n-representation-finite algebras and n-APR tilting

We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting, and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure, and use this to completely describe a class of n-representation-finite algebras called"type A".

1. Introduction 1 Acknowledgments 2 2. Background and notation 2 2.1. n-representation-finiteness 2 2.2. Derived categories and n-cluster tilting 3 2.3. n-Amiot-cluster categories and (n + 1)-preprojective algebras 4 3. n-APR tilting 4 3.1. n-APR tilting modules 4 3.2. n-APR tilting as one-point extension 7 3.3. Quivers for 2-APR tilts 8 3.4. n-APR tilting complexes 10 4. n-APR tilting for n-representation-finite algebras 13 4.1. n-APR tilting modules preserve n-representation-finiteness 13 4.2. n-APR tilting complexes preserve n-representation-finiteness One of the highlights in representation theory of algebras is given by representation-finite algebras, which provide a prototype of the use of functorial methods in representation theory. In 1971, Auslander gave a one-to-one correspondence between representation-finite algebras and Auslander algebras, which was a milestone in modern representation theory leading to later Auslander-Reiten theory. Many categorical properties of module categories can be understood as analogues of homological properties of Auslander algebras, and vice versa.
To study higher Auslander algebras, the notion of n-cluster tilting subcategories (=maximal (n − 1)-orthogonal subcategories) was introduced in [Iya3], and a higher analogue of Auslander-Reiten theory was developed in a series of papers [Iya1,Iya2,IO], see also the survey paper [Iya4]. Recent results (in particular [Iya1], but also this paper and [HI,HZ1,HZ2,HZ3,IO]) suggest, that n-cluster tilting modules behave very nicely if the algebra has global dimension n. In this paper, we call such algebras n-representation-finite and study them from the viewpoint of APR (=Auslander-Platzeck-Reiten) tilting theory (see [APR]).
For the case n = 1, 1-representation-finite algebras are representation-finite hereditary algebras. In the representation theory of path algebras, the notion of Bernstein-Gelfand-Ponomarev reflection functors play an important role. Nowadays they are formulated in terms of APR tilting modules from a functorial viewpoint (see [APR]). A main property is that the class of representation-finite hereditary algebras is closed under taking endomorphism algebras of APR tilting modules. By iterating the APR tilting process, we get a family of path algebras with the same underlying graph with different orientations We follow this idea to construct from one given n-representation-finite algebra a family of n-representation-finite algebras. We introduce the general notion of n-APR tilting modules, which are explicitly constructed tilting modules associated with simple projective modules. The difference from the case n = 1 is that we need a certain vanishing condition of extension groups, but this is always satisfied if Λ is n-representation-finite.
In Section 3 we introduce n-APR tilting. We first introduce n-APR tilting modules. We give descriptions of the n-APR tilted algebra in terms of one-point (co)extensions (see Subsection 3.2, in particular Theorem 3.8), and for n = 2 also in terms of quivers with relations (see Subsection 3.3, in particular Theorem 3.11). Finally we introduce n-APR tilting in derived categories.
In Section 4 we apply n-APR tilts to n-representation-finite algebras. The first main result is that n-APR tilting preserves n-representation-finiteness (Theorems 4.2 and 4.7). In Subsections 4.3 and 4.4 we introduce the notions of slices and admissible sets in order to gain a better understanding of what algebras are iterated n-APR tilts of a given n-representation-finite algebra. More precisely we show that the iterated n-APR tilts are precisely the quotients of an explicitly constructed algebra by admissible sets (Theorem 4.23).
As an application of our general n-APR tilting theory, in Section 5, we give a family of nrepresentation-finite algebras by an explicit quivers with relations, which are iterated n-APR tilts of higher Auslander algebras given in [Iya1]. We call them n-representation-finite algebras of type A, since, for the case n = 1, they are path algebras of type A s with arbitrary orientation. As shown in Section 4 in general, they form a family indexed by admissible sets. In contrast to the general setup, for type A we have a very simple combinatorial description of admissible sets (we call sets satisfying this description cuts until we can show that they coincide with admissible sets -see Definition 5.3 and Remark 5.13). Then the n-APR tiling process can be written purely combinatorially in terms of 'mutation' of admissible sets, and we can give a purely combinatorial proof of the fact that all admissible sets are transitive under successive mutation.
Summing up with results in [IO], we obtain self-injective weakly (n + 1)-representation-finite algebras as (n + 1)-preprojective algebras of n-representation-finite algebras of type A. This is a generalization of a result of Geiss, Leclerc, and Schröer [GLS1], saying that preprojective algebras of type A are weakly 2-representation-finite.
Note that a 1-cluster tilting object is just an additive generator of the module category.
Definition 2.2. Let Λ be a finite dimensional algebra. We say Λ is weakly n-representationfinite if there exists an n-cluster tilting object in mod Λ. If moreover gl.dim Λ ≤ n we say that Λ is n-representation-finite.
The main aim of this paper is to understand better n-representation-finite algebras, and to construct larger families of examples.
(See [ARS] for definitions and properties of the functors Tr, D, and τ 1 .) Iya3]). Let M be an n-cluster tilting object in mod Λ.
• We have an equivalence τ n : add M add M with a quasi-inverse τ − n : add M add M . • We have functorial isomorphisms Hom Λ (τ − n Y, X) ∼ = D Ext n Λ (X, Y ) ∼ = Hom Λ (Y, τ n X) for any X, Y ∈ add M . • If gl.dim Λ ≤ n then add M = add{τ −i n Λ | i ∈ N} = add{τ i n DΛ | i ∈ N}. We have the following criterion for n-representation-finiteness: Proposition 2.4 ([Iya3, Theorem 5.1(3)]). Let Λ be a finite dimensional algebra and n ≥ 1. Let M be an n-rigid generator-cogenerator. The following conditions are equivalent.
(1) M is an n-cluster tilting object in mod Λ.
(3) For any indecomposable object X ∈ add M , there exists an exact sequence 0 M n · · · M 0 f X with M i ∈ add M and a right almost split map f in add M .

2.2.
Derived categories and n-cluster tilting. Let Λ be a finite dimensional algebra of finite global dimension. We denote by D Λ := D b (mod Λ) the bounded derived category of mod Λ. We denote by the Nakayama-functor in D Λ . Clearly ν restricts to the usual Nakayama functor ν : add Λ add DΛ.
We denote by ν n the n-th desuspension of ν, that is Iya1,Theorem 1.22]). Let Λ be an algebra of gl.dim Λ ≤ n, such that τ −i n Λ = 0 for sufficiently large i. Then the category U is an n-cluster tilting subcategory of D Λ .
In particular, if Λ is n-representation-finite, then U is n-cluster tilting.
We have the following criterion for n-representation-finiteness in terms of the derived category: Theorem 2.6 ([IO, Theorem 3.1]). Let Λ be an algebra with gl.dim Λ ≤ n. Then the following are equivalent.
Definition 2.7 (see [Ami1,Ami2]). We denote by D Λ /ν n the orbit category, that We denote by C n Λ the n-Amiot-cluster category, that is the triangulated hull (see [Ami1,Ami2] we do not give a definition because for the purposes in this paper it does not matter if we think of the orbit category or the n-Amiot-cluster category). We denote by π : D Λ C n Λ the functor induced by projection onto the orbit category.
Lemma 2.8 (Amiot [Ami1,Ami2]). Let Λ be an algebra with gl.dim Λ ≤ n. The n-Amiot-cluster category C n Λ is Hom-finite if and only if τ −i n Λ = 0 for sufficiently large i. In particular it is Hom-finite for any n-representation-finite algebra.
Theorem 2.9 (Amiot [Ami1,Ami2]). Let Λ be an algebra with gl.dim Λ ≤ n such that C n Λ is Hom-finite. Then πΛ is an n-cluster tilting object in C n Λ . Observation 2.10. Note that add πΛ is the image of U under the functor of the derived category to the n-Amiot-cluster category as indicated in the following diagram.

n-APR tilting
In this section we introduce n-APR tilting, and prove some general properties. In Subsection 3.1 we introduce the notion of (weak) n-APR tilting modules and study their basic properties.
In Subsection 3.2 we will give a concrete description of the n-APR tilted algebra in terms of one-point (co)extensions. Namely, if Λ is a one-point coextension of End Λ (Q) op by a module M , then the n-APR tilt is the one-point extension of End Λ (Q) op by Tr n−1 M . This result will allow us to give an explicit description of the quivers and relations in case n = 2 in Subsection 3.3.
Finally, in Subsection 3.4, we introduce a version version of APR tilting in the language of derived categories.
Definition 3.1. Let Λ be a basic finite dimensional algebra and n ≥ 1. Let P be a simple projective Λ-module satisfying Ext i Λ (DΛ, P ) = 0 for any 0 ≤ i < n. We decompose Λ = P ⊕ Q. We call T := (τ − n P ) ⊕ Q the weak n-APR tilting module associated with P . If moreover id P = n, then we call T an n-APR tilting module, and we call End Λ (T ) op an n-APR tilt of Λ.
The more general notion of n-BB tilting modules has been introduced in [HX].
The following result shows that weak n-APR tilting modules are in fact tilting Λ-modules.
Theorem 3.2. Let Λ be a basic finite dimensional algebra, and let T be a weak n-APR tilting Λ-module (as in Definition 3.1). Then T is a tilting Λ-module with pd Λ T = n.
We also have the following useful properties.
(2) If moreover T is n-APR tilting, then Hom Λ (τ − n P, Λ) = 0. For the proof of Theorem 3.2 and Proposition 3.3, we use the following observation on tilting mutation due to Riedtmann-Schofield [RS].
Lemma 3.4 (Riedtmann-Schofield [RS]). Let T be a Λ-module and Y g T ′ f X an exact sequence with T ′ ∈ add T . Then the following conditions are equivalent.
• T ⊕ X is a tilting Λ-module and f is a right (add T )-approximation.
• T ⊕ Y is a tilting Λ-module and g is a left (add T )-approximation.
Proof of Theorem 3.2 and Proposition 3.3. Take a minimal injective resolution (1) 0 P I 0 · · · I n−1 g I n .
By definition the homology in its rightmost term is τ − n P , and since Ext i Λ (DΛ, P ) = 0 for 0 ≤ i < n all other homologies vanish. Since (DI 0 ) * is an indecomposable projective Λ-module with top(DI 0 ) * = Soc I 0 = P , we have (DI 0 ) * = P . Thus we have an exact sequence Thus we have pd Λ T = n. Since P is a simple projective Λ-module, we have (DI i ) * ∈ add Q for 0 < i ≤ n.
Applying the functor (−) * to the sequence (3), we have an exact sequence (2). Thus we have Proposition 3.3(1). If id P = n, then g in (1) is surjective and Dg in (2) is injective. Since (Dg) * * = Dg we have Note that we have a functorial isomorphism Apply the functors − ⊗ Λ Q and Hom Λ (−, Q) to Sequences (2) and (3) respectively, the above isomorphism gives rise to to a commutative diagram of exact sequences. Thus (3) is a left (add Q)-approximation sequence of P , and we have that T is a tilting Λ-module by using Lemma 3.4 repeatedly.
We recall the following result from tilting theory [Hap]: For a tilting Λ-module T with Γ := End Λ (T ) op , we have functors j (Y, T ) = 0 for any j = i}. Lemma 3.5 (Happel [Hap]). • We now prove the following result, saying that the class of algebras of global dimension at most n is closed under n-APR tilting.
Proof. We only have to show that pd Γ (top F 0 X) ≤ n for any indecomposable X ∈ add T .
(i) First we consider the case X ∈ add Q. Since gl.dim Λ = n, we can take a minimal projective resolution 0 P n · · · P 1 f X top X 0. Since Hom Λ (τ − n P, Λ) = 0 by Proposition 3.3(2), we have that any morphism T X which is not a split epimorphism factors through f .
Applying Hom Λ (T, −), we have an exact sequence since we have Ext i Λ (T, Λ) = 0 for any 0 < i < n by Proposition 3.3(1). Moreover the above observation implies Cok F 0 f = top F 0 X. Thus we have pd Γ (top F 0 X) ≤ n.
(ii) Next we consider the case X = τ − n P . We will show that pd Γ (top F 0 τ − n P ) is precisely n. Applying F 0 to the sequence (3) in the proof of Theorem 3.2, we have an exact sequence Λ (T, Λ) = 0 for any 0 < i < n by Proposition 3.3(1). Since Q, (DI n ) * , and τ − n P are in F 0 , we have a commutative diagram (since Ext i Λ (DΛ, P ) = 0 ∀i ∈ {1, . . . , n − 1}, see for instance [AB]), any non-zero endomorphism of F 0 τ − n P is an automorphism. Thus F 0 f is a right almost split map in add Γ, and we have Cok F 0 f = top F 0 τ − n P and pd Γ (top F 0 τ − n P ) = n. Later we shall use the following observation.
Lemma 3.7. Under the circumstances in Theorem 3.2, we have the following.
(2) F n P is a simple Γ-module. If id P = n, then F n P is an injective Γ-module. Proof.
(1) Follows immediately from Proposition 3.3 and the fact that P is simple.
(2) By AR-duality we have , any composition factor of the Γ-module F n P is isomorphic. Thus we only have to show that End Γ (F n P ) is a division ring. By Lemma 3.5, we have End Γ (F n P ) ∼ = End Λ (P ). Thus the assertion follows.
3.2. n-APR tilting as one-point extension. Let Λ be a finite dimensional algebra, M ∈ mod Λ op and N ∈ mod Λ. Slightly more general than "classical" one-point (co)extensions, we consider the algebras K M Λ and K N Λ if K is a finite skew-field extension of our base field k, such that K ⊆ End Λ op (M ) and K ⊆ End Λ (N ) op , respectively. Now let Λ be a basic algebra which has a simple projective module P . We set K P = End Λ (P ) op . Let Q be the direct sum over the other indecomposable projective Λ-modules, that is Λ = P ⊕ Q. We set Λ P := End Λ (Q) op and M P := Hom Λ (P, Q) ∈ mod(K P ⊗ k Λ op P ). Then we have an isomorphism Λ ∼ = K P M P Λ P and P is identified with the module K P 0 .
Theorem 3.8. Assume Λ is a basic finite dimensional algebra with simple projective module P , and n > 1. Then the following are equivalent: (i) P gives rise to an n-APR tilting module, (ii) M P has the following properties Moreover, if the above conditions are satisfied and Γ = End Λ ((τ − n P ) ⊕ Q) op , then Γ ∼ = K P Tr n−1 M P Λ P Remark 3.9. The object Tr n−1 M P is uniquely determined only up to projective summands. In this section we always understand Tr n−1 M P to be constructed using a minimal projective resolution, or, equivalently, Tr n−1 M P to not have any non-zero projective summands.
Proof of Theorem 3.8. Let be an injective resolution of the Λ P -module DM P . Then the injective resolution of the Λ-module Hence pd Λ op P M P = id Λ P DM P = id Λ P − 1. In particular we have id Λ P = n ⇐⇒ pd Λ op P M P = n − 1.
Moreover, for any i ≥ 1 and any I ∈ inj Λ P we have (Note that the first equality also holds for n = 1, since there are no non-zero maps from 0 I to the injective Λ-module K P DM P .) Finally we look at extensions between P and the corresponding injective module. For i > 1 we have For i = 1 we obtain This proves the equivalence of (i) and (ii).
For the second claim note that by Proposition 3.3(2) we have Hom Λ (τ − n P, Q) = 0. Therefore it only remains to verify Hom Λ (Q, τ − n P ) = Tr n−1 M P and End Λ (τ − n P ) = K P . This follows by looking at the injective resolution of P above and applying D to it to obtain (a projective resolution of) τ − n P .
3.3. Quivers for 2-APR tilts. In this subsection we give an explicit desctription of 2-APR tilts in terms of quivers with relations.
Remark 3.10. For comparison, recall the classical case (n = 1): Assume Λ = kQ/(R), and the set of relations R is minimal (∀r ∈ R : r ∈ (R \ {r})). Simple projective modules correspond to sources of Q. Let P be a simple projective, i ∈ Q 0 the corresponding vertex. Then id P = 1 ⇐⇒ no relation in R involves a path starting in i. In this situation we have where Q ′ is the quiver obtained from Q by reversing all arrows starting in i.
For n = 2 we have to take into account the second cosyzygy of P , which corresponds to relations involving the corresponding vertex of the quiver.
Let Λ = kQ/(R) be a finite dimensional k-algebra presented by a quiver Q = (Q 0 , Q 1 ) with relations R (which is assumed to be a minimal set of relations). Let P be a simple projective Λ-module associated to a source i of Q. We define a quiver Q ′ = (Q ′ 0 , Q ′ 1 ) with relations R ′ as follows.
where r * is a new arrow associated to each r ∈ R with s(r) = i. We write r ∈ R with s(r) = i as and define a * ∈ kQ ′ for each a ∈ Q 1 with s(a) = i by Theorem 3.11. Let Λ = kQ/(R) and P a simple projective Λ-module. Assume that P gives rise to a 2-APR tilting Λ-module T . Then End Λ (T ) is isomorphic to kQ ′ /(R ′ ) (with Q ′ and R ′ as explained above).
Remark 3.12. Roughly speaking, Theorem 3.11 means that arrows in Q starting in i become relations, and relations become arrows.
Let us start with the following general observation.
Observation 3.13. Let ∆ = kQ/(R) be a finite dimensional k-algebra presented by a quiver Q with relations R. Let M be a ∆-module with a projective presentation Now we are ready to prove Theorem 3.11.
Proof of Theorem 3.11. We can write Λ = k M P Λ P as in Subsection 3.2. Let Q P be the quiver obtained from Q by removing the vertex i, and R P := {r ∈ R | s(r) = i}. Then we have (4) Λ P ∼ = kQ P /(R P ).
By Theorem 3.8 we have Since we have a minimal projective resolution Tr M P 0 of the Λ P -module Tr M P . Applying Observation 3.13 to the one-point coextension (5), we have the assertion from (4) and (6).
For example we could take Q to be the Auslander Reiten quiver of A 3 and R to be the mesh relations. Then kQ/(R) is the Auslander algebra. See Tables 1 (linear oriented A 3 ) and 2 (nonlinear oriented A 3 ) for the iterated 2-APR tilts of these Auslander algebras. In the pictures a downward line is a 2-APR tilt. Vertices labeled T are sources that have an associated 2-APR tilt, vertices labeled C are sinks having an associated 2-APR cotilt. Sources and sinks that do not admit a 2-APR tilt or cotilt are marked X.  Note that there are no Xes occurring in Table 1. In fact, by [Iya1, Theorem 1.18] (see Theorem 5.7) the Auslander algebras of linear oriented A n are 2-representation-finite, and hence every source and sink has an associated 2-APR tilt and cotilt, respectively. We will more closely investigate n-APR tilts on n-representation-finite algebras in Section 4, and the particular algebras coming from linear oriented A n in Section 5.
3.4. n-APR tilting complexes. As in Section 2.2, throughout this section we assume Λ to be a basic finite dimensional algebra of finite global dimension. We will constantly use the functors ν and ν n introduced in the first paragraph of Section 2.2.
We call T := (ν − n P ) ⊕ Q the n-APR tilting complex associated with P .
By abuse of notation (see Remark 3.15 below for a justification), we also call End D Λ (T ) op an n-APR tilt of Λ.
(1) Any n-APR tilting module (τ − n P ) ⊕ Q in the sense of Definition 3.1 is an n-APR tilting comples, since in that case ν − n P = τ − n P holds (under the assumption that Λ has finite global dimension).
(2) Weak n-APR tilting modules are in general not n-APR tilting complexes.
Remark 3.16. In the setup of Definition 3.14 there is no big difference between tilting and cotilting: the n-APR tilting complex (ν − n P ) ⊕ Q associated to P , and the n-APR cotilting complex νP ⊕ ν n νQ associated to (the injective module) νQ are mapped to each other by the autoequivalence ν n ν of the derived category.
In the rest of this subsection we will show that n-APR tilting complexes are indeed tilting complexes, and that they preserve gl.dim ≤ n.  Theorem 3.17. Let Λ be an algebra of finite global dimension, and T an n-APR tilting complex (as in Definition 3.14). Then T is a tilting complex in D Λ .
Remark 3.18. More generally, in Theorem 3.17 it is possible to replace the assumption that Λ has finite global dimension by the weaker assumption that P has finite injective dimension. (In this case ν − n P = R Hom Λ (DΛ, P ) is still in K b (proj Λ), the homotopy category of complexes of finitely generated projective Λ-modules.) Proof of Theorem 3.17. We have to check that T has no self-extensions, and that T generates the derived category D Λ . We first check that T has no self-extensions. Clearly for all i = 0 we have Finally Hom D Λ (Q, ν − n P [i]) = Ext n+i Λ (νQ, P ), which vanishes for i = 0 by assumption (2) of the definition. Now we prove that T generates D Λ . Let X ∈ D Λ such that Hom D Λ (ν − P [i], X) = 0 and Hom D Λ (Q[i], X) = 0 for all i. By the latter property we see that the homology of X does not contain any composition factors in add(top Q). We can assume that X is a complex . The following result generalizes Proposition 3.6 to the setup af n-APR tilting complexes.
Proposition 3.19. If gl.dim Λ ≤ n and T is an n-APR tilting complex in D Λ then, for Γ : Proof. By [Ric] the algebra Γ has finite global dimension, and hence , which is non-zero only for i = n. Finally . Since νQ ∈ mod Λ and gl.dim Λ ≤ n it follows that ν 2 Q has non-zero homology only in degrees −n, . . . , 0. Hence the above Hom-space vanishes for i > n, since gl.dim Λ ≤ n.
Summing up we obtain Hom D Λ (νT, T [i]) = 0 for i > n, which implies the claim of the theorem by the remark at the beginning of the proof.
Recall the definition of the subcategory Proposition 3.20. Let Λ be n-representation-finite, and T an n-APR tilting complex in D Λ . Let Proof. This is clear since the derived equivalence R Hom Λ (T, −) commutes with ν n and T ∈ U n Λ . An application of Proposition 3.20 we will use in Subsection 4.4 is the following.
Proof. By Propositions 3.20 and 2.12 we have

n-APR tilting for n-representation-finite algebras
In this section we study the effect of n-APR tilts on n-representation-finite algebras. The first main result is that n-APR tilting preserves n-representation-finiteness (Theorems 4.2 and 4.7). We give two independant proofs for this fact. In Subsection 4.1 we study n-APR tilting modules for n-representation-finite algebras. We give an explicit description of a cluster tilting object in the new module category in terms of the cluster tilting object of the original algebra (Theorem 4.2). In Subsection 4.2 we give an independant proof (which is less explicit and relies heavily on a result from [IO]) that the more general procedure of tilting in n-APR tilting complexes also preserves n-representation-finiteness.
In Subsections 4.3 and 4.4 we introduce the notions of slices and admissible sets, which classify, for a given n-representation-finite algebra, all iterated n-APR tilts (see Theorem 4.23).
Throughout this Section, let Λ be an n-representation-finite algebra. For simplicity of notation we assume Λ to be basic. 4.1. n-APR tilting modules preserve n-representation-finiteness. The following proposition shows that the setup of n-representation-finite algebras is particularly well-suited for looking at n-APR tilts.
Proof. We have id P ≤ n by gl.dim Λ ≤ n. Since M is an n-rigid generator-cogenerator, we have Ext i Λ (DΛ, P ) = 0 for any 0 < i < n. This proves (1), the proof of (2) is dual.
Throughout this subsection, we denote by M the unique basic n-cluster tilting object in mod Λ (see the last point of Proposition 2.3). Now let P be a simple projective and non-injective Λ-module. We decompose Λ = P ⊕ Q. Since P ∈ add M we can also decompose M = P ⊕ M ′ . By Observation 4.1 we have an n-APR tilting Λ-module T := (τ − n P ) ⊕ Q.
Theorem 4.2. Under the above circumstances, we have the following.
Before we prove the theorem let us note the following immediate consequence.
Corollary 4.3. Any iterated n-APR tilt of an n-representation-finite algebra is n-representationfinite.
In the rest we shall show Theorem 4.2. The assertion (1) follows immediately from the first part of Proposition 2.3.
Proposition 3.6 proves that gl.dim Γ = n in Theorem 4.2. We shall show that N in Theorem 4.2(2) is an n-cluster tilting object. We will use the subcategories F i ⊆ mod Λ, and the functors F i which were introduced in Section 3.1 (see in particular Lemma 3.5).
Proof. By Theorem 4.2(1) we know that T ∈ add M . Hence, since M is an n-rigid Λ-module, we have Ext i Λ (T, M ) = 0 for any 0 < i < n. Since gl.dim Λ ≤ n, we only have to check Ext n Λ (T, M ′ ) = 0. Of course, we have Ext n Λ (Q, M ′ ) = 0 since Q is projective. By Proposition 2.3(2), we have Ext n Λ (τ − n P, M ′ ) ∼ = Hom Λ (M ′ , P ), and the latter Hom-space vanishes since P is simple projective.
Proof. We have Ext i Γ (−, F n P ) = 0 for any i > 0 since F n P is injective (Lemma 3.7(2)). Since M ′ ∈ F 0 and P ∈ F n by (1) and Lemma 3.7(1) respectively, we can check the assertion as follows by using Lemma 3.5: For 0 < i < n both of the above vanish, since M is n-rigid.
We now complete the proof of Theorem 4.2.
Proof of Theorem 4.2(2). By Lemma 4.5 we know that N is n-rigid, and hence we may apply Proposition 2.4. We will show that N is n-cluster tilting by checking the third of the equivalent conditions in Proposition 2.4(3).
(i) First we consider F n P . Take a minimal injective resolution 0 P I 0 · · · I n 0.
By Proposition 3.3 we have Ext i Λ (T, P ) = 0 for 0 ≤ i < n. Hence, applying Hom Λ (T, −), we have an exact sequence We shall show that f is a right almost split map in add N . By Lemma 3.5, we have Ext j Γ (F 0 M ′ , F 0 I i ) ∼ = Ext j Λ (M ′ , I i ) = 0 for any i and any j > 0. Using this, we see that the map is surjective. Since F n P is a simple injective Λ-module by Lemma 3.7, any non-zero endomorphism of F n P is an automorphism. Thus f is a right almost split map in add N .
(ii) Next we consider F 0 X for any indecomposable X ∈ add M ′ . Since M is an n-cluster tilting object in mod Λ, we have an exact sequence since we have Ext i Λ (T, M ) = 0 for any 0 < i < n. Since F 0 M i ∈ add N , we only have to show that F 0 f is a right almost split map in add N .
Since F n P is a simple injective Λ-module by Lemma 3.7, there is no non-zero map from F n P to F 0 X. Thus we only have to show that any morphism g : F 0 M ′ F 0 X which is not a split epimorphism factors through f . By Lemma 3.5, we can put g = F 0 h for some h : M ′ X which is not a split epimorphism. Since h factors through f , we have that g = F 0 h factors through F 0 f . Thus we have shown that F 0 f is a right almost split map in add N . 4.2. n-APR tilting complexes preserve n-representation-finiteness. Similar to Observation 4.1 we have the following result for n-representation finite algebras.
We have the following result.
Theorem 4.7. Let Λ be n-representation-finite, and T be an n-APR tilting complex in D Λ . Then End D Λ (T ) op is also n-representation-finite.
Proof. We set Γ = End D Λ (T ) op . By Proposition 3.19 we know that, since gl.dim Λ ≤ n we also have gl.dim Γ ≤ n.
By Proposition 3.20 we know that the derived equivalence D Λ D Γ induces an equivalence U Λ U Γ . Hence, by Theorem 2.6 we have Slices. In this subsection we introduce the notion of slices in the n-cluster tilting subcategory U (see Definition 4.8). The aim is to provide a bijection between these slices and the iterated n-APR tilting complexes of Λ (Theorem 4.15). This will be done by introducing a notion of mutation of slices (Definition 4.12), and by proving that this mutation coincides with n-APR tilts. Throughout, let Λ be an n-representation-finite algebra. We consider the n-cluster tilting subcategory U = U n Λ ⊆ D Λ given in Section 2.2. Definition 4.8. An object S ∈ U is called slice if (1) for any indecomposable projective module P there is exactly one i such that ν i n P ∈ add S, and (2) add S is convex, that means any path (that is, sequence of non-zero maps) in ind U , which starts and ends in add S, lies entirely in add S.
The following two observations give us the slices we are interested in here.
Observation 4.9. In the setup above, Λ ∈ U is a slice, since we have Hom D Λ (ν i n Λ, ν j n Λ) = H 0 (ν j−i n Λ) = 0 if i < j. Similarly, any iterated n-APR tilting complex of Λ is a slice in U , by Theorem 4.7 and Proposition 3.20.
Proposition 4.10. Let S be a slice. Then Hom D Λ (S, ν i n S) = 0 for any i > 0. For the proof we will need the following observation: Lemma 4.11. Assume Λ is indecomposable (as a ring) and not semi-simple. For any indecomposable X ∈ U there is a path ν n X X in U .
Proof. Assume first that X is a non-projective Λ-module. By [Iya1,Theorem 2.2] there is an n-almost split sequence ν n X = τ n X X n−1 · · · X 1 X with X i ∈ U ∩ mod Λ. This sequence gives rise to the desired path ν n X X in U . Now let X ∈ U be arbitrary indecomposable. By [IO,Lemma 4.9] there exists i ∈ Z such that ν i X is a non-projective Λ-module. Then there exists a path ν n ν i X ν i X in U . Since ν is an autoequivalence of U by Theorem 2.6, we have a path ν n X X in U .
Proof of 4.10. We may assume Λ to be connected, and not semi-simple. Then, by the above lemma, for any indecomposable S ′ ∈ add S there is a path ν n S ′ S ′ in U . Hence there are also a paths ν i n S ′ S ′ for i > 0. If Hom D Λ (S, ν i n S ′ ) = 0 for some i > 0, then we have ν i n S ′ ∈ add S by Definition 4.8(2), contradicting 4.8(1).
Proof. We restrict to the case of µ + S ′ (S). It is clear that it satisfies Condition (1) of Definition 4.8. To see that µ + S ′ (S) is convex, let p be a path in ind U starting and ending in µ + S ′ (S). We have the following four cases with respect to where p starts and ends: (1) If p starts and ends in S ′′ then it lies entirely in S. Since Hom D Λ (S ′′ , S ′ ) = 0 it lies entirely in S ′′ .
(2) Similarly, if p starts and ends in ν − n S ′ then it lies entirely in ν − n S ′ . (3) By Proposition 4.10 p cannot start in ν − n S ′ and end in S ′′ . (4) Assume that p starts in S ′′ and ends in ν − n S ′ . Hence, by Proposition 4.10 the path p to lies entirely in S ⊕ ν − n S. Then, since Hom D Λ (S ′′ , S ′ ) = 0, it can pass neither through S ′ nor through ν − n S ′′ . Therefore it lies entirely in our slice. Thus also Condition (2) of Definition 4.8 is satisfied.
(1) Any two slices in U are iterated mutations of each other.
(2) If moreover the quiver of Λ contains no oriented cycles, then any two slices are iterated mutations with respect to sinks or sources of each other.
Proof. Let Λ = P i be a decomposition into indecomposable projectives. We choose d i and e i such that the two slices are ν d i n P i and ν e i n P i , respectively. Since µ + S (S) = ν − n S, we can assume e i > d i for all i. We set I = {i | e i − d i is maximal}, Now for i ∈ I and j ∈ I we have Since by our choice of I we have (e i − d i ) − (e j − d j ) > 0, the above space vanishes by Proposition 4.10. Hence we may mutate, and obtain Repeating this procedure we see that any two slices are iterated mutations of each other. For the proof of the second claim first note that if the quiver of Λ containes no oriented cycles, then neither does the quiver of U . So we can number the indecomposable direct summands of S ′ as S ′ = S ′ 1 ⊕ · · · ⊕ S ′ ℓ such that Hom D Λ (S ′ i , S ′ j ) = 0 for any i > j. Then we have Theorem 4.15. Assume that Λ is n-representation-finite.
(1) The iterated n-APR tilting complexes of Λ are exactly the slices in U .
(2) If moreover the quiver of Λ contains no oriented cycles, then any iterated n-APR tilting complex can be obtained by a sequence of n-APR (co)tilts in the sense of Definition 3.1. Proof.
(1) By Observation 4.9 any iterated n-APR tilt comes from a slice. The converse follows from Lemma 4.14(1) and Observation 4.6.

Admissible sets.
In this subsection we will see that all the endomorphism rings of slices, and hence all the iterated n-APR tilts, of an n-representation-finite algebra are obtained as quotients of the (n + 1)-preprojective algebra (see Definition 2.11).
Lemma 4.16. Let S be a slice in U . Then Hom U (S, ν −i n S) ⊆ Rad i U (S, ν −i n S). Proof. By Theorem 4.15 we may assume S to be the slice Λ. Then the claim follows from Proposition 2.12.
Construction 4.17. For P, Q ∈ add Λ indecomposable we choose Rad End U (Q) op -bimodule, and H(P, Q) ⊆ Rad U (P, Q) such that H(P, Q) is a minimal generating set of We set A(P, Q) = C 0 (P, Q) H(P, Q) ⊆ Hom C n Λ (P, Q) We write C 0 = P,Q C 0 (P, Q) and A = P,Q A(P, Q). Note that by Definition 2.11 the set A(P, Q) generates Rad C n Λ (P, Q)/ Rad 2 C n Λ (P, Q).
If k is algebraically closed, then H consists of the arrows in the quiver of Λ, and C 0 consist of the additional arrows in the quiver of Λ. Thus A consist of all arrows in the quiver of Λ. Proof. This follows from Proposition 2.12 and the definition of C 0 above.
(1) We call C 0 as above the standard admissible set.
We call this set a mutation of C.
(3) An admissible set is a subset of A which is an iterated mutation of the standard admissible set.
We will now investigate the relation of slices in U and admissible sets.
Construction 4.20. Let S = ν s i n P i be a slice in U . We set C S (P i , P j ) = {ϕ ∈ A(P i , P j ) | ϕ is a map P i ν s j −s i −r n P j for some r > 0}. (1) The map C ? : S C S sends slices in U to admissible sets. Moreover any admissible set is of the form C S for some slice S.
(2) C ? commutes with mutations in the following way: whenever S = S ′ ⊕ S ′′ and Λ = Λ ′ ⊕ Λ ′′ such that π(S ′ ) ∼ = π(Λ ′ ) and π(S ′′ ) ∼ = π(Λ ′′ ) (recall that π denotes the map from the derived category to the n-Amiot cluster category -see Definition 2.7). In particular the mutations of slices are defined if and only if the mutations of admissible sets are.
Proof. By definition Λ is a slice, and C Λ = C 0 is the standard admissible set. We now proceed by checking that all these properties are preserved under mutation. Assume we are in the setup of (2), that is S = S ′ ⊕ S ′′ is a slice, Λ = Λ ′ ⊕ Λ ′′ , such that π(S ′ ) ∼ = π(Λ ′ ) and π(S ′′ ) ∼ = π(Λ ′′ ). We may further inductively assume that C S is an admissible set. Therefore, the "in particular"-part of (2) holds. Similar to the arguments above one sees that C µ + S ′ (S) = µ + Λ ′ (C S ). Now the surjectivity in (1) follows from the fact that, by definition, any admissible set is an iterated mutation of the standard admissible set.
Theorem 4.23. Let Λ be n-representation-finite. Then the iterated n-APR tilts of Λ are precisely the algebras of the form Λ/(C), where C is an admissible set.
In particular all these algebras are also n-representation-finite.
Proof. The first part follows from Propositions 4.21, 4.22 and Theorem 4.15. The second part then follows by Theorem 4.7.

n-representation-finite algebras of type A
The aim of this section is to construct n-representation-finite algebras of 'type A'. The starting point (and the reason we call these algebras type A) is the construction of higher Auslander algebras of type A s in [Iya1] (we recall this in Theorem 5.7 here). The main result of this section is Theorem 5.6, which gives an explicit combinatorial description of all iterated n-APR tilts of these higher Auslander algebras by removing certain arrows from a given quiver (see also Definitions 5.1 and 5.3 for the notation used in that theorem).
(2) For n ≥ 1 and s ≥ 1, we define the k-algebra Λ (n,s) to be the quiver algebra of Q (n,s) with the following relations: For any x ∈ Q (n,s) 0 otherwise.
(We will show later that this notation is justified: In Subsection 5.1 we construct algebras Λ (n,s) , such that Λ (n,s) is the (n + 1)-preprojective algebra of Λ (n,s) -see also Proposition 5.48.) Example 5.2.

Remark 5.4.
(1) We will show later (see Remark 5.13) that cuts coincide with admissible sets (as introduced in Definition 4.19).
(1) Clearly the cuts of Q (1,s) correspond bijectively to orientations of the Dynkin diagram A s .
We are now ready to state the main result of this section.
(1) Let Q (n,s) as in Definition 5.1, and let C be a cut. Then the algebra Λ (n,s) C
(2) All these algebras (for fixed (n, s)) are iterated n-APR tilts of one another.
We call the algebras of the form Λ (n,s) C as in the theorem above n-representation-finite of type A. Note that 1-representation-finite algebras of type A are exactly path algebras of Dynkin quivers of type A. See Tables 1, 3, and 4 for the examples (n, s) = (2, 3), (2, 4), and (3, 3), respectively. 5.1. Outline of the proof of Theorem 5.6.
Step 1. Let C 0 be the set of all arrows of type n + 1. This is clearly a cut. We set Λ (n,s) := Λ (n,s) C 0 . For example, Λ (1,s) is a path algebra of the linearly oriented Dynkin quiver A s , and Λ (2,s) is the Auslander algebra of Λ (1,s) . More generally, the following result is shown in [Iya1].
Theorem 5.7 (see [Iya1]). The algebra Λ (n,s) is n-representation-finite. In particular mod Λ (n,s) has a unique basic n-cluster tilting object M (n,s) . We have that is Λ (n+1,s) is the n-Auslander algebra of Λ (n,s) .
Step 2. We now introduce mutation on cuts.
For simplicity of notation, we fix n and s for the rest of this section, and omit all superscripts − (n,s) whenever there is no danger of confusion. (That is, by Q we mean Q (n,s) , by Λ we mean Λ (n,s) and similar.) Definition 5.8. Let C be a cut of Q.
(1) We denote by Q C the quiver obtained by removing all arrows in C from Q.
(2) Let x be a source of the quiver Q C . Define a subset µ + x (C) of Q 1 by removing all arrows in Q ending at x from C and adding all arrows in Q starting at x to C.
(3) Dually, for each sink x of Q C , we get another subset µ − x (C) of Q 1 . We call the process of replacing a cut C by µ + x (C) or µ − x (C), when the conditions of (2) or (3) above are satisfied, mutation of cuts.
We will show in Proposition 5.14 in Subsection 5.2 that mutations of cuts are again cuts.
Observation 5.9. The quiver Q C is the quiver of the algebra kQ/(C).
The following remark explains the relationship between cuts and admissible sets.
(1) Whenever we mention admissible sets, it is implicitly understood that we choose A = Q 1 the set of arrows in Q in Definition 4.19. (It is shown in Subsection 4.4 that the choice of A does not matter there, but with this choice we can easier compare admissible sets and cuts.) (2) When C is a cut and an admissible set, and x is a source of Q C , then the mutations µ + x (C) of C as cut and as admissible set coincide. (3) The standard admissible set C 0 , as defined in Construction 4.17 and Definition 4.19, is identical to the set C 0 defined in Step 1. In particular it is a cut. (4) By (3) and (2) we know that any admissible set is a cut. The converse follows when we have shown that all cuts are iterated mutations of one another (see Theorem 5.11 and Remark 5.13). = =  We need the following purely combinatorial result, which will be proven in Subsections 5.3 to 5.5.
Theorem 5.11. All cuts of Q are successive mutations of one another.
Step 3. Finally, we need the following result which will also be shown in Subsections 5.3 to 5.5.
Proposition 5.12. ( is n-representation-finite if and only if so is Λ C . Now Theorem 5.6 follows: Proof of Theorem 5.6. By Theorem 5.7, there is a cut C 0 such that Λ C 0 is strictly n-representation-finite. By Proposition 5.12 this property is preserved under mutation of cuts, and by Theorem 5.11 all cuts are iterated mutations of C 0 . Remark 5.13. Theorem 5.11, together with Remark 5.10(2) and (3), shows that in the setup of Definition 5.1 the set of cuts (as defined in 5.3) and the set of admissible sets (as defined in 4.19), coincide.

Mutation of cuts.
In this subsection we show that the mutations µ + x (C) (or µ − x (C)) as in Definition 5.8 for a cut C are cuts again.
Proposition 5.14. In the setup of Definition 5.8 we have the following: (1) Any arrow in Q ending at x belongs to C, and any arrow in Q starting at x does not belong to C.
(2) µ + x (C) is again a cut. (3) x is a sink of the quiver Q µ + x (C) . For the proof we need the following observation, which tells us that any sequence of arrows of pairwise different type may be completed to an (n + 1)-cycle.
Proof of Proposition 5.14.
(1) The former condition is clear since x is a source of Q C . Assume that an arrow a starting at x belongs to C. By Lemma 5.15 we know that a is part of an (n + 1)-cycle c. Then c contains at least two arrows which belong to C, a contradiction.
(2) Let c be an (n + 1)-cycle. We only have to check that exactly one of the (n + 1) arrows in c is contained in µ + x (C). This is clear if x is not contained in c. Assume that x is contained in c, and let a and b be the arrows in c ending and starting in x, respectively. Since C is a cut a is the unique arrow in c contained in C. Thus b is the unique arrow in c contained in µ + x (C). (3) Clear from (1). 5.3. n-cluster tilting in derived categories. This and the following two subsections are dedicated to the proofs of Theorem 5.11 and Proposition 5.12.
We consider a covering Q of Q, then introduce the notion of slices (see Definition 5.20) in Q, and their mutation. Then we construct a correspondence between cuts and ν n -orbits of slices (Theorem 5.24) and show that slices are transitive under mutations (Theorem 5.27). These results are the key steps of the proofs of Theorem 5.11 and Proposition 5.12.
We give the conceptual part of the proof in this subsection, and postpone the proof of the combinatorial parts (Theorems 5.24 and 5.27) to Subsection 5.4.
We recall the subcategory U = add{ν i n Λ | i ∈ Z} of D Λ (see Subsection 2.2).
(1) The n-cluster tilting subcategory U of D Λ is presented by the quiver Q with relations as in 5.16.
We now carry over the concept of slices to the quiver-setup.
Definition 5.20. A slice of Q is a full subquiver S of Q satisfying the following conditions.
(1) Any ν n -orbit in Q contains precisely one vertex which belongs to S.
(2) S is convex, i.e. for any path p in Q connecting two vertices in S, all vertices appearing in p belong to S.
Remark 5.21. Definition 5.20 is just a "quiver version" of Definition 4.8. In particular it is clear that slices in Q and slices in U are in natural bijection.
Next we carry over Construction 4.20 to this combinatorial situation, that is, we produce from any slice in Q a cut C S .
(1) For any slice S in Q, we have a cut (2) π gives an isomorphism S Q C S of quivers. Proof.
(1) Let x 1 be an (n + 1)-cycle in Q. We only have to show that there exists precisely one i ∈ {1, . . . , n + 1} such that the arrow a i does not lie in π(S 1 ).
Now we state the first main assertion of this subsection, which will be proven in the next one.
Theorem 5.24. The correspondence S C S in Proposition 5.22 gives a bijection between ν norbits of slices in Q and cuts in Q.
Let us introduce the following notion.
Definition 5.25. Let S be a slice in Q.
(1) Let x be a source of S. Define a full subquiver µ + x (S) of Q by removing x from S and adding ν − n x.  Table 5. Some slices and corresponding cuts for n = 2 and s = 3 We call the process of replacing a slice S by µ + x (S) or µ − x (S) mutation of slices. Proposition 5.26. In the setup of Definition 5.25(1) we have the following.
(1) Any successor of x in Q belongs to S, and any predecessor of x in Q does not belong to S.
(2) Any successor of ν − n x in Q does not belong to µ + x (S), and any predecessor of ν − n x in Q belongs to µ + x (S).
(1) Let C S be the cut given in Proposition 5.22. Then x is a source of Q C S . By Remark 5.10(1), we have the assertion.
(2) The former assertion follows from the former assertion in (1) and the definition of slice. Take a predecessor y of ν − n x and an integer i such that ν i n y ∈ S 0 . If i > 0, then we have i = 1 since there exists a path from ν i n y to ν − n x passing through ν n y. This is a contradiction to the latter assertion of (1) since ν n y is a predecessor of x. Thus we have i ≤ 0. Since there exists a path from x to ν i n y passing through y, we have y ∈ S 0 . (3) By (2), ν − n x is a sink of µ + x (S). We only have to show that µ + x (S) is convex. We only have to consider paths p in Q starting at a vertex in µ + x (S) and ending at ν − n x. Since any predecessor of ν − n x in Q belongs to S by (2) and S is convex, any vertex appearing in p belongs to µ + x (S). (4) This is clear from (1) and (2).
The following is the second main statement in this section, which will be proven in the next one.
Theorem 5.27. The slices in Q are transitive under successive mutation.
Remark 5.28. Note that one can prove Theorem 5.27 by using the categorical argument in Lemma 4.14. But we will give a purely combinatorial proof in the next subsection since it has its own interest.
Clearly Theorem 5.11 is an immediate consequence of Proposition 5.26(4) and Theorems 5.24 and 5.27 above.
We now work towards a proof of Proposition 5.12. We identify a slice S in Q with the direct sum of all objects in D Λ corresponding to vertices in S.
(2) Let x be a source of S. If S is a tilting complex in D Λ , then µ + x (S) is an n-APR tilting Λ C S -module.
Proof. (1) π gives an isomorphism S C S . It is easily checked that the relations for U correspond to those for Λ.
(2) This is clear from the definition.
Proposition 5.30. For any slice S in Q, the corresponding object S ∈ D Λ is an iterated n-APR tilting complex.
Proof of Proposition 5.12. By Theorem 5.24 there exists a slice S in Q such that C = C S . Take a source y of S such that x = π(y). By Lemma 5.29(1) we can identify Λ C with S. By Lemma 5.29(2) and Proposition 5.30, µ + x (S) is an n-APR tilting Λ C -module with . Thus the assertion follows. 5.4. Proof of Theorems 5.24 and 5.27. In this subsection we give the proofs of Theorems 5.24 and 5.27 which were postponed in Subsection 5.3. We postpone further (to Subsection 5.5) the proof of Proposition 5.33, a technical classification result needed in the proofs here.
We need the following preparation.
(1) We denote by walk(Q) the set of walks in Q (that is, finite sequences of arrows and inverse arrows such that consecutive entries involve matching vertices). For a walk p we denote by s(p) and e(p) the starting and ending vertex of p, respectively. A walk p is called cyclic if s(p) = e(p).
(2) We define an equivalence relation ∼ on walk(Q) as the transitive closure of the following relations: (a) aa −1 ∼ e s(a) and a −1 a ∼ e e(a) for any a ∈ Q 1 . (b) If p ∼ q, then rpr ′ ∼ rqr ′ for any r and r ′ . Similarly we define walk( Q) and the equivalence relation ∼ on walk( Q).
For a walk p = a 1 · · · a n we denote by p −1 := a −1 n · · · a −1 1 the inverse walk. Any map ω : Q 1 A with an abelian group A is naturally extended to a map ω : walk(Q) A by putting ω(a −1 ) := −ω(a) for any a ∈ Q 1 and for any walk p = b 1 · · · b ℓ . We define ω : walk( Q) A by ω(p) := ω(π(p)). Clearly these maps ω : walk(Q) A and ω : walk( Q) A are invariant under the equivalence relation ∼. In particular, we define maps by setting φ i (a) := δ ij for any arrow a of type j in Q.
We will prove Theorems 5.24 and 5.27 by using the following result, which will be shown in the next subsection.
Proposition 5.33. Any cyclic walk on Q belongs to G.
Using this, we will now prove the following proposition, telling us that on Q C the value Φ(p) depends only on s(p) and e(p).
Proposition 5.34. Let C be a cut of Q.
(1) For any cyclic walk p on Q C , we have Φ(p) = 0.
(2) For any walks p and q on Q C satisfying s(p) = s(q) and e(p) = e(q), we have Φ(p) = Φ(q).
To prove Proposition 5.34, we define a map Lemma 5.35. For any cyclic walk p on Q, we have φ C (p) = 0.
We define a map ℓ C : walk(Q) Z by putting ℓ C (a) := 0 if a / ∈ C 1 if a ∈ C for any arrow a ∈ Q 1 .
The following result is clear.
The fact that Q Q is a Galois covering is reflected by the following lemma on lifting of walks.
Lemma 5.37. Fix x 0 ∈ Q 0 and x 0 ∈ Q 0 such that π( x 0 ) = x 0 . For any walk p in Q with s(p) = x 0 , there exists a unique walk p in Q such that s( p) = x 0 and π( p) = p.
Proof. For any x ∈ Q 0 and y ∈ Q 0 such that π(y) = x, the morphism π : Q Q gives a bijection from the set of arrows starting (respectively, ending) at y to the set of arrows starting (respectively, ending) at x. Thus the assertion follows.
We have the following key observation.
Lemma 5.38. Fix x 0 ∈ Q 0 and x 0 ∈ Q 0 such that π( x 0 ) = x 0 . For any walks p and q in Q C satisfying s(p) = s(q) = x 0 and e(p) = e(q), then p and q as given in Lemma 5.37 satisfy e( p) = e( q).
Proof. By our definition of Φ, we have that φ i ( p) counts the number of arrows of type i appearing in p. Since we have φ i ( p) = φ i ( q) by Proposition 5.34, we have that the number of arrows of type i appearing in p is equal to that in q. Since s( p) = s( q), we have e( p) = e( q). Now Theorem 5.24 follows from the following result, which allows us to construct slices from cuts.
Proposition 5.39. Let C be a cut in Q. Fix a vertex x 0 ∈ Q 0 and x 0 ∈ π −1 (x 0 ).
(1) There exists a unique morphism ι : Q C Q of quivers satisfying the following conditions. Proof.
(1) To give the desired morphism ι : Q C Q of quivers, we only have to give a map ι : Q 0 Q 0 between the sets of vertices, satisfying the following conditions. • ι(x 0 ) = x 0 , • the composition π • ι : Q C Q is the identity on Q 0 , • for any arrow a : x y in Q C , there is an arrow ι(x) ι(y) in Q.
We define ι : Q 0 Q 0 as follows: Fix any x ∈ Q 0 . We take any walk p in Q C from x 0 to x. By Lemma 5.37, there exists a unique walk p in Q such that s( p) = x 0 and π( p) = p. Then we put ι(x) := e( p). By Lemma 5.38, ι(x) does not depend on the choice of the walk p.
We only have to check the third condition above. Fix an arrow a : x y in Q C . Take any walk p in Q C from x 0 to x. The walk pa : x 0 y in Q C gives the corresponding walk pa : x 0 ι(y) in Q. Then pa has the form p b for an arrow b : ι(x) ι(y) and a walk p : x 0 ι(x) in Q. Thus the third condition is satisfied.
The uniqueness of ι is clear.
(2) Fix vertices x, y ∈ ι(Q C ) 0 and a path p in Q from x to y. We only have to show that p is a path in ι(Q C ).
Since Q C is connected, we can take a walk q on ι(Q C ) from x to y. Then we have Φ(π(p)) = Φ(π(q)). We have by Lemma 5.36. Since we have φ C (p) = φ C (q) by Lemma 5.35, we have ℓ C (p) = 0. Thus any arrow appearing in p belongs to ι(Q C ).
This completes the proof of Theorem 5.24. In the remainder of this subsection we give a purely combinatorial proof of Theorem 5.27. For a slice S, we denote by S + 0 the subset of Q 0 consisting of sources in S. Lemma 5.40. The correspondence S S + 0 is injective. Proof. We denote by S ′ 0 be the set of vertices x of Q satisfying the following conditions. • there exists a path in Q from some vertex in S + 0 to x, • there does not exist a path in Q from any vertex in S + 0 to ν n x. To prove the assertion, we only have to show S 0 = S ′ 0 . It is easily seen from the definition of S ′ 0 that each ν n -orbit in Q 0 contains at most one vertex in S ′ 0 . Since S 0 is a slice, we only have to show S 0 ⊂ S ′ 0 . For any x ∈ S 0 , there exists a path in S from some vertex in S + 0 to x since S is a finite acyclic quiver. Assume that there exists a path p in Q from y ∈ S + 0 to ν n x. Since there exists a path q in Q from ν n x to x, we have a path pq from x to y. Since S is convex, we have ν n x ∈ S 0 , a contradiction to x ∈ S 0 .
Clearly we have ( Q ≥0 Lemma 5.41. Let S be a slice in Q. Then there exists a numbering S 0 = {x 1 , · · · , x N } of vertices of S such that the following conditions are satisfied. ( Proof. When we have x 1 , · · · , x i−1 ∈ S 0 , then we define x i as a source of the quiver S \ {x 0 , · · · , x i−1 }. It is easily checked that the desired conditions are satisfied. . Now we are ready to prove Theorem 5.27. Let S and T be slices. We can assume S ≤ T by Lemma 5.41. We use the induction on d(S, T ). If d(S, T ) = 0, then we have S = T . Assume d(S, T ) > 0. By Lemma 5.40, there exists a source x of S such that x / ∈ T 0 . Then we have µ + x (S) ≤ T and d(µ + x (S), T ) = d(S, T ) − 1. By our assumption on induction, µ + x (S) is obtained from T by a successive mutation. Thus S is obtained from T by a successive mutation. 5.5. Proof of Proposition 5.33. We complete the proof of Theorem 5.6 by filling the remaining gap, that is by proving Proposition 5.33.
For a walk p, we denote by |p| the length of p. For x, y ∈ Q 0 , we denote by d(x, y) the minimum of the length of walks on Q from x to y.
It is easily checked (similarly to the proof of Lemma 5.15) that d(x, y) = d(x ′ , y ′ ) whenever x − y = x ′ − y ′ .
Lemma 5.42. Let p be a cyclic walk. Assume that, for any decomposition p = p 1 p 2 p 3 of p, d(s(p 2 ), e(p 2 )) = min{|p 2 |, |p 3 p 1 |} holds. Then one of the following conditions holds.
Proof. (i) Assume that p contains an arrow of type i and an inverse arrow of type i at the same time. Take any decomposition p = q 1 aq 2 b −1 with arrows a, b of type i and walks q 1 and q 2 . If |q 2 | < |q 1 |, then we have d(s(q 1 ), e(q 1 )) = d(e(q 2 ), s(q 2 )) < min{|q 1 |, |aq 2 b −1 |}, a contradiction. Similarly, |q 1 | < |q 2 | cannot occur. Consequently, we have |q 1 | = |q 2 |. This equality also implies that q 1 and q 2 do not contain arrows or inverse arrows of type i. Consequently, p satisfies Condition (2).
(ii) In the rest, we assume that p does not satisfy Condition (2). By (i), we have that p does not contain an arrow of type i and an inverse arrow of type i at the same time. Without loss of generality we may assume φ i (p) > 0. Then p contains exactly φ i (p) arrows of type i for each i, and does not contain inverse arrows.
This implies that at most n+1 2 types of arrows appear in ap 1 (respectively, bp 2 ). Since all kinds of arrows appear in p, we have that ap 1 and bp 2 contain exactly n+1 2 types of arrows, and there is no common type of arrows in ap 1 and bp 2 . By the same argument, we have that p 1 b and p 2 a contain exactly n+1 2 types of arrows, and there is no common type of arrows in p 1 b and p 2 a. Since φ > 1, either p 1 or p 2 contains an arrow of same type with a. Assume that p 1 contains an arrow of same type with a. Then p 1 b contains more than n+1 2 types of arrows, a contradiction. Similarly, p 2 does not contain an arrow of same type with a, a contradiction.
Lemma 5.44. Let pa ǫ b ǫ ′ q and pc ǫ ′ d ǫ q be cyclic walks on Q, with ǫ, ǫ ′ ∈ {±1}, such that a and d are arrows of the same type, and b and c are arrows of the same type. Then one of them belongs to G if and only if the other does.
We now look at the following special case of Proposition 5.33.
Lemma 5.46. Any cyclic walk satisfying the condition in Lemma 5.42(2) belongs to G.
Now we are ready to prove Proposition 5.33.