_{C}(T) ≃ P(Q, W) for a QP (Q, W) and no 2-cycles start in the vertex k of Q, then End_{C}(μ_{k}(T)) ≃ P(μ_{k}(Q, W)).

Theorem 5.2. Let Λ be a 3-CY algebra with a basic tilting module T of projective dimension at most one. If End_{Λ}(T) ≃ P(Q, W) for a QP (Q, W) and no 2-cycles start in the vertex k of Q, then End_{Λ}(μ_{k}(T)) ≃ P(μ_{k}(Q, W)).

To prove these theorems at the same time, we start with the following general setup. Let C be an additive category satisfying (C1) and (C2). Let T = T_{1} ⊕ ⋯ ⊕ T_{n} ∈ C be a basic object and k = 1, …, n. We assume that there exists an indecomposable object T_{k}^{∗} ∈ C which does not belong to add T and that the following conditions for μ_{k}(T) = (T/T_{k}) ⊕ T_{k}^{∗} are satisfied.

(1) For any i = 1, …, n, there exists a weak 2-almost split sequence in add T:

T_{i} ⟶^{f_{i2}} U_{i1} ⟶^{f_{i1}} U_{i0} ⟶^{f_{i0}} T_{i}.

(2k) There exist complexes

T_{k} ⟶^{f_{k2}} U_{k1} ⟶^{g_{k0}} T_{k}^{∗} and T_{k}^{∗} ⟶^{g_{k2}} U_{k0} ⟶^{f_{k0}} T_{k}

in C such that U_{k0}, U_{k1} ∈ add(T/T_{k}), f_{k1} = g_{k0}g_{k2} and

T_{k}^{∗} ⟶^{g_{k2}} U_{k0} ⟶^{f_{k0}f_{k2}} U_{k1} ⟶^{g_{k0}} T_{k}^{∗}

is a weak 2-almost split sequence in add μ_{k}(T). Moreover the following sequences are exact:

(T_{k}^{∗}, T_{k}^{∗}) ⟶^{g_{k2}} (T_{k}^{∗}, U_{k0}) ⟶^{f_{k0}} (T_{k}^{∗}, T_{k}),

(T_{k}^{∗}, T_{k}^{∗}) ⟶^{g_{k0}} (U_{k1}, T_{k}^{∗}) ⟶^{f_{k2}} (T_{k}, T_{k}^{∗}).

(3k) For any i ≠ k, we have that T_{k} ∉ (add U_{i1}) ∩ (add U_{i0}) and the following sequences are exact:

(T_{k}^{∗}, U_{i1}) ⟶^{f_{i1}} (T_{k}^{∗}, U_{i0}) ⟶^{f_{i0}} (T_{k}^{∗}, T_{i}),

(U_{i0}, T_{k}^{∗}) ⟶^{f_{i1}} (U_{i1}, T_{k}^{∗}) ⟶^{f_{i2}} (T_{i}, T_{k}^{∗}).

Our key result is the following.

Theorem 5.3. Assume that the conditions (1), (2k) and (3k) are satisfied. If End_{C}(T) ≃ P(Q, W) for a QP (Q, W), then End_{C}(μ_{k}(T)) ≃ P(μ_{k}(Q, W)).

Before proving Theorem 5.3, we notice that our main Theorems 5.1 and 5.2 follow immediately from Theorem 5.3 and the following observation.

Lemma 5.4. Let T be as in Theorem 5.1 or 5.2. Then the conditions (1), (2k) and (3k) are satisfied.

Proof. Take exchange triangles/sequences T_{i} ⟶^{f_{i2}} U_{i1} ⟶^{g_{i0}} T_{i}^{∗} and T_{i}^{∗} ⟶^{g_{i2}} U_{i0} ⟶^{f_{i0}} T_{i} given in sections 1.3 and 1.5.

(1)(2k) By Proposition 4.5, we have weak 2-almost split sequences T_{i} ⟶^{f_{i2}} U_{i1} ⟶^{g_{i0}g_{i2}}

U_{i0} ⟶^{f_{i0}} T_{i} in add T and T_{i}^{∗} ⟶^{g_{i2}} U_{i0} ⟶^{f_{i0}f_{i2}} U_{i1} ⟶^{g_{i0}} T_{i}^{∗} in add μ_{i}(T). The sequences

(T_{k}^{∗}, T_{k}^{∗}) ⟶^{g_{k2}} (T_{k}^{∗}, U_{k0}) ⟶^{f_{k0}} (T_{k}^{∗}, T_{k}) and (T_{k}^{∗}, T_{k}^{∗}) ⟶^{g_{i0}} (U_{k1}, T_{k}^{∗}) ⟶^{f_{k2}} (T_{k}, T_{k}^{∗}) are

clearly exact (use Proposition 1.3(b) for Theorem 5.2).

(3k) Since no 2-cycle starts at the vertex k, we have T_{k} ∉ (add U_{i1}) ∩ (add U_{i0}). We only show that the upper sequence (T_{k}^{∗}, U_{i1}) ⟶^{g_{i0}g_{i2}} (T_{k}^{∗}, U_{i0}) ⟶^{f_{i0}} (T_{k}^{∗}, T_{i}) is ex-

act since the proof for the lower sequence is dual. Since (T_{k}^{∗}, T_{i}^{∗}) ⟶^{g_{i2}} (T_{k}^{∗}, U_{i0}) ⟶^{f_{i0}} (T_{k}^{∗}, T_{i}) is exact, we only have to show that (T_{k}^{∗}, U_{i1}) ⟶^{g_{i0}} (T_{k}^{∗}, T_{i}^{∗}) is surjective. This is clear for the case Theorem 5.1 since we have Ext_{C}^{1}(T_{k}^{∗}, T_{i}) = 0.

We consider the case Theorem 5.2. Fix any a ∈ Hom_{Λ}(T_{k}^{∗}, T_{i}^{∗}). Since 0 →

(T, T_{i}) ⟶^{f_{i2}} (T, U_{i1}) ⟶^{g_{i0}} (T, T_{i}^{∗}) → 0 is exact by Proposition 1.3(c), there exists b ∈ Hom_{Λ}(U′_{k}, U′_{i}) and c ∈ Hom_{Λ}(T_{k}^{∗}, T_{i}^{∗}) which make the diagram

\xymatrix{0 \ar[r] & T_{k} \ar[r]^{f_{k2}} \ar[d]^{c} & U_{k1} \ar[r]^{g_{k0}} \ar[d]^{b} & T_{k}^{∗} \ar[d]^{a} \\ 0 \ar[r] & T_{i} \ar[r]^{f_{i2}} & U_{i1} \ar[r]^{g_{i0}} & T_{i}^{∗}}

commutative. Since 0 → (T_{k}^{∗}, T) ⟶^{g_{k0}} (U_{k1}, T) ⟶^{f_{k2}} J_{mod Λ}(T_{k}, T) → 0 is exact by Proposition 1.3(c), there exist s ∈ Hom_{Λ}(U_{k1}, T_{i}) such that c = f_{k2}s and t ∈ Hom_{Λ}(T_{k}^{∗}, U_{i1}) such that b = sf_{i2} + g_{k0}t. Then we have g_{k0}(a − tg_{i0}) = 0. Since g_{k0}(a − tg_{i0})g_{i2} = 0, we have (a − tg_{i0})g_{i2} = 0. Since g_{i2} is injective, we have a = tg_{i0}. □

Now we shall prove Theorem 5.3. The idea of the proof is to use Theorem 4.6 and construct weak 2-almost split sequences in add μ_{k}(T) using the following result.

Proposition 5.5. Assume that the conditions (1), (2k) and (3k) are satisfied for some k. Fix i ≠ k. If we have a commutative diagram

T_{i} ⟶^{f_{i2}}  U_{i1}  ⟶^{f_{i1}=(f′_{i1} f′′_{i1})} U_{i0} = T_{k}^{ℓ} ⊕ U′′_{i0} ⟶^{f_{i0}=\binom{f{′ \atop i0}}{f{′′ \atop i0}}} T_{i}
∥  ↑ \binom{0}{1}  ↑ \binom{f_{k0}^{ℓ} 0}{0 1}  ∥
T_{i} ⟶^{(s f_{i2})} T_{k}^{∗ℓ} ⊕ U_{i1}  \underrightarrow{\binom{g{ℓ \atop k2} 0}{t f{′′ \atop i1}}}  U_{k0}^{ℓ} ⊕ U′′_{i0}  \underrightarrow{\binom{f{ℓ \atop k0}f{′ \atop i0}}{f{′′ \atop i0}}} T_{i}

of complexes such that - U′′_{i0} and U_{i1} belong to add(T/T_{k}),

- s ∈ C(T_{i}, T_{k}^{∗ℓ}) induces a bijection s: (C /J_{C})(T_{k}^{∗ℓ}, T_{k}^{∗}) → (J_{C}/J_{add μ_{k}(T)}^{2})(T_{i}, T_{k}^{∗}), then the lower sequence is a weak 2-almost split sequence in add μ_{k}(T).

Proof. By our assumptions (1), (2k) and (3k) we have exact sequences

(μ_{k}(T), T_{k}^{∗}) ⟶^{g_{k2}} (μ_{k}(T), U_{k0}) ⟶^{f_{k0}} (μ_{k}(T), T_{k}),  (6)
(μ_{k}(T), U_{i1}) ⟶^{f_{i1}} (μ_{k}(T), U_{i0}) ⟶^{f_{i0}} (μ_{k}(T), T_{i}),  (7)
(U_{i0}, μ_{k}(T)) ⟶^{f_{i1}} (U_{i1}, μ_{k}(T)) ⟶^{f_{i2}} (T_{i}, μ_{k}(T)).  (8)

(i) We will show that \binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}} is right almost split in add μ_{k}(T).

First we will show that any morphism a ∈ J_{C}(T/T_{k}, T_{i}) factors through \binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}}.

Since f_{i0} = \binom{f′_{i0}}{f′′_{i0}} is right almost split in add T, there exists (a_{1} a_{2}) ∈ C(T/T_{k}, T_{k}^{ℓ} ⊕ U′′_{i0}) such that a = a_{1}f′_{i0} + a_{2}f′′_{i0}. Since f_{k0} is right almost split in add T, there exists b ∈ C(T/T_{k}, U_{k0}^{ℓ}) such that a_{1} = bf_{k0}^{ℓ}. Then we have a = (b a_{2})\binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}}.

Next we take any a ∈ J_{C}(T_{k}^{∗}, T_{i}). Since g_{k2} is left almost split in add μ_{k}(T), there exists b ∈ C(U_{k0}, T_{i}) such that a = g_{k2}b. Since T_{k} ∉ add U_{i1}, we have T_{i} ∉ add U_{k0}. Thus b ∈ J_{C}(U_{k0}, T_{i}) holds, and the first case implies that b factors through \binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}}.

Hence a factors through \binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}}.

(ii) We will show that \binom{g_{k2}^{ℓ} 0}{t f′′_{i1}} is a pseudo-kernel of \binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}} in add μ_{k}(T).

Assume (a_{1} a_{2}) ∈ C(μ_{k}(T), U_{k0}^{ℓ} ⊕ U′′_{i0}) satisfies (a_{1} a_{2})\binom{f_{k0}^{ℓ}f′_{i0}}{f′′_{i0}} = 0. Since

(a_{1} a_{2})\binom{f_{k0}^{ℓ} 0}{0 1}\binom{f′_{i0}}{f′′_{i0}} = 0 holds and (7) is exact, there exists b ∈ C(μ_{k}(T), U_{i1}) such that b(f′_{i1} f′′_{i1}) = (a_{1} a_{2})\binom{f_{k0}^{ℓ} 0}{0 1}. Hence we have bf′_{i1} = a_{1}f_{k0}^{ℓ} and bf′′_{i1} = a_{2}. Since (a_{1} − bt)f_{k0}^{ℓ} = bf′_{i1} − bf′_{i1} = 0 and (6) is exact, there exists c ∈ C(μ_{k}(T), T_{k}^{∗ℓ}) such that a_{1} − bt = cg_{k2}^{ℓ}. Then we have (a_{1} a_{2}) = (c b)\binom{g_{k2}^{ℓ} 0}{t f′′_{i1}}.

(iii) We will show that (s f_{i2}) is left almost split in add μ_{k}(T).

Since f_{i2} is left almost split in add T, any morphism in J_{C}(T_{i}, T/T_{k}) factors through (s f_{i2}). Then take any a ∈ J_{C}(T_{i}, T_{k}^{∗}). By our assumption on s, there exists a_{1} ∈ C(T_{k}^{∗ℓ}, T_{k}^{∗}) such that a − sa_{1} ∈ J_{add μ_{k}(T)}^{2}(T_{i}, T_{k}^{∗}). Since g_{k0} is right almost split in add μ_{k}(T), there exists b ∈ J_{C}(T_{i}, U_{k1}) such that a − sa_{1} = bg_{k0}. Since f_{i2} is left almost split in add T, there exists c ∈ C(U_{i1}, U_{k1}) such that b = f_{i2}c. Then we have a = sa_{1} + f_{i2}cg_{k0} = (s f_{i2})\binom{a_{1}}{cg_{k0}}.

(iv) We will show that \binom{g_{k2}^{ℓ} 0}{t f′′_{i1}} is a pseudo-cokernel of (s f_{i2}) in add μ_{k}(T). Assume \binom{a_{1}}{a_{2}} ∈ C(T_{k}^{∗ℓ} ⊕ U_{i1}, T′) with T′ ∈ add μ_{k}(T) satisfies (s f_{i2})\binom{a_{1}}{a_{2}} = 0. We first show that there exists b ∈ C(U_{k0}^{ℓ}, T′) such that a_{1} = g_{k2}^{ℓ}b. We have to consider the cases T′ = T/T_{k} and T′ = T_{k}^{∗}. If T′ = T/T_{k}, then this is clear since g_{k2} is left almost split in add μ_{k}(T). If T′ = T_{k}^{∗}, then we have sa_{1} = −f_{i2}a_{2} ∈ J_{add μ_{k}(T)}^{2}. Thus our assumption on s implies that a_{1} ∈ J_{C}(T_{k}^{∗ℓ}, T_{k}^{∗}). Since g_{k0} is right almost split in add μ_{k}(T), we have that a_{1} factors through g_{k0} ∈ C(U_{k1}, T_{k}^{∗}). Since U_{k1} ∈ add(T/T_{k}), the assertion follows from the case T′ = T/T_{k}.

Since f_{i2}(a_{2} − tb) = f_{i2}a_{2} + sg_{k2}^{ℓ}b = f_{i2}a_{2} + sa_{1} = 0 and (8) is exact, there exists \binom{b_{1}}{b_{2}} ∈ C(T_{k}^{ℓ}⊕U′′_{i0}, T′) such that a_{2}−tb = (f′_{i1} f′′_{i1})\binom{b_{1}}{b_{2}}. Then a_{2} = tb+f′_{i1}b_{1}+f′′_{i1}b_{2} =

s(b + f_{k0}^{ℓ}b_{1}) + f′′_{i1}b_{2} holds. Thus we have \binom{a_{1}}{a_{2}} = \binom{g_{k2}^{ℓ} 0}{t f′′_{i1}}\binom{b+f_{k0}^{ℓ}b_{1}}{b_{2}}. □

We need the following information about mutation of QP’s.

Lemma 5.6. Let (Q, W) be a QP and (Q′, W′) = μ̃_{k}(Q, W). Let a and a′ be arrows in Q with e(a) = e(a′) = k, let b and b′ be arrows in Q with s(b) = s(b′) = k, and let c and c′ be arrows in Q ∩ Q′. Then we have the following equalities.

(a) ∂_{(c,c′)}W′ = ∂_{(c,c′)}[W].
(b) ∂_{(a^{∗},c)}W′ = 0 and ∂_{(c,b^{∗})}W′ = 0.
(c) ∂_{(c,[ab])}W′ = ∂_{(c,[ab])}[W] and ∂_{([ab],c)}W′ = ∂_{([ab],c)}[W].
(d) ∂_{(a^{∗},[ab])}W′ = b^{∗} and ∂_{(a^{′∗},[ab])}W′ = 0 if a ≠ a′.
(e) ∂_{([ab],b^{∗})}W′ = a^{∗} and ∂_{([ab],b^{′∗})}W′ = 0 if b ≠ b′.
(f) ∂_{(b^{∗},a^{∗})}W′ = [ab].
(g) ∂_{([ab],[a′b′])}W′ = 0.
(h) For other pairs d, d′ ∈ Q′_{1}, we have ∂_{(d,d′)}W′ = 0.

Proof. Immediate from the definition W′ = [W] + Δ. □

Proof of Theorem 5.3. Let (Q′, W′) = μ̃_{k}(Q, W). We have a surjective K-algebra homomorphism ϕ: K̂Q → End_{C}(T) which induces an isomorphism P(Q, W) ≃ End_{C}(T). Then ϕ gives a K-linear functor Φ: K̂Q → C such that T = ⨁_{i∈Q_{0}} Φi. By Lemma 3.5, we can define a K-linear functor Φ′: K̂Q′ → C in the following way.

(i) Φ′ coincides with Φ on Q ∩ Q′.

(ii) Φ′[ab] = ΦaΦb for each pair of arrows a: i → k and k → j in Q.

(iii) Φ′k^{∗}, Φ′a^{∗} (e(a) = k) and Φ′b^{∗} (s(b) = k) are defined by sequences

Φk ⟶^{(Φb)_{b}} ⨁_{b∈Q_{1}, s(b)=k} Φe(b) ⟶^{(Φ′b^{∗})_{b}} Φ′k^{∗},

Φ′k^{∗} ⟶^{(Φ′a^{∗})_{a}} ⨁_{a∈Q_{1}, e(a)=k} Φs(a) ⟶^{(Φa)_{a}} Φk

which are given in (2k).

We simply denote Φp (respectively, Φ′p) by p for any path p in Q (respectively, Q′).

Let us calculate weak 2-almost split sequences in add μ_{k}(T) by dividing into 3 cases.

Lemma 5.7. We have the following weak 2-almost split sequence in add μ_{k}(T).

k^{∗} ⟶^{(a^{∗})_{a}} ⨁_{a∈Q_{1}, e(a)=k} s(a) ⟶^{([ab])_{a,b}} ⨁_{b∈Q_{1}, s(b)=k} e(b) ⟶^{(b^{∗})_{b}} k^{∗}.

Proof. Immediate from our assumption (2k) and the construction (iii). □

Lemma 5.8. For a vertex i in Q′ with i ≠ k^{∗}, assume that there is no arrow i → k in Q. Then i has a weak 2-almost split sequence

i ⟶^{((b^{∗})_{b} (d)_{d})} ( ⨁_{\begin{matrix}b∈Q_{1} \\ b: k→i \end{matrix}} k^{∗}) ⊕ ( ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \end{matrix}} e(d)) \underrightarrow{\binom{f_{1} 0}{f_{2} f_{3}}} ( ⨁_{\begin{matrix}a,b∈Q_{1} \\ e(a)=k \\ b: k→i \end{matrix}} s(a)) ⊕ ( ⨁_{\begin{matrix}c∈Q_{1} \\ s(c)≠k \\ e(c)=i \end{matrix}} s(c)) \underrightarrow{\binom{([ab])_{(a,b)}}{(c)c}} i

in add μ_{k}(T), where

f_{1} = ⨁_{\begin{matrix}b∈Q_{1} \\ b: k→i \end{matrix}} (k^{∗} ⟶^{(a^{∗})} ⨁_{\begin{matrix}a∈Q_{1} \\ e(a)=k \end{matrix}} s(a))

f_{2} = (∂_{([ab],d)}[W])_{d,(a,b)} : ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \end{matrix}} e(d) → ⨁_{\begin{matrix}a,b∈Q_{1} \\ e(a)=k \\ b: k→i \end{matrix}} s(a)

f_{3} = (∂_{(c,d)}[W])_{d,c}: ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \end{matrix}} e(d) → ⨁_{\begin{matrix}c∈Q_{1} \\ s(c)≠k \\ e(c)=i \end{matrix}} s(c)

Proof. Since End_{C}(T) ≃ P(Q, W), we have a weak 2-almost split sequence

i ⟶^{(d)_{d}} ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \end{matrix}} e(d) ⟶^{(g_{1} g_{2})} ( ⨁_{\begin{matrix}b∈Q_{1} \\ b: k→i \end{matrix}} k) ⊕ ( ⨁_{\begin{matrix}c∈Q_{1} \\ s(c)≠k \\ e(c)=i \end{matrix}} s(c)) \underrightarrow{\binom{(b)_{b}}{(c)c}} i

in add T by Theorem 4.6, where

g_{1} = (∂_{(b,d)}W)_{d,b}: ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \end{matrix}} e(d) → ⨁_{\begin{matrix}b∈Q_{1} \\ b: k→i \end{matrix}} k,

g_{2} = (∂_{(c,d)}W)_{d,c}: ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \end{matrix}} e(d) → ⨁_{\begin{matrix}c∈Q_{1} \\ s(c)≠k \\ e(c)=i \end{matrix}} s(c).

Consider the following diagram of complexes in C.

[figure: i; (d)_{d}; ⨁_{{d∈Q_{1} \atop s(d)=i}} e(d); (g_{1} g_{2}); ( ⨁_{{b∈Q_{1} \atop b: k→i}} k) ⊕ ( ⨁_{c∈Q_{1} ; {s(c)≠k \atop e(c)=i}} s(c)); ^{\binom{(b)_{b}}{(c)c}}; i; i; ((b^{∗})_{b} (d)_{d}); ( ⨁_{{b∈Q_{1} \atop b: k→i}} k^{∗}) ⊕ ( ⨁_{{d∈Q_{1} \atop s(d)=i}} e(d)); _{\binom{f_{1} 0}{f_{2} f_{3}}}; _{\binom{0}{1}}; ( ⨁_{a,b∈Q_{1} ; {e(a)=k \atop b: k→i}} s(a)) ⊕ ( ⨁_{c∈Q_{1} ; {s(c)≠k \atop e(c)=i}} s(c)); _{\binom{(a) 0}{0 1}}; _{\binom{([ab])_{(a,b)}}{(c)c}}; i]

This is commutative since we have f_{1}(a) = 0 by ∑_{a∈Q_{1}, e(a)=k} a^{∗}a = 0, and f_{2}(a) = g_{1} by

∑_{a∈Q_{1}, e(a)=k} ∂_{([ab],d)}[W]a = ∂_{(b,d)}W,

and f_{3} = g_{2} clearly.

By virtue of Proposition 5.5, we only have to show that

(b^{∗})_{b}: (C /J_{C})( ⨁_{b: k→i} k^{∗}, k^{∗}) → (J_{C}/J_{add μ_{k}(T)}^{2})(i, k^{∗})

is bijective. By the assumption (C2), we have K = (C /J_{C})(k^{∗}, k^{∗}). By Lemma 5.7 we have that (b^{∗})_{b}: ⨁_{b∈Q_{1}, s(b)=k} e(b) → k^{∗} is minimal right almost split in add μ_{k}(T) since the middle morphism in the sequence in Lemma 5.7 belongs to J_{C}. Thus we have that (J_{C}/J_{add μ_{k}(T)}^{2})(i, k^{∗}) is a K-vector space with the basis {b^{∗} | b: k → i}. Thus the above map is bijective. □

The following result is just a dual of Lemma 5.8.

Lemma 5.9. For a vertex i in Q′ with i ≠ k^{∗}, assume that there is no arrow k → i in Q. Then i has a weak 2-almost split sequence

i ⟶^{(([ab])_{(a,b)} (d)_{d})} ( ⨁_{\begin{matrix}a,b∈Q_{1} \\ s(b)=k \\ a: i→k \end{matrix}} e(b)) ⊕ ( ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \\ e(d)≠k \end{matrix}} e(d)) \underrightarrow{\binom{f_{1} f_{2}}{0 f_{3}}} ( ⨁_{\begin{matrix}a∈Q_{1} \\ a: i→k \end{matrix}} k^{∗}) ⊕ ( ⨁_{\begin{matrix}c∈Q_{1} \\ e(c)=i \end{matrix}} s(c)) \underrightarrow{\binom{(a^{∗})a}{(c)c}} i

in add μ_{k}(T), where

f_{1} = ⨁_{\begin{matrix}a∈Q_{1} \\ a: i→k \end{matrix}} ( ⨁_{\begin{matrix}b∈Q_{1} \\ s(b)=k \end{matrix}} e(b) ⟶^{(b^{∗})} k^{∗})

f_{2} = (∂_{(c,[ab])}[W])_{(a,b),c}: ⨁_{\begin{matrix}a,b∈Q_{1} \\ s(b)=k \\ a: i→k \end{matrix}} e(b) → ⨁_{\begin{matrix}c∈Q_{1} \\ e(c)=i \end{matrix}} s(c)

f_{3} = (∂_{(c,d)}[W])_{d,c}: ⨁_{\begin{matrix}d∈Q_{1} \\ s(d)=i \\ e(d)≠k \end{matrix}} e(d) → ⨁_{\begin{matrix}c∈Q_{1} \\ e(c)=i \end{matrix}} s(c)

Consequently we have the following observation.

Proposition 5.10. Any vertex i in Q′ has the following weak 2-almost split sequence in add μ_{k}(T).

i ⟶^{(d)_{d}} ⨁_{d∈Q′_{1}, s(d)=i} e(d) ⟶^{(∂_{(c,d)}W′)_{d,c}} ⨁_{c∈Q′_{1}, e(c)=i} s(c) ⟶^{(c)_{c}} i.

Proof. In Lemma 5.6, we have calculated the ∂_{(c,d)}W′. They coincide with the middle morphism of weak 2-almost split sequences given in Lemmas 5.7, 5.8 and 5.9. Note that we assumed that k is not contained in 2-cycles, so Lemmas 5.7, 5.8 and 5.9 cover all possible cases. □

Now Theorem 4.6 and Proposition 5.10 imply End_{C}(μ_{k}(T)) ≃ P(Q′, W′). By [DWZ1, Theorem 4.6], we have P(Q′, W′) ≃ P(μ_{k}(Q, W)). Thus we have proved Theorem 5.3. □

We have the following direct consequences of Theorems 5.1 and 5.2.

Corollary 5.11. Let C be one of the following categories.

(a) C is a 2-CY triangulated category with a basic cluster-tilting object T.

(b) C = mod Λ for a 3-CY algebra Λ with a basic tilting Λ-module T of projec-

tive dimension at most one. If End_{C}(T) ≃ P(Q, W) for a QP (Q, W), then for any sequence k_{1}, …, k_{ℓ} of vertices in Q such that k_{i} does not lie on a 2-cycle in μ_{k_{i−1}} ∘ ⋯ μ_{k_{1}}(Q, W), we have

End_{C}(μ_{k_{ℓ}} ∘ ⋯ ∘ μ_{k_{1}}(T)) ≃ P(μ_{k_{ℓ}} ∘ ⋯ ∘ μ_{k_{1}}(Q, W)).

Remark If C is a 2-CY triangulated category with cluster structure and End_{C}(T) is Jacobian, then none of the Jacobian algebras in the same mutation class will have 2-cycles in their quivers. Hence they are all non-degenerate QP’s in the sense of [DWZ1].

The next result is of special interest.

Corollary 5.12. Any cluster-tilted algebra is isomorphic to a Jacobian algebra of a rigid QP.

Proof. We here use Theorem 5.1 and the fact that any cluster-tilted algebra belongs to the same mutation class as a hereditary algebra, which is trivially given by a QP. That the QP is rigid follows from [DWZ1]. □

It is not known in general whether the algebra End_{C}(T^{∗}) can be defined directly from the algebra End_{C}(T) when T^{∗} is a cluster-tilting object in the triangulated 2-CY category C obtained from the cluster tilting object T by mutation. When the potential (Q′, W′) is obtained by mutation from the potential (Q, W), it is also not known if the Jacobian algebra P(Q′, W′) can be defined directly from the Jacobian algebra P(Q, W).

However, we can prove this for Jacobian algebras of Jacobi-finite QP’s by the following result together with a result of Amiot [A]. We call a QP (Q, W) Jacobifinite if the factor algebra KQ/⟨∂_{a}W | a ∈ Q_{1}⟩ of the ordinary path algebra KQ is finite dimensional [A]. This condition implies that P(Q, W) is finite dimensional.

Theorem 5.13. Let Λ be an algebra isomorphic to a Jacobian algebra P(Q, W) for some QP (Q, W) and to a 2-CY-tilted algebra End_{C}(T) for some cluster-tilting object T in a 2-CY category C. Assume k is a vertex in Q not lying on any 2-cycle.

(a) The Jacobian algebra P(μ_{k}(Q, W)) is determined by the algebra Λ and does not depend on the choice of a QP (Q, W).

(b) The 2-CY-tilted algebra End_{C}(μ_{k}(T)) is determined by the algebra Λ and does not depend on the choice of a 2-CY category C and a cluster-tilting object T.

Proof. This is a direct consequence of Theorem 5.1. □

The following gives a partial answer to Question 12.2 in [DWZ1].

Corollary 5.14. Let P(Q, W) be a Jacobian algebra of a Jacobi-finite QP. Assume k is a vertex in Q not lying on any 2-cycle. If P(Q, W) ≃ P(Q′, W′) for some QP (Q′, W′), then Q_{0} ≃ Q′_{0} and P(μ_{k}(Q, W)) ≃ P(μ_{k}(Q′, W′)).

Proof. Clearly we have Q_{0} ≃ Q′_{0}. The second assertion follows from Theorem 5.13(a) and the fact that any Jacobian algebra of a Jacobi-finite QP is a 2-CY-tilted algebra [A]. □

We end this section with using the results in this section to give an alternative proof of the following.

Corollary 5.15. Cluster-tilted algebras are determined by their quiver.

Proof. If a cluster-tilted algebra Λ has an acyclic quiver Q, then Λ is isomorphic to the path algebra KQ by [ABS, KR1], in particular it is determined by its quiver.

Assume that T_{1} and T_{2} are cluster-tilting objects in the cluster categories C_{Q_{1}} and C_{Q_{2}} respectively, such that the associated quivers are isomorphic. Let μ = μ_{k_{ℓ}} ∘ ⋯ ∘ μ_{k_{1}} be a sequence of mutations such that for the projective KQ_{1}-module KQ_{1} we have μ(KQ_{1}) ≃ T_{1}, and hence μ(Q_{1}) ≃ Q_{T_{1}}. Then let T be the clustertilting object in C_{Q_{2}} such that μ(T) ≃ T_{2} and hence μ(Q_{T}) ≃ Q_{T_{2}}. Then Q_{1} ≃ Q_{T}, and hence KQ_{1} ≃ End_{C_{Q_{2}}} (T). Since cluster-tilted algebras are QP’s by Corollary 5.12, it follows from Theorem 5.13(b) that End_{C_{1}}(T_{1}) ≃ End_{C_{2}}(T_{2}). □

Remark It follows from the work in [DWZ1] that if Λ = P(Q, W) is a Jacobian algebra given by a rigid potential (Q, W) where Q belongs to the mutation class of an acyclic quiver, then Λ is determined by its quiver. Since by Corollary 5.12 any cluster-tilted algebra has this property, this gives yet another way of seeing that cluster-tilted algebras are determined by their quiver.

6. 2-CY-tilted algebras associated with elements in Coxeter groups

In this section we show that a large class of 2-CY-tilted algebras, including the cluster-tilted algebras and the class of 2-CY-tilted algebras coming from stable categories of preprojective algebras of Dynkin type, are given by QP’s, by finding an explicit description of the potentials for some of them. Hence by Theorem 5.1 we get that all 2-CY-tilted algebras in the mutation class are given by QP’s. We prove that these QP’s are rigid in the sense of [DWZ1].

These 2-CY-tilted algebras come from 2-CY triangulated categories constructed from elements in the Coxeter groups associated with connected quivers with no loops, as investigated in [IR, BIRS]. (See [GLS2] for an alternative approach to the construction of a subclass of these 2-CY categories). We start with recalling the relevant results from [BIRS], including a description of the quivers of some special 2-CY-tilted algebras.

Let Q be a finite connected quiver with no loops, with vertices 1, …, n and set of arrows Q_{1}. The associated preprojective algebra over the algebraically closed field K is defined as follows. For each arrow a ∈ Q_{1} from i to j, we add a corresponding arrow a^{∗} from j to i to get a new quiver \overline{Q}. Then the preprojective algebra is defined by

Λ = K\overline{Q}/⟨ ∑_{a∈Q_{1}}(aa^{∗} − a^{∗}a)⟩.

We shall write (a^{∗})^{∗} = a, and ϵ(a) = 1, ϵ(a^{∗}) = −1.

The Coxeter group W_{Q} is presented by generators s_{1}, …, s_{n} with relations s_{i}s_{j} = s_{j}s_{i} if there is no arrow in Q between i and j and s_{i}s_{j}s_{i} = s_{j}s_{i}s_{j} if there is precisely one arrow in Q between i and j. Let w be an element in the associated Coxeter group W_{Q}, and w = s_{u_{1}} ⋯ s_{u_{m}} a reduced expression, where the u_{1}, …, u_{m} are integers in {1, …, n}. For each integer i in {1, …, n}, consider the ideal I_{i} = Λ(1 − e_{i})Λ in Λ, where e_{i} denotes the trivial path at the vertex i. Let I_{w} = I_{u_{1}} ⋯ I_{u_{m}}, and let Λ_{w} = Λ/I_{w}. Then I_{w} and Λ_{w} are independent of the choice of reduced expression for w, and Λ_{w} is a finite dimensional K-algebra. Denote by Sub Λ_{w} the full subcategory of mod Λ_{w} whose objects are the submodules of finitely generated projective Λ_{w}- modules. We have that the injective dimensions id_{Λ_{w}}Λ_{w} and idΛ_{w}_{Λ_{w}} are at most one, and hence the stable category C = \underline{Sub}Λ_{w} is a Hom-finite triangulated 2-CY category [H]. Associated with each reduced expression s_{u_{1}} ⋯ s_{u_{m}} of w is the object T = Λ/I_{u_{1}} ⊕ Λ/I_{u_{1}}I_{u_{2}} ⊕ ⋯ ⊕ Λ/I_{u_{1}} ⋯ I_{u_{m}} in Sub Λ_{w}. Consider the algebra End_{Λ_{w}}(T) and the associated 2-CY-tilted algebra End_{C}(T). Alternatively we can write T as a sum of indecomposable objects

\begin{aligned}T & = & T_{1} & ⊕ & T_{2} & ⊕ ⋯ ⊕ & T_{m} \\ & = & \frac{P_{u_{1}}}{I_{u_{1}} P_{u_{1}}} & ⊕ & \frac{P_{u_{2}}}{(I_{u_{1}} I_{u_{2}} )P_{u_{2}}} & ⊕ ⋯ ⊕ & \frac{P_{um}}{(I_{u_{1}} I_{u_{2}} ...I_{um})P_{um}} , \end{aligned}

where P_{i} is the indecomposable projective Λ-module associated with the vertex i. The structure of T_{v} is determined by the sequence of integers u_{1}, …, u_{v}. Let K_{v,1} be the smallest submodule of P_{u_{v}} such that P_{u_{v}}/K_{v,1} is a sum of copies of S_{u_{v}} (in this case only S_{u_{v}}). Then let K_{v,2} be the smallest submodule of K_{v,1} such that K_{v,1}/K_{v,2} is a sum of copies of S_{u_{v−1}} etc. Then T_{v} = P_{u_{v}}/K_{v,v}.

We also recall [BIRS] the following description of the quiver Q′ = Q(u_{1}, …, u_{m}) of End_{Λ}(T). The vertices are 1, 2, ⋯, m ordered from left to right. We often denote the vertex v by i_{r} if it is the r-th vertex of type i. In this case we write |i_{r}| = v. There are two kinds of arrows in Q′.

(i) Arrows going to the left: For two consecutive vertices of type i, for 1 ≤ i ≤ n, draw an arrow from the right one to the left one.

(ii) Arrows going to the right: For each arrow i ⟶^{a} j in \overline{Q}, draw an arrow u ⟶^{a} v

whenever the following is satisfied - u is of type i and v is of type j

- there is no vertex of type i between u and v

- if there is a vertex v′ of type j after v, then there is a vertex of type i

between v and v′

The following picture gives an illustration.

\xymatrix{ & i_{1} & i_{2} \ar[l] & i_{3} \ar[l] \ar[drrr]^{a} &  &  &  & i_{4} \ar[llll] & i_{5} \ar[l] \ar[dr]^{a} \\ j_{1} \ar[urrr]^{a^{∗}} &  &  &  & j_{2} \ar[llll] & j_{3} \ar[l] & j_{4} \ar[l] \ar[urr]^{a^{∗}} &  &  & j_{5} \ar[lll]}

Example We give a concrete example. Let Q be the quiver \xymatrix{1 \ar[r]^{a} & 2 \ar[r]^{b} & 3 \ar[ll]_{c}}, and let w = s_{1}s_{2}s_{1}s_{3}s_{1}s_{2}s_{3}s_{1}s_{2}s_{3}s_{2} be a reduced expression. The associated object T in Sub Λ_{w} is the following.

1 ⊕ {2 \atop 1} ⊕ {1 \atop 2} ⊕ 3 1 2 2  1 ⊕ 1 2 3 1 2 1 ⊕ 2 3 1 1 2 3 2  1 2 1 ⊕ 3 1 2 2 3 1 1 2 3 1 2 1 ⊕ 1 2 3 3 1 2 1 2 3 1 2  1 2 3 1 2 1

⊕ 2 3 1 1 2 3 2 3 1 2 1 2 3 1 1 2 3 1 2 1 ⊕ 3 1 2 2 3 1 3 1 2 3 1 2 3 1 2 2  1 2 3 1 1 2 3 1 2 1 ⊕ 2 3 1 1 2 3 2 3 1 2 3 1 2 3 1 1 2  1 2 3 2  1 2 1

The quiver Q(1, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2) is the following.

\xymatrix{1_{1} \ar[dr]_{a_{1}} &  & 1_{2} \ar[ll]_{p_{2}} \ar[ddr]^{c_{2}^{∗}} &  & 1_{3} \ar[ll]_{p_{3}} \ar[dr]^{a_{3}} \ar[ddrr]_{c_{3}^{∗}} &  &  & 1_{4} \ar[lll]_{p_{4}} \ar[drrr]^{a_{4}} \ar[ddrr]_{c_{4}^{∗}} \\  & 2_{1} \ar[urrr]^{a_{1}^{∗}} \ar[drr]_{b_{1}} &  &  &  & 2_{2} \ar[llll]^{q_{2}} \ar[urr]^{a_{2}^{∗}} \ar[dr]^{b_{2}} &  &  & 2_{3} \ar[lll]^{q_{3}} \ar[dr]^{b_{3}} &  & 2_{4} \ar[ll]^{q_{4}} \\  &  &  & 3_{1} \ar[urr]^{b_{1}^{∗}} \ar[uur]^{c_{1}} &  &  & 3_{2} \ar[lll]^{r_{2}} \ar[urr]^{b_{2}^{∗}} \ar[uur]^{c_{2}} &  &  & 3_{3} \ar[lll]^{r_{3}} \ar[ur]_{b_{3}^{∗}}}

Recall that the corresponding maps are given as follows.

Lemma 6.1. (a) Let i_{r} →^{a} i_{r−1} be an arrow in Q′ going to the left. Then the map T_{i_{r}} →^{a} T_{i_{r−1}} is given by the natural surjection.

(b) Let i_{r} →^{b} j_{s} be an arrow in Q′ corresponding to an arrow i →^{b} j in \overline{Q}. Then

the map T_{i_{r}} →^{b} T_{j_{s}} is given by multiplication with b.

For each i, the last (i.e. rightmost) vertex of type i corresponds to the indecomposable projective Λ_{w}-module \frac{P_{i}}{I_{w}P_{i}}. By dropping the last vertex of type i for each i = 1, …, n, where possible, we obtain the quiver \underline{Q}′ = \underline{Q}(u_{1}, …, u_{m}) of End_{C}(T) from the quiver Q′ of End_{Λ}(T), and T is a cluster-tilting object in \underline{Sub}Λ_{w}.

Our aim is to show that the 2-CY-tilted algebra End_{C}(T) is a Jacobian algebra P(\underline{Q}′, W), by giving an explicit description of the potential W. For each arrow i_{r} →^{b} j_{s} in \underline{Q}′, we let W_{b} = ϵ(b)bb^{∗}p if there is a (unique) arrow j_{s} →^{b^{∗}} i_{t} in \underline{Q}′, where p denotes the path i_{t} → i_{t−1} → ⋯ → i_{r}. Otherwise we let W_{b} = 0. Then let

W = ∑_{b∈\underline{Q}′_{1}} W_{b}.

Our strategy is to show that all the relations ∂_{a}W are satisfied for End_{C}(T), and then apply Proposition 4.7 to show End_{C}(T) ≃ P(\underline{Q}′, W).

Example Again we consider the above example. Deleting vertices 1_{4}, 3_{3} and 2_{4}, we have the quiver \underline{Q}(1, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2) as follows.

\xymatrix{1_{1} \ar[dr]_{a_{1}} &  & 1_{2} \ar[ll]_{p_{2}} \ar[ddr]^{c_{2}^{∗}} &  & 1_{3} \ar[ll]_{p_{3}} \ar[dr]^{a_{3}} \ar[ddrr]_{c_{3}^{∗}} \\  & 2_{1} \ar[urrr]^{a_{1}^{∗}} \ar[drr]_{b_{1}} &  &  &  & 2_{2} \ar[llll]^{q_{2}} \ar[dr]^{b_{2}} &  & 2_{3} \ar[ll]^{q_{3}} \\  &  &  & 3_{1} \ar[urr]^{b_{1}^{∗}} \ar[uur]^{c_{1}} &  &  & 3_{2} \ar[lll]^{r_{2}} \ar[ur]^{b_{2}^{∗}}}

The associated potential is given by

W = a_{1}a_{1}^{∗}p_{3}p_{2} − a_{1}^{∗}a_{3}q_{2} + b_{1}b_{1}^{∗}q_{2} − b_{1}^{∗}b_{2}r_{2} + b_{2}b_{2}^{∗}q_{3} − c_{2}^{∗}c_{1}p_{3} + c_{1}c_{3}^{∗}r_{2}.

Let us start with the following information.

Lemma 6.2. Let i_{r} → i_{r−1} be an arrow in Q′ going to the left. For any arrow i →^{b} j in \overline{Q}, there exists a map T_{i_{r−1}} →^{bb^{∗}} T_{i_{r}} of Λ-modules given by multiplication with bb^{∗}. Moreover precisely one of the following holds.

(a) There do not exist vertices i_{u} and j_{t} in Q′ satisfying |i_{u}| < |j_{t}| < |i_{r}|. In

this case the map T_{i_{r−1}} →^{bb^{∗}} T_{i_{r}} is zero.

(b) There exists a path i_{u} →^{b} j_{t} →^{b^{∗}} i_{v} in Q′ with u ≤ r − 1 < r ≤ v. In this case such a path is unique, and the map T_{i_{r−1}} →^{bb^{∗}} T_{i_{r}} is equal to the

composition T_{i_{r−1}} →^{p} T_{i_{u}} →^{b} T_{j_{t}} →^{b^{∗}} T_{i_{v}} →^{q} T_{i_{r}} where p denotes a path

i_{r−1} → i_{r−2} → ⋯ → i_{u} and q denotes a path i_{v} → i_{v−1} → ⋯ → i_{r}.

Proof. We can write T_{i_{r−1}} = P_{i}/IP_{i} and T_{i_{r}} = P_{i}/(II′I_{i})P_{i}, where I′ is a product of ideals I_{1}, ⋯, I_{n} except I_{i}. Then we have bb^{∗} ∈ I′I_{i}. Thus we have Ibb^{∗} ⊂ II′I_{i}, and the map T_{i_{r−1}} →^{bb^{∗}} T_{i_{r}} is well-defined.

(a) Since bb^{∗} is zero in T_{i_{r}}, we have the assertion.

(b) The uniqueness of the path is clear from the definition of Q′. The latter assertion is clear from Lemma 6.1. □

We now show that the 2-CY-tilted algebra End_{C}(T) satisfies the relations for the Jacobian algebra P(\underline{Q}′, W).

Proposition 6.3. ∂_{a}W belongs to the kernel of the surjection K̂\underline{Q}′ → End_{C}(T) for any arrow a in \underline{Q}′.

Proof. There are two cases to consider.

(1) Consider an arrow i_{r} →^{b} j_{s} going to the right in \underline{Q}′. If there is an arrow

j_{s} →^{b^{∗}} i_{t} in \underline{Q}′, we have the cycle ϵ(b)bb^{∗}p as part of the potential W, where p is the path i_{t} → i_{t−1} → ⋯ → i_{r}. And if there is an arrow j_{u} →^{b^{∗}} i_{r} in \underline{Q}′, we have the cycle ϵ(b^{∗})b^{∗}bq, where q is the path j_{s} → j_{s−1} → ⋯ → j_{u}. The relation ∂_{b}W corresponding to the arrow i_{r} →^{b} j_{s} is one of the following four possibilities up to sign.

(i) b^{∗}p − qb^{∗}, if both j_{s} →^{b^{∗}} i_{t} and j_{u} →^{b^{∗}} i_{r} exist,

(ii) b^{∗}p, if j_{s} →^{b^{∗}} i_{t} exists, but not j_{u} →^{b^{∗}} i_{r}, (iii) qb^{∗}, if j_{u} →^{b^{∗}} i_{r} exists, but not j_{s} →^{b^{∗}} i_{t}, (iv) 0, if neither j_{s} →^{b^{∗}} i_{t} nor j_{u} →^{b^{∗}} i_{r} exist.

In each case, both T_{j_{s}} ⟶^{b^{∗}p} T_{i_{r}} and T_{j_{s}} →^{qb^{∗}} T_{i_{r}} are given by multiplication with b^{∗} by Lemma 6.1. Thus the cases (i) and (iv) are clear. For the case (ii), there is no vertex j_{u} of type j satisfying |j_{u}| < |i_{r}|. Since then T_{i_{r}} has no composition

factor S_{j}, we have that T_{j_{s}} →^{b^{∗}p} T_{i_{r}} is zero. For the case (iii), there exists an arrow

j_{s} →^{b^{∗}} i_{t} in Q′ such that T_{i_{t}} is a projective Λ_{w}-module. Thus T_{j_{s}} →^{qb^{∗}} T_{i_{r}} is equal to

T_{j_{s}} →^{b^{∗}p} T_{i_{r}} which is zero in C.

(2) Consider an arrow i_{r} →^{a} i_{r−1} going to the left in \underline{Q}′.

By definition of W we have

∂_{a}W = ∑_{b} ϵ(b)pbb^{∗}q,

where b is an arrow in \overline{Q} satisfying the condition in Lemma 6.2(b) and the additional

condition that the path pbb^{∗}q is in \underline{Q}′. In this case the map T_{i_{r−1}} ⟶^{ϵ(b)pbb^{∗}q} T_{i_{r}} is given by multiplication with ϵ(b)bb^{∗} by Lemma 6.2(b).

If b is an arrow in \overline{Q} satisfying the condition in Lemma 6.2(b) and the path pbb^{∗}q

is not in \underline{Q}′, then the map T_{i_{r−1}} ⟶^{ϵ(b)pbb^{∗}q} T_{i_{r}} is zero in C and equal to the map which is multiplication with ϵ(b)bb^{∗}.

If b is an arrow in \overline{Q} which does not satisfy the condition in Lemma 6.2(b), then the multiplication map T_{i_{r−1}} →^{bb^{∗}} T_{i_{r}} is zero by Lemma 6.2(a).

Consequently the map T_{i_{r−1}} →^{∂_{a}W} T_{i_{r}} is equal to multiplication with ∑_{b∈\overline{Q}_{1}} ϵ(b)bb^{∗}, which is zero by the definition of the preprojective algebra. □

Now we are ready to prove the main result in this section.

Theorem 6.4. Let Q be a finite quiver without loops and W_{Q} the Coxeter group associated with Q. Let Λ be the preprojective algebra over the algebraically closed field K and w ∈ W_{Q}. Let T be a cluster-tilting object associated with a reduced expression of w in the 2-CY triangulated category C = \underline{Sub}Λ_{w}. Then the 2-CY-tilted algebra End_{C}(T) is given by the QP (\underline{Q}′, W) above.

Proof. By Proposition 6.3 we have that End_{C}(T) is a factor algebra of the Jacobian algebra P(\underline{Q}′, W). We want to apply Proposition 4.7.

We first claim that each cycle C in W is full. Recall that C is of the form

\xymatrix{i_{u} \ar[drr] & … \ar[l] & … & i_{r−1} \ar[l] & i_{r} \ar[l] \\  &  & j_{s} \ar[urr]}

There are no more arrows between the vertices of type i in C. Further i_{r} is the unique vertex of type i where there is an arrow from j_{s}, and i_{u} is the unique vertex of type i with an arrow to j_{s}. This shows that C is full.

For any cycle C of W and any arrow a in C, it is clear from the definition of W that the path ∂_{a}C determines a. Hence for any distinct arrows a and b contained in cycles of W, we have that ∂_{a}W and ∂_{b}W do not have common paths. Consequently the ∂_{a}W for all arrows a contained in cycles of W are linearly independent over K.

Since Γ = End_{C}(T) is 2-CY-tilted, it follows from [KR1] that dim_{K} Ext_{Γ}^{2}(S, S′) ≤ dim_{K} Ext_{Γ}^{1}(S′, S) for simple End_{C}(T)-modules S and S′. It now follows from Proposition 4.7 that End_{C}(T) is isomorphic to the Jacobian algebra P(\underline{Q}′, W). □

Since the 2-CY categories \underline{Sub}Λ_{w} with the cluster-tilting objects determine a cluster structure [BIRS], we have the following consequence of Corollary 5.11.

Corollary 6.5. With the previous notation, all the 2-CY-tilted algebras belonging to the mutation class of cluster-tilting objects associated with reduced expressions are given by QP’s.

We now show that the potential W is rigid.

Theorem 6.6. With the previous notation, the potential W on \underline{Q}′ is rigid.

Proof. Let C be any cycle in the quiver \underline{Q}′. We want to show that C belongs to J (W) up to cyclic equivalences.

We say that a vertex v in C is a right turning point if an arrow in C going from left to right ends at v, and an arrow in C going from right to left starts at v. Then define r(C) = ∑_{v} 3^{|v|}, where we sum over all right turning points v of C. Consider a vertex i_{u} on C, with |i_{u}| minimal. Consider the last right turning point i_{v} preceding i_{u}, and let j_{s} → i_{v} be the preceding arrow in C. Since |i_{u}| < |j_{s}| < |i_{v}|, we can choose i_{r} with |i_{r}| < |j_{s}| < |i_{r+1}|.

Assume first that there is some j_{t} with |j_{t}| < |i_{r}|. Then there is an arrow j_{t} →^{b} i_{r} if j_{t} is chosen with |j_{t}| largest possible. We have the following subquiver of \underline{Q}′:

\xymatrix{i_{u} & · \ar[l] & ⋯ & i_{r} &  & i_{r+1} \ar[ll] & ⋯ & · & i_{v} \ar[l] \\  & j_{t} \ar[urr]^{b} & ⋯ \ar[l] & ⋯ & j_{s} \ar[urrrr]^{b} \ar[l]}

The composition j_{s} →^{b} i_{v} → i_{v−1} → ⋯ → i_{r} coincides with the composition j_{s} → j_{s−1} → ⋯ → j_{t} →^{b} i_{r} by Lemma 6.1. So we replace the first path by the second one to get a new cycle C′ satisfying C − C′ ∈ J (W). The right turning point i_{v} in C is replaced by at most two right turning points i_{r} and j_{s} in C′, where |i_{r}| < |i_{v}| and |j_{s}| < |i_{v}|, and hence 3^{|i_{r}|} + 3^{|j_{s}|} < 3^{|i_{v}|}. This shows that r(C′) < r(C), and we are done by induction.

Assume now that there is no j_{t} with |j_{t}| < |i_{r}|. Then T_{i_{r}} has no composition factor S_{j}, so the composition T_{j_{s}} → ⋯ → T_{i_{r}} must be 0. Hence C ∈ J (W) in this case. □

Note that it follows from [DWZ1] that all the potentials in the mutation class of (\underline{Q}′, W) are rigid, and hence we obtain a large class of rigid QP’s. Some examples of rigid QP’s where the quiver Q is not mutation equivalent to an acyclic quiver were given in [DWZ1].

7. Nearly Morita equivalence for neighboring Jacobian algebras

Let T be a cluster-tilting object in a triangulated 2-CY category C, and T^{∗} = μ_{k}(T) another cluster-tilting object obtained by mutation. Then we have a nearly Morita equivalence between Λ = End_{C}(T) and Λ′ = End_{C}(T^{∗}), that is an equivalence \frac{mod Λ}{[add S_{k}]} → \frac{mod Λ′}{[add S′_{k}]}, where S_{k} and S′_{k} denote the simple modules at the vertex k [BMR1, KR1]. This generalizes the equivalence of [BGP] using reflection functors at sinks or sources. Since there are Jacobian algebras which are not 2-CY-tilted, it is natural to ask if we have a nearly Morita equivalence in general when performing mutations of potentials. It is the aim of this section to show that this is the case, when working with finite length modules.

Let (Q, W) be a quiver with potential and Λ = P(Q, W) the associated Jacobian algebra. Denote by μ̃_{k}(Q, W) the quiver with potential and Λ′ = P(μ̃_{k}(Q, W)) the Jacobian algebra obtained by mutation at the vertex k. We show that there is an equivalence of categories \frac{f.l. Λ}{[add S_{k}]} → \frac{f.l. Λ′}{[add S′_{k}]}, where S_{k} and S′_{k} denote the simple modules at the vertex k.

Our starting point is the map G from objects in f. l. Λ to objects in f. l. Λ′ used in [DWZ1], which we now recall.

Given an Λ-module M and a vertex k, we let a_{1}, ..., a_{s} be all arrows in Q with e(a_{p}) = k and b_{1}, ..., b_{t} be all arrows such with s(b_{q}) = k. We write

M_{in} = ⨁_{p=1}^{s} M_{e(a_{p})}, M_{out} = ⨁_{q=1}^{t} M_{s(b_{q})}.

Let α: M_{in} → M_{k} and β: M_{k} → M_{out} be the maps given in matrix form by

α = (a_{1}, ..., a_{s}), β = ( b_{1} ; ⋮ ; b_{t} ) .

Also, using the potential W, we define a map γ: M_{out} → M_{in}

γ = ( \begin{matrix}γ_{1,1} & ⋯ & γ_{1,t} \\ ⋮ & ... & ⋮ \\ γ_{s,1} & ⋯ & γ_{s,t} \end{matrix} )

where γ_{p,q} = ∂_{(a_{p},b_{q})}W in the notation of section 4. Thus, locally, we get a triangle

\xymatrix{ & M_{k} \ar[dr]^{β} \\ M_{in} \ar[ur]^{α} &  & M_{out}. \ar[ll]^{γ}}

Note that γ_{p,q} is a linear combination of paths in Q from e(b_{q}) to s(a_{p}), and that there may be other paths from M_{out} to M_{in}.

Now, starting with M, we define a representation M̃ of Λ′ = P(μ̃_{k}(Q, W)) as follows: first, we let i: Ker γ → M_{out} be the inclusion and φ: Ker α → \frac{Ker α}{Im γ} the natural surjection, and choose two K-linear splittings,

ρ: M_{out} → Ker γ such that iρ = id_{Ker γ}, and

σ: \frac{Ker α}{Im γ} → Ker α such that σφ = id_{\frac{Ker α}{Im γ}}.

Then, locally, the representation M̃ is given by:

\xymatrix{ & M̃_{k} \ar[dl]_{β̃} \\ M_{in} \ar[rr]^{αβ} &  & M_{out} \ar[ll]^{γ} \ar[ul]_{α̃}}

where

M̃_{k} = \frac{Ker γ}{Im β} ⊕ Im γ ⊕ \frac{Ker α}{Im γ} , α̃ = (−ρπ, −γ, 0) and β̃ = ( \begin{matrix}0 \\ ι \\ σ \end{matrix} ) ,

where ι: Im γ → Ker α is the natural injection and π: Ker γ → \frac{Ker γ}{Im β} is the natural projection.

By [DWZ1, Proposition 10.7], this makes M̃ a representation of Λ′. Moreover, by [DWZ1, Proposition 10.9], the isomorphism class of M̃ does not depend on the choice of the splittings ρ and σ.

We next want to define an associated map on morphisms. Assume that we have two representations M and M′ of Λ, and a map f: M → M′, that is locally a commutative diagram

\xymatrix{ &  & M_{k} \ar[dr]^{β} \ar[dd]_{f_{k}} \\ M \ar[dd]_{f} & M_{in} \ar[ur]^{α} \ar[dd]_{f_{in}} &  & M_{out} \ar[ll]^{γ} \ar[dd]^{f_{out}} \\  &  & M′_{k} \ar[dr]^{β′} \\ M′ & M′_{in} \ar[ur]^{α′} &  & M′_{out}. \ar[ll]^{γ′}}  (9)

We want to define a map M̃ → M̃′. To do so, we need to define a map f̃_{k}: M̃_{k} → M̃′_{k} such that the squares of the following diagram commute:

\xymatrix{ &  & M̃_{k} \ar[dl]_{β̃} \ar[dd]_{f̃_{k}} \\ M̃ \ar[dd]_{f̃} & M_{in} \ar[rr]^{αβ} \ar[dd]_{f_{in}} &  & M_{out} \ar[ll]^{γ} \ar[dd]^{f_{out}} \ar[ul]_{α̃} \\  &  & M̃′_{k} \ar[dl]_{β̃′} \\ M̃′ & M′_{in} \ar[rr]^{α′β′} &  & M′_{out} \ar[ll]^{γ′} \ar[ul]_{α̃′}}

Remark From the first diagram we have induced maps Ker γ → Ker γ′, Ker α → Ker α′, Im γ → Im γ′, Im β → Im β′, \frac{Ker γ}{Im β} → \frac{Ker γ′}{Im β′} and \frac{Ker α}{Im γ} → \frac{Ker α′}{Im γ′}. This again

induces componentwise a natural map M̃_{k} → M̃′_{k}. However, this map turns out not to work for our purposes, as the following example shows.

Example Consider the Jacobian algebra P(Q, W) given by the quiver

\xymatrix{ & • \ar[dr]^{β} \\ • \ar[ur]^{α} &  & • \ar[ll]^{γ}}

and the potential W = αβγ. Then μ̃_{k}(P(Q, W)) = (Q̃, W̃), where Q̃ is the quiver

\xymatrix{ & • \ar[dl]_{α̃} \\ • \ar[rr]^{[αβ]} &  & • \ar[ll]^{γ} \ar[ul]_{β̃}}

and W̃ is the the potential [αβ]γ + [αβ]α̃β̃. Consider a nonzero map between P(Q, W)-modules

[figure: 0; 0; K; K; id; 0; K]

The image of this map in f. l. P(Q̃, W̃) would be

[figure: (0, K, 0); h; (0, 0, K); K; 0; K; id; 0; K]

where h is the zero-map, which indicates that this is not the map we are looking for.

In view of the above, we now proceed to define a new map. Considering the expression

\frac{Ker γ}{Im β} ⊕ Im γ ⊕ \frac{Ker α}{Im γ},

we observe that the two first terms extend to Cok β, while the two last terms extend to Ker α. We have a natural induced map Cok β → Ker α. Taking these features into account we consider the knitting of short exact sequences

[figure: 0; \frac{Ker γ}{Im β}; ĩ; Cok β; γ̃ι; γ̃; Ker α; φ; \frac{Ker α}{Im γ}; 0; Im γ; ι; 0; 0]

Moreover, ĩ: \frac{Ker γ}{Im β} → Cok β and γ̃: Cok β → Im γ are the induced maps.

Considering the induced map ρ̃: Cok β → \frac{Ker γ}{Im β}, we get a knitting of (K-split) short exact sequences

[figure: 0; \frac{Ker γ}{Im β}; ĩ; Cok β; ρ̃; γ̃ι; γ̃; Ker α; φ; ε; \frac{Ker α}{Im γ}; σ; 0; Im γ; ι; j; 0; 0]

where the maps j and ε are induced from the splittings ρ̃ and σ respectively, so that id_{Cok β} = ρ̃ĩ+ γ̃j and id_{Ker α} = ει + φσ.

The idea is then to use the diagram

[figure: \frac{Ker α}{Im γ}; σ; \frac{Ker α′}{Im γ′}; ⊕; Ker α; f_{in}; Ker α′; φ′; ε′; ⊕; Im γ; ι; j; Im γ′; ⊕; Cok β; f̃_{out}; γ̃; Cok β′; γ̃′; γ̃′; ρ̃′; ⊕; \frac{Ker γ}{Im β}; ĩ; \frac{Ker γ′}{Im β′}]

where f̃_{out} is the induced quotient map and where the middle part commutes, that is

ι ∘ f_{in} ∘ε′ = j ∘ f̃_{out} ∘ γ̃′.

We let

f̃_{k} = ( \begin{matrix}ĩ∘ f̃_{out} ∘ ρ̃′ & ĩ∘ f̃_{out} ∘ γ̃′ & 0 \\ j ∘ f̃_{out} ∘ ρ̃′ & ι ∘ f_{in} ∘ε′ & ι ∘ f_{in} ∘φ′ \\ 0 & σ ∘ f_{in} ∘ε′ & σ ∘ f_{in} ∘φ′ \end{matrix} ) =

ĩ∘ f̃_{out} ∘ ρ̃′  0  0
j ∘ f̃_{out} ∘ ρ̃′ ι ∘ f_{in} ∘ε′  0
0  σ ∘ f_{in} ∘ε′ σ ∘ f_{in} ∘φ′

since ι ∘ f_{in} ∘φ′ = 0 and ĩ ∘ f̃_{out} ∘ γ̃′ = 0. It is however often convenient to use the first expression of f̃_{k}.

We need to verify that β̃ ∘ f_{in} = f̃_{k} ∘ β̃′ and α̃ ∘ f̃_{k} = f_{out} ∘α̃′.

MUTATION OF CLUSTER-TILTING OBJECTS AND POTENTIALS 33

First,

f̃_{k} ∘ β̃′ = ( \begin{matrix}ĩ∘ f̃_{out} ∘ ρ̃′ & ĩ∘ f̃_{out} ∘ γ̃′ & 0 \\ j ∘ f̃_{out} ∘ ρ̃′ & ι ∘ f_{in} ∘ε′ & ι ∘ f_{in} ∘φ′ \\ 0 & σ ∘ f_{in} ∘ε′ & σ ∘ f_{in} ∘φ′ \end{matrix} ) ( \begin{matrix}0 \\ ι′ \\ σ′ \end{matrix} )
= ( \begin{matrix}0 \\ ι ∘ f_{in} ∘(ε′ ∘ ι′ + φ′ ∘ σ′) \\ σ ∘ f_{in} ∘(ε′ ∘ ι′ + φ′ ∘ σ′) \end{matrix} )
= ( \begin{matrix}0 \\ ι ∘ f_{in} \\ σ ∘ f_{in} \end{matrix} )
= β̃ ∘ f_{in}

On one hand, we have

f_{out} ∘α̃′ = (− f_{out} ∘ρ′ ∘ π′, − f_{out} ∘γ′, 0).

On the other hand,

α̃ ∘ f̃_{k} = (−ρπ, −γ, 0) ( \begin{matrix}ĩ∘ f̃_{out} ∘ ρ̃′ & 0 & 0 \\ j ∘ f̃_{out} ∘ ρ̃′ & ι ∘ f_{in} ∘ε′ & 0 \\ 0 & σ ∘ f_{in} ∘ε′ & σ ∘ f_{in} ∘φ′ \end{matrix} )
= (−ρ ∘ π ∘ ĩ∘ f̃_{out} ∘ ρ̃′ − γ ∘ j ∘ f̃_{out} ∘ ρ̃′, −γ ∘ ι ∘ f_{in} ∘ε′, 0)

Since −γ ∘ ι ∘ f_{in} ∘ε′ = −γ ∘ f_{in} ∘ι′ ∘ ε′ = −γ ∘ f_{in} = − f_{out} ∘γ′, it suffices (to get α̃ ∘ f̃_{k} = f_{out} ∘α̃′) to show that

−ρ ∘ π ∘ ĩ∘ f̃_{out} ∘ ρ̃′ − γ ∘ j ∘ f̃_{out} ∘ ρ̃′ = − f_{out} ∘ρ′ ∘ π′.

But, by the definition of ρ̃ and γ̃, the diagrams

[figure: M_{out}; ρ; π; Ker γ; π; Cok β; ρ̃; \frac{Ker γ}{Im β}; and; M_{out}; γ; π; Im γ; Cok β; γ̃]

are commutative (where, by abuse of notation, we also denote by π the natural projection M_{out} → Cok β). Therefore,

−ρ ∘ π ∘ ĩ∘ f̃_{out} ∘ ρ̃′ − γ ∘ j ∘ f̃_{out} ∘ ρ̃′ = −(ρ ∘ π ∘ ĩ+ γ ∘ j) ∘ f̃_{out} ∘ ρ̃′
= −(π ∘ ρ̃∘ ĩ+ π ∘ γ̃ ∘ j) ∘ f̃_{out} ∘ ρ̃′
= −π ∘ (ρ̃∘ ĩ+ γ̃ ∘ j) ∘ f̃_{out} ∘ ρ̃′
= −π ∘ id_{Cok β} ∘ f̃_{out} ∘ ρ̃′
= −π ∘ f̃_{out} ∘ ρ̃′
= − f_{out} ∘ρ′ ∘ π′

where the last equality follows from the commutativity of

[figure: M_{out}; f_{out}; π; M′_{out}; ρ′; π′; Ker γ′; π′; Cok β; f̃_{out}; Cok β′; ρ̃′; \frac{Ker γ′}{Im β′}]

Note that the image f̃apparently depends on the choice of the choice of splittings ρ, σ, ρ′ and σ′. However, we only aim to show that we obtain a functor f. l. Λ → \frac{f.l. Λ′}{[add S′_{k}]}.

In [DWZ1] it is proved that different choice of splittings will give isomorphic objects, and for each vertex i ≠ k, the isomorphism is given by id: M_{i} → M̃_{i}. Therefore, different choices of splittings will give maps which are equal in \frac{f.l. Λ′}{[add S′_{k}]}.

So, we get a well defined map of morphisms from f. l. Λ to \frac{f.l. Λ′}{[add S′_{k}]}.

We next want to show that the definition of G on objects and morphisms from f. l. Λ to \frac{f.l. Λ′}{[add S′_{k}]} actually gives rise to a functor, and that we get an induced equivalence

\frac{f. l. Λ}{[add S_{k}]} → \frac{f. l. Λ′}{[add S′_{k}]}.

We first observe that for M in f. l. Λ we have F (id_{M}) = id_{F M}. This follows from σφ = id_{\frac{Ker α}{Im γ}}, ĩρ̃ = id_{\frac{Ker γ}{Im β}} and ιε = id_{Im γ}. Then, let f: M → M′ and f′: M′ → M′′

be two maps in f. l. Λ. We will compare { \atop f ∘ f′} and f̃∘ f̃′. By construction, these maps are obtained from the following diagrams:

[figure: { \atop f ∘ f′}; f̃∘ f̃′; \frac{Ker γ}{Im β}; ĩ; ⊕; Im γ; j; ι; ⊕; \frac{Ker α}{Im γ}; σ; \frac{Ker γ}{Im β}; ĩ; ⊕; Im γ; j; ι; ⊕; \frac{Ker α}{Im γ}; σ; Cok β;  ; f_{out} ∘ f_{out}′; Ker α; f_{in} ∘ f_{in}′; Cok β; f̃_{out}; Ker α; f_{in}; Cok β′; ρ̃′; γ̃′; Ker α′; ε′; φ′; \frac{Ker γ′}{Im β′}; ĩ′; ⊕; Im γ′; j′; ι′; ⊕; \frac{Ker α′}{Im γ′}; σ′; Cok β′; f̃_{out}′; Ker α′; _{f_{in}_{′}}; Cok β′′; ρ̃′′; γ̃′′; Ker α′; ε′′; φ′′; Cok β′′; ρ̃′′; γ̃′′; Ker α′; ε′′; φ′′; \frac{Ker γ′′}{Im β′′}; ⊕; Im γ′′; ⊕; \frac{Ker α′′}{Im γ′′}; \frac{Ker γ′′}{Im β′′}; ⊕; Im γ′′; ⊕; \frac{Ker α′′}{Im γ′′}]

and easy computations show that, locally, the morphism f̃∘ f̃′ − { \atop f ∘ f′} is given by

\xymatrix{ & \frac{Ker γ}{Im β} ⊕ Im γ ⊕ \frac{Ker α}{Im γ} \ar@{-->}[dl]_{β̃} \ar[dd]^{F_{k}} \\ M_{in} \ar[dd]_{0} \ar[rr]^{αβ} &  & M_{out} \ar[ll]^{γ} \ar[ul]_{α̃} \ar[dd]^{0} \\  & \frac{Ker γ′′}{Im β′′} ⊕ Im γ′′ ⊕ \frac{Ker α′′}{Im γ′′} \ar@{-->}[dl]_{β̃′′} \\ M′′_{in} \ar[rr]^{α′′β′′} &  & M′′_{out} \ar[ll]^{γ′′} \ar[ul]_{α̃′′}}

where F_{k} = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ σ ∘ f_{in} ∘ ε′ ∘ j′ ∘ f̃_{out}′ ∘ ρ̃′′ & 0 & 0 \end{matrix} ) . Therefore, the image of f̃∘ f̃′ −

{ \atop f ∘ f′} lies in add S′_{k}, showing that we have a functor f. l. Λ → \frac{f.l. Λ′}{[add S′_{k}]}.

We have F (S_{k}) = 0, hence there is a functor

\frac{f. l. Λ}{[add S_{k}]} → \frac{f. l. Λ′}{[add S′_{k}]}.

We now verify that this functor is an equivalence. By (the proof of) [DWZ1, Theorem 10.13], this functor is a bijection at the level of representations: in particular it is dense. To show that it is a bijection at the level of the morphisms, suppose

that we have a morphism of representations as in diagram (9). By construction ˜ f̃ is of the form:

\xymatrix{ &  & ˜M̃_{k} \ar[dr]^{β=˜β̃} \ar[dd]_{˜f̃_{k}} \\ ˜M̃ \ar[dd]_{˜f̃} & M_{in} \ar[rr]^{αβ} \ar[ur]^{α=˜α̃} \ar[dd]_{f_{in}} &  & M_{out} \ar[ll]|{γ} \ar[ll]^{_{α̃β̃}} \ar[dd]^{f_{out}} \\  &  & ˜M̃′_{k} \ar[dr]^{β′= ˜β̃′} \\ M′ & M′_{in} \ar[rr]^{α′β′} \ar[ur]^{α′= ˜α̃′} &  & M′_{out}. \ar[ll]^{γ′} \ar[ll]^{_{α̃′β̃′}}}

By [DWZ1], M is isomorphic to the reduced form μ_{k}(M) of ˜ M̃. To obtain this reduced form, one applies three steps; see [DWZ1, Theorem 5.7, Steps 1-3]. It turns out that only the first step has an impact on α, β and γ, and this modification consists of changing the map β to −β. So, applying these steps does not change the commutativity of the above diagram, saying that after the reduction, ˜ f̃ is still a morphism for the reduced representation.

Now, since f − ˜ f̃ is (possibly) not zero only at position k, then f and ˜ f̃ coincide in \frac{f.l. Λ}{[add S_{k}]}, showing the bijection at the level of morphisms.

Hence we have proved the following.

Theorem 7.1. The Jacobian algebras Λ = P(Q, W) and Λ′ = P(μ_{k}(Q′, W′)) are nearly Morita equivalent.

For Jacobi-finite QP’s this result can be obtained with a very different approach by combining the result of [A] saying that these algebras are 2-CY-tilted with the corresponding result for 2-CY-tilted algebras [BMR1, KR1].

References

[A] Amiot C. Cluster categories for algebras of global dimension 2 and quivers with potential, arXiv:0805.1035.

[ABS] Assem I., Brüstle T., Schiffler R. Cluster-tilted algebras as trivial extensions, Bull. London Math. Soc. 40 (2008), no.1, 151–162

[APR] Auslander M, Platzeck M.I., Reiten I. Coxeter functors without diagrams, Trans. Amer. Math. Soc. 259 (1979), 1–46

[BGP] Bernstein I.N., Gelfand I.M., Ponomarev V.A Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no.2 (170), 19–33

[Boc] Bocklandt R. Graded Calabi-Yau algebras of dimension 3, J. Pure Appl. Algebra 212, (2008), no.1, 14–32

[Bon] Bongartz K. Algebras and quadratic forms, J. London Math. Soc. (2) 28 (1983), no.3, 461–469

[BG] Bongartz K., Gabriel P. Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378

[BIRS] Buan A., Iyama O., Reiten I., Scott J. Cluster structures for 2-Calabi-Yau categories and unipotent groups, arXiv:math/0701557.

[BMR1] Buan A., Marsh R., Reiten I. Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323–332

[BMR2] Buan A., Marsh R., Reiten I. Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2008), no. 1, 143–177

[BMR3] Buan A., Marsh R., Reiten I. Cluster-tilted algebras of finite representation type, J. Algebra, Volume 306 (2006), Issue 2, 412–431

[BMRRT] Buan A., Marsh R., Reineke M., Reiten I., Todorov G. Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618

[BR] Buan A., Reiten I. From tilted to cluster-tilted algebras of Dynkin type, preprint arxiv:math/0510445

[BIKR] Burban I., Iyama O., Keller B., Reiten I. Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 (2008), no. 6, 2443–2484.

[CR] Chuang J., Rouquier R., in progress

[DWZ1] Derksen H., Weyman J., Zelevinsky A. Quivers with potentials and their representations I: Mutations, arXiv:0704.0649, to appear in Selecta Math. (N.S.)

[DWZ2] Derksen H., Weyman J., Zelevinsky A. Quivers with potentials and their representations II, in preparation

[FZ] Fomin S., Zelevinsky A. Cluster Algebras I: Foundations, J. Amer. Math. Soc. 15, no. 2 (2002), 497–529

[GLS1] Geiss C., Leclerc B., Schröer J. Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589–632

[GLS2] Geiss C., Leclerc B., Schröer J. Cluster algebra structures and semicanonical bases for unipotent groups, arXiv:math/0703039

[G] Ginzburg V. Calabi-Yau algebras, arXiv:math/0612139 [H] Happel D. Triangulated categories in the representation theory of finite-dimensional alge-

bras, London Mathematical Society Lecture Note Series, 119. Cambridge University Press, Cambridge, 1988

[HU] Happel D., Unger L. On the set of tilting objects in hereditary categories, Representations of algebras and related topics, Fields Inst. Commun., 45, Amer. Math. Soc., Providence, RI, , 141–159, (2005)

[IT1] Igusa K., Todorov G. Radical layers of representable functors, J. Algebra 89 (1984), no. 1, 105–147

[IT2] Igusa K., Todorov G. A characterization of finite Auslander-Reiten quivers, J. Algebra 89 (1984), no. 1, 148–177.

[I1] Iyama O. τ-categories. I,II,III. Algebr. Represent. Theory 8 (2005), no. 3, 297–321; ibid. 8 (2005), no. 4, 449–477; ibid. 8 (2005), no. 5, 601–619

[I2] Iyama O. Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22–50

[I3] Iyama O. Cluster tilting in 2-Calabi-Yau categories II, Oberwolfach Report no.9 (2008)

[IR] Iyama O., Reiten I. Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, arXiv:math/0605136, to appear in Amer. J. Math.

[IY] Iyama O., Yoshino Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), no. 1, 117–168

[K1] Keller B. On triangulated orbit categories, Doc. Math. 10 (2005), 551–581

[K2] Keller B. CY-completions and their duals, in preparation

[KR1] Keller B., Reiten I. Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math. 211 (1), (2007), 123–151

[KR2] Keller B., Reiten I. Acyclic Calabi-Yau categories, preprint arxiv:math.RT/0610594, to appear in Compos. Math.

[MRZ] Marsh R., Reineke M., Zelevinsky A. Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4171–4186

[Re] Reiten I. Cluster tilting in 2-Calabi-Yau categories I, Oberwolfach Report no.9 (2008)

[Rie] Riedtmann C. Algebren, Darstellungskocher, Uberlagerungen und zuruck, Comment. Math. Helv. 55 (1980), no. 2, 199–224

[Rin] Ringel C.M. Some Remarks Concerning Tilting Modules and Tilted Algebras. Origin. Relevance. Future. (An appendix to the Handbook of Tilting Theory), Cambridge University Press (2007), LMS Lecture Notes Series 332

[Y] Yoshino Y. Universal lifts of chain complexes over non-commutative parameter algebras, arXiv:0712.3110.

Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway

   E-mail address: aslakb@math.ntnu.no

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

   E-mail address: iyama@math.nagoya-u.ac.jp

Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway

   E-mail address: idunr@math.ntnu.no

Department of Mathematics, Bishop’s University, Sherbrooke, QC, Canada, J1MOC8

   E-mail address: dsmith@ubishops.ca