Gluing of $n$-cluster tilting subcategories for representation-directed algebras

Given $n\leq d<\infty$, we investigate the existence of algebras of global dimension $d$ which admit an $n$-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras $A$ and $B$, a projective $A$-module $P$ and an injective $B$-module $I$ satisfying certain conditions, we show how we can construct a new representation-directed algebra $\Lambda$ in such a way that the representation theory of $\Lambda$ is completely described by the representation theories of $A$ and $B$. Next we introduce $n$-fractured subcategories which generalize $n$-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an $n$-cluster tilting subcategory for $\Lambda$ by using $n$-fractured subcategories of $A$ and $B$. As an application of our construction, we show that if $n$ is odd and $d\geq n$ then there exists an algebra admitting an $n$-cluster tilting subcategory and having global dimension $d$. We show the same result if $n$ is even and $d$ is odd or $d\geq 2n$.


Introduction
For a representation-finite algebra Λ, classical Auslander-Reiten theory gives a complete description of the module category mod Λ, see for example [ARS95]. However in general the whole module category of an algebra is very hard to study. In Osamu Iyama's higher-dimensional Auslander-Reiten theory ( [Iya07], [Iya08]) one moves the focus from mod Λ to a suitable subcategory C ⊆ mod Λ satisfying certain homological properties. Such a subcategory C is called an n-cluster tilting subcategory for some positive integer n; if moreover C = add(M ) for some M ∈ mod Λ, then M is called an n-cluster tilting module.
An n-cluster tilting subcategory C is the setting for a higher-dimensional analogue of the classical Auslander-Reiten theory: it admits an n-Auslander-Reiten translation, n-almost split sequences and an n-Auslander-Reiten duality generalising the classical Auslander-Reiten translation, almost split sequences and Auslander-Reiten duality when n = 1. However, in general it is not easy to find n-cluster tilting subcategories. If we set d := gl. dim Λ, then the existence of an n-cluster tilting subcategory for n > d implies that Λ is semisimple. Hence we may restrict to the case n ≤ d.
The extreme case n = d is of special interest and has been studied extensively before, for example in [IO13] and [HI10b]. If C is given by a d-cluster tilting module M , it follows that C is unique and given by where τ − d denotes the d-Auslander-Reiten translation. In this case Λ is called d-representation-finite (see [HI10a], [IO10]). It is an open question whether all d-cluster tilting subcategories are given by d-cluster tilting modules. Nevertheless, if we assume the existence of a d-cluster tilting module M we can obtain further results about C = add(M ). In particular in this case C is directed if and only if add(Λ) is directed. Furthermore it is asked in [HIO14] if the mere existence of a d-cluster tilting module implies that add(Λ) is directed.
Cases where n < d have also been studied before. For the case where Λ is selfinjective, and so d = ∞, see for example [EH08] and [DI20]. Note that in this case C is never directed. A class of examples satisfying n ≤ d < ∞ with d ∈ nZ first appeared in [Jas16] and many more were constructed recently in [JKPK19]. To our knowledge, the only known examples where n ∤ d appear in [Vas18, Theorem 2] for n even and d = n + 2k(n − 1) + ⌊ 2k l ⌋ + ⌈ 2k l ⌉ where k ∈ Z ≥1 , and in [Vas18, Example 3.7] for n = 2 and d = 3.
Recall that an algebra Λ is called representation-directed if there exists no sequence of nonzero nonisomorphisms f k : M k → M k+1 between indecomposable modules M 0 , . . . , M t with M 0 ∼ = M t . For representation-directed algebras, a characterization of n-cluster tilting subcategories was given in [Vas18, Theorem 1] (see Theorem 3.2). Using this characterization, it is easy to check the existence of n-cluster tilting subcategories. Moreover, in this case Λ is representation-finite and so any n-cluster tilting subcategory admits an additive generator. Finally, since mod Λ is directed, we have that C is also directed. As a consequence, it turns out that there is a unique choice for C. It follows that one of the simplest cases to consider when trying to find n-cluster tilting subcategories is that of Λ being representation-directed.
In this paper we address the general question of whether for a pair of positive integers (n, d) with n < d there exists an algebra Λ of global dimension d, admitting an n-cluster tilting subcategory; we call such an algebra (n, d)-representation-finite. We show that for n odd and any d we can find an (n, d)-representation-finite algebra. Moreover, for n even and d odd or d ≥ 2n we again answer the question affirmatively.
To construct (n, d)-representation-finite algebras we first introduce the method of gluing. Our method takes as input a representation-directed algebra A with a certain kind of projective module P and a representation-directed algebra B with a certain kind of injective module I and returns a new representation-directed algebra Λ := B P ⊲I A. The representation theory of Λ can be completely described in terms of the representation theories of A and B. In particular, the Auslander-Reiten quiver Γ(Λ) of Λ is given as the union of the Auslander-Reiten quivers Γ(A) and Γ(B) of A and B, identified over a common piece. In general there may be several choices of P and I, but choosing P and I to be simple modules always works. Similar ideas to this method have appeared before in the literature: in [IPTZ87], the authors are interested in the case where the common piece of Γ(A) and Γ(B) is a point or a triangle mesh. Our construction generalizes some of their results.
If A and B admit n-cluster tilting subcategories, in general it is not true that B P ⊲I A admits an n-cluster tilting subcategory. To this end we modify the characterization of n-cluster tilting subcategories given in [Vas18, Theorem 1] and introduce the more general notion of n-fractured subcategories.
We show that under some compatibility conditions gluing of algebras admitting n-fractured subcategories gives rise to an algebra admitting an n-fractured subcategory. Moreover, by repeating this process sufficiently many times, one can arrive at an algebra which admits an actual n-cluster tilting subcategory, as desired.
Let us call an algebra Λ strongly (n, d)-representation-directed if Λ is representation-directed and (n, d)-representation-finite. As a corollary of our previous results we show that if A is strongly (n, d 1 )-representation-directed, B is strongly (n, d 2 )-representation-directed, P is a simple projective A-module and I is a simple injective B-module then Λ = B P ⊲I A is strongly (n, d)-representation-directed for some d. By iterating this result, many new examples can be constructed. Moreover, while the global dimension d of Λ in general is difficult to compute, we show that in some simple cases we This paper is divided into four parts. In the first part of the paper we introduce some basic notation and give a motivating example in detail. In the second part, given two representation-directed algebras A and B, we describe our method of gluing of algebras and the associated results. In the third part we introduce n-fractured subcategories and describe how they are affected by gluing under certain conditions. In the fourth part of this paper we use these constructions to prove our results about the existence of (n, d)-representation-finite algebras. Most results are proved using standard techniques of representation theory: see for example the books [ARS95], [ASS06] as well as the survey article [Rin16]. Many examples are given throughout. We also include a list of terminology with reference to their definition in the text as well as an index of symbols at the end of this paper.

Part I: Preliminaries
1.1. Conventions. Let us introduce some conventions and notation that we use throughout this paper. Let K be an algebraically closed field and n ≥ 1 an integer. In this paper by an algebra Λ we mean a basic finite-dimensional unital associative algebra over K and by a Λ-module we mean a right Λ-module. We denote the category of right Λ-modules by mod Λ. We write M Λ for a module M ∈ mod Λ when the algebra is not clear from the context.
For a quiver Q we denote by Q 0 the set of vertices and by Q 1 the set of arrows. For an arrow α ∈ Q 1 we denote by s(α) its source and by t(α) its target. We compose arrows in quivers from the left to the right, that is if α i ∈ Q 1 , 1 ≤ i ≤ n are arrows in Q, then α 1 α 2 · · · α n−1 α n is a path in Q if s(α i ) = t(α i−1 ).
Throughout we use quivers with relations and their representations; for details we refer to [ASS06, Chapter III]. Contrary to the notation in [ASS06], we use R to denote an admissible ideal of the path algebra KQ of a quiver Q. If KQ/R is a bound quiver algebra, for a vertex k ∈ Q 0 , we denote by P (k) (respectively I(k)) the corresponding indecomposable projective (respectively injective) KQ/Rmodule.
By a subcategory of an additive category we always mean a full subcategory closed under isomorphisms, direct sums and summands unless specified otherwise. Now let A i ⊆ mod Λ be subcategories and M j ∈ mod Λ be modules indexed by some i ∈ I and j ∈ J. We set • A i -the set of isomorphism classes of indecomposable modules in A i , • |A i | -the cardinality of A i , • add{A i } i∈I -the subcategory of mod Λ containing all direct sums of modules M such that M ∈ A i for some i ∈ I, • add(M i ) -the subcategory of mod Λ containing all direct sums of direct summands of M i , • add{A i , M j } i∈I,j∈J := add{A i , add(M i )} i∈I,j∈J , For an algebra Λ, we denote by D the standard duality D = Hom K (−, K) between mod Λ and mod Λ op . By an ideal of Λ we mean a two-sided ideal, unless mentioned otherwise. For X ∈ mod Λ we denote by Ω(X) the syzygy of X, that is the kernel of P ։ X, where P is the (unique, up to isomorphism) minimal projective cover of X and by Ω − (X) the cosyzygy of X, that is the cokernel of X ֒→ I where I is the (unique, up to isomorphism) minimal injective hull of X. Note that Ω(X) and Ω − (X) are unique up to isomorphism. We denote by τ and τ − the Auslander-Reiten translations and by Γ(Λ) the Auslander-Reiten quiver of Λ. If M ∈ mod Λ is indecomposable, we denote by [M ] the corresponding vertex in the Auslander-Reiten quiver Γ(Λ). For more details on Auslander-Reiten theory we refer to [ASS06,Chapter IV]. Following [Iya08], we denote by τ n and τ − n the n-Auslander-Reiten translations defined by τ n (X) = τ Ω n−1 (X) and τ − n (X) = τ − Ω −(n−1) (X). Let φ : Λ → Γ be an algebra homomorphism. We denote by φ * : mod Γ → mod Λ the restriction of scalars functor that turns a Γ-module M into a Λ-module via m · λ = m · φ(λ) for m ∈ M and λ ∈ Λ. We denote by φ * : mod Λ → mod Γ the induced module functor, given by φ * (−) = − ⊗ Λ Γ. Finally, we denote by φ ! : mod Λ → mod Γ the coinduced module functor, given by φ ! (−) = Hom Λ (Γ, −). Note that (φ * , φ * ) and (φ * , φ ! ) form adjoint pairs.
We denote by A h the quiver Let Λ = KA m /R where R is an admissible ideal. Then Λ is called an acyclic Nakayama algebra and its representation theory is well known, see for example [ASS06,Chapter V]. We also introduce some notation from [Vas18]. In particular, recall that the isomorphism classes of the indecomposable Λ-modules can be described by the representations M (i, j) of the form (1, 1) (2, 1) (3, 1) (h − 2, 1) (h − 1, 1) (h, 1) .
Following [Mil71], given two algebra homomorphisms f : A → C and g : B → C, we define the pullback algebra (Λ, φ, ψ) of A f −→ C g ←− B to be the subalgebra with φ : Λ → A and ψ : Λ → B being induced by the natural projections. When clear from context, we will identify the pullback (Λ, φ, ψ) with the underlying algebra Λ. Notice that whenever we have f (a) = g(b) for some a ∈ A and b ∈ B, then there exists a unique λ ∈ Λ such that φ(λ) = a and ψ(λ) = b. It turns out that this is the pullback in the category of K-algebras. Note that if g is a surjection, then so is φ (but the converse is not true). In this case, the diagram is called a Milnor square of algebras (see [Mil71]).
1.2. A motivating example. Let us first give a motivating example that illustrates the theory that is developed in this paper.
( Notice that we can identify Γ(Λ) with the amalgamated sum Γ(B) △ Γ(A), under the identification △ (7,3) ≡ △(3) ≡ (1,3) △. Under this identification we also see that much of the representation theory of Λ is given by the representation theory of B and A: indecomposable Λ-modules correspond to indecomposable B-modules or to indecomposable A-modules, almost split sequences in mod Λ correspond to almost split sequences either in mod B or in mod A and similarly for syzygies and cosyzygies of indecomposable Λ-modules. Moreover C Λ is the additive closure of C A and C B viewed inside mod Λ. Notice that the indecomposable modules in C A and C B corresponding to the identified part match. In this case C Λ turns out to be a 2-cluster tilting subcategory. In particular, in mod Λ we have τ − 2 (M (7, 1)) ∼ = M (9, 4) and τ − 2 (M (7, 2)) ∼ = M (10, 3), since these functors can be computed in the subquiver corresponding to mod A.
In Example 1.1 we managed to get a 2-cluster tilting subcategory by identifying the "problematic" piece △ (7,3) of Γ(B) with the "well-behaved" piece (1,3) △ of Γ(A). In this paper we explain how this process can be defined rigorously and under which conditions it can be applied.
2. Part II: Gluing 2.1. Glued subcategories. Let us first recall some definitions from [AS81]. Let Λ be an algebra and A ⊆ L ⊆ mod Λ be subcategories. Recall that a morphism g : M → N in A is called right almost split if g is not a retraction and any non-retraction v : V → N with V ∈ A factors through g. Dually we can define left almost split morphisms. A short exact sequence 0 We say that A has almost split sequences if for any non-A-projective indecomposable module N ∈ A there is an almost split sequence in A ending at N and for any non-A-injective indecomposable module L ∈ A there is an almost split sequence starting at L.
Next we introduce the notion of gluing of subcategories.
Definition 2.1. Assume that there exist subcategories A and B of L such that the following are satisfied.
(i) L = add{A, B}, (ii) A and B have almost split sequences, (iii) If M ∈ A \ B and M is indecomposable, then Hom Λ (M, B) = 0, (iv) If N ∈ B and M ∈ A, then for all g : N → M , there exists an X ∈ A ∩ B such that g = g 1 • g 2 for some g 1 : X → N and g 2 : M → X.
In that case L is called the gluing of B and A and we write L = B ⊲ A. Note that gluing is not a commutative operation.
is an almost split sequence in B, then it is also an almost split sequence in L.

Proof.
(i) Since L and N are indecomposable in A, they are also indecomposable in L. Hence it is enough to show that g is right almost split in L. Clearly g is not a retraction in L. Let v : V → N be a morphism in L which is not a retraction and without loss of generality assume that V is indecomposable. By Definition 2.1(ii), we have that V ∈ A or V ∈ B. If V ∈ A, then v factors through g because g is right almost split in A. If V ∈ B \ A, then by Definition 2.1(iv) there exists some Hence V ∈ B and v factors through g since g is right almost split in B.

2.2.
Glued representation-directed algebras. Throughout this subsection we only consider representation-directed algebras. Our aim is to construct algebras A, B and Λ such that mod Λ = mod B ⊲ mod A. Our construction gives Λ as a certain pullback of A and B over KA h for a specific h, called gluing. This gluing is based on the existence of certain modules which we call left abutments and right abutments. We then show how we can describe completely the representation theory of Λ using the representation theories of A and B.
Pullbacks similar to gluing have been considered before in [IPTZ87, Section 3]. In particular, the authors of that paper assume the existence of such a pullback and are interested in the cases h = 1 and h = 2. After finishing this paper, it has come to the author's attention that such pullbacks have been also considered in [L08,BCW15], citing the work of [IPTZ87]. To make this article self-contained as well as to establish notation, we include full proofs of some results which can also be found in [IPTZ87,L08,BCW15]. Moreover, we include many additional properties that we will use later.
2.2.1. Abutments. If M is a Λ-module, then M is said to be uniserial if it has a unique composition series. In this case M has simple top and socle and hence M is indecomposable. Being uniserial is equivalent to the radical series Definition 2.3. Let Λ be a representation-directed algebra. We call a uniserial projective Λ-module P a left abutment if every submodule of P is projective and for any indecomposable projective Λ-module P ′ not isomorphic to a submodule of P , we have that all morphisms U → P ′ with U ⊆ P factor through P .
We call an indecomposable injective Λ-module I a right abutment if D(I) is a left abutment as a Λ op -module.
Let P be a left abutment with composition series 0 ⊆ P h ⊆ · · · ⊆ P 2 ⊆ P 1 = P.
Then the modules P i are also uniserial and so indecomposable. Hence there exist primitive orthogonal idempotents e 1 , . . . , e h such that P i ∼ = e i Λ and hence the composition series of P corresponds to a diagram where f i ∈ Hom Λ (e i+1 Λ, e i Λ) = e i Λe i+1 . We call such a choice of (e i , f i ) h i=1 a realization of the left abutment P and we denote e · = h i=1 e i . Note that h is the length l(P ) of P and that f h = 0. We will call h the height of the left abutment P .
For a right abutment I such that D(I) has a realization ( a realization of the right abutment I and h the height of the right abutment I. Diagrammatically, we have a sequence of factor modules Note that simple projective modules are the same as left abutments of height 1 and simple injective modules are the same as right abutments of height 1. Note also that since Λ is representation-directed, there exists at least one simple projective module and one simple injective module.
The following lemma will be used to characterize algebras admitting abutments in terms of a quiver with relations.
Proof. We only prove (a); then (b) follows from the definition and (a). If i = j then by [ASS06, Proposition IX.1.4] we have End Λ (e i Λ, e i Λ) ∼ = K. Notice that by definition, if i > j, we have Hom Λ (e i Λ, e j Λ) = 0. Since Λ is representation-directed, it follows that for i < j we have Hom Λ (e i Λ, e j Λ) = 0. It remains to show that for i > j, we have Hom Λ (e i Λ, e j Λ) ∼ = K. Since the morphism f i : e i+1 Λ → e i Λ corresponds to the radical inclusion rad(e i Λ) ⊆ e i Λ, it follows that any homomorphism g i : e i+1 Λ → e i Λ factors through f i . Since End Λ (e i Λ, e i Λ) ∼ = K, it follows that g i = λf i for some λ ∈ K and so Hom Λ (e i+1 Λ, e i Λ) ∼ = K. The result follows by a simple induction.
Remark 2.5. The requirement of P being a left abutment in Lemma 2.4(a) is stronger than what is used in the proof. Specifically, Lemma 2.4(a) holds for any uniserial projective module such that all submodules are projective and dually for Lemma 2.4(b).
Recall that a presentation of an algebra Λ is an isomorphism Φ : KQ/R ∼ −→ Λ where Q is a quiver and R is an admissible ideal of KQ. Recall also that Q is unique (up to isomorphism of quivers) but R is in general not unique. In particular, if {e 1 , . . . e k } is a complete set of primitive orthogonal idempotents of Λ then the quiver Q = Q Λ is the quiver with (Q Λ ) 0 = {1, . . . , k} and with arrows i → j being in bijection with a basis of the K-vector space e i (rad Λ/ rad 2 Λ)e j . For more details we refer to [ASS06,Chapter II.3]. The following proposition describes abutments in terms of quivers with relations.
Proposition 2.6. Let Λ be a representation-directed algebra.
(a) P is a left abutment realized by (e i , f i ) h i=1 if and only if there exists a presentation Φ : and such that no path of the form α i · · · α i+k is in R, and for 1 ≤ i ≤ h we have Φ(a i ) = e i and, moreover, we have Φ( if and only if there exists a presentation Φ : and such that no path of the form β i · · · β i+k is in R, and for 1 ≤ i ≤ h we have Φ(b i ) = e i , and, moreover, Proof.
(a) Throughout this proof let e ′ = 1 Λ − e · and identify Hom Λ (e i Λ, e j Λ) with e j Λe i . In particular, we have that Λ = e · Λ ⊕ e ′ Λ.
Assume first that P is a left abutment realized by (e i , f i ) h i=1 . Notice that by the uniqueness of the composition series of e 1 Λ it follows that rad(e i Λ) ∼ = e i+1 Λ (under the convention e h+1 = 0). In particular, we have that e i Λ/e i+1 Λ ∼ = top(e i Λ) = S i .
Let us first show that the quiver Q Λ has the required form. We extend the set {e 1 , . . . , e h } to a complete set of primitive orthogonal idempotents of Λ. Then the idempotents {e 1 , . . . , e h } correspond to vertices Q e · = {1, . . . , h} in the quiver Q Λ . We set Q e ′ := (Q Λ ) 0 \ Q e · . For 1 ≤ i ≤ h and for x ∈ (Q Λ ) 0 we have under the convention S h+1 = 0. In particular, we have dim K (e i (rad Λ/ rad 2 Λ)e x ) = δ i+1,x since Λ is basic. It follows that there is no arrow from the vertices Q e · to the vertices Q e ′ and that for 1 ≤ i, j ≤ h there is an arrow α i : i → j if and only if j = i + 1, and in that case it is the only arrow. Let 2 ≤ i ≤ h. It remains to show that there are no arrows from Q e ′ to e i . Let x ∈ Q e ′ and let e x ae i + rad 2 Λ ∈ e x (rad Λ/ rad 2 Λ)e i for some a ∈ rad Λ. It is enough to show that e x ae i ∈ rad 2 Λ. Since e x ae i ∈ e x (rad Λ)e i = Hom Λ (e i Λ, e x rad Λ), we have that e x ae i corresponds to a morphism e i Λ → e x rad Λ = rad(e x Λ). Composing with the inclusion rad(e x Λ) → e x Λ, we obtain a morphism e i Λ → rad(e x Λ) → e x Λ. By the factorization property in the definition of an abutment, this morphism factors through e i−1 Λ. In particular, we get a commutative diagram where the arrow u 2 exists because the morphism e i−1 Λ → e x Λ factors through rad(e x Λ). We now study the morphisms u 1 and u 2 . For u 1 notice that u 1 ∈ Hom Λ (e i Λ, e i−1 Λ) = e i−1 Λe i and we claim that e i−1 Λe i = e i−1 (rad Λ)e i . Indeed, we have e i−1 (rad Λ)e i ⊆ e i−1 Λe i and both vector spaces have dimension equal to 1 since by Lemma 2.4(a) we have Hence u 1 ∈ e i−1 (rad Λ)e i ⊆ rad Λ since rad Λ is a two-sided ideal. Moreover, we have u 2 ∈ Hom Λ (e 1 Λ, e x rad Λ) = e x (rad Λ)e i ⊆ rad Λ, and so u 2 ∈ rad Λ as well. Hence e x ae i = u 2 u 1 ∈ rad 2 Λ as required.
Next define an algebra homomorphism F : KQ Λ → Λ in the following way. For each pair of vertices x, y ∈ (Q Λ ) 0 we pick a basis {z α + rad 2 Λ | α : x → y} of e x (rad Λ/ rad 2 Λ)e y . In particular, for α i : i → i + 1 we pick z αi = f i + rad 2 Λ. For x ∈ (Q Λ ) 0 we define F (a x ) = e x and for α ∈ (Q Λ ) 0 we define F (α) = z α . We define F on paths in KQ Λ to be the multiplication of the corresponding elements in Λ and extend by K-linearity. By [ASS06, Theorem 3.7] this induces an isomorphism Φ : and so α i · · · α i+k ∈ R, which proves this direction. For the other direction, we may identify KQ Λ /R-modules with representations of Q Λ bound by R. We then have by a direct computation that P (h) is a simple projective module and for 1 ≤ i ≤ h − 1 we have rad(P (i)) ∼ = P (i + 1). Therefore the element α i ∈ a i (KQ Λ /R)a i+1 = Hom KQΛ/R (P (i + 1), P (i)) corresponds to the inclusion rad(P (i)) ⊆ P (i). Hence the radical series of P (1) is its composition series and so P (1) is uniserial. Moreover this composition series corresponds to the diagram Since there are no other arrows with target j for 2 ≤ j ≤ h, then for k ∈ {1, . . . , h} we have Hom KQΛ/R (P (j), P (k)) = a k (KQ Λ /R)a j = a k (KQ Λ /R)a 1 α 1 · · · α j−1 a j = Hom KQΛ/R (P (1), P (k))α 1 · · · α j−1 .
It follows that P (1) is a left abutment with realization (a i , α i ) h i=1 . By the assumptions on Φ it follows that Φ(P (1)) ∼ = e 1 Λ is a left abutment with realization (e i , f i ) h i=1 . (b) This follows immediately from the definition and (a), since Q Λ op = Q op Λ .
Proposition 2.6 shows that abutments are linearly oriented arms in the sense of Ringel [Rin16].
Remark 2.7. It follows from Proposition 2.6 that if (e i , f i ) h i=1 is a realization of a left abutment of height h, then (e i , f i ) h i=k is a realization of a left abutment of height h − k + 1, for any 1 ≤ k ≤ h. In particular, a submodule of a left abutment is also a left abutment.
Similarly, if (e i , g i−1 ) h i=1 is a realization of a right abutment of height h, then (e i , g i−1 ) k i=1 is a realization of a right abutment of height k, for any 1 ≤ k ≤ h. In particular, a quotient module of a right abutment is also a right abutment. If Λ is given by a quiver with relations, it is easy to find all abutments using Proposition 2.6, as the following examples show.
Example 2.9. It follows from Proposition 2.6(a) that the algebra KA h has exactly h left abutments, , and that the height of t i KA h is h − i + 1. By Proposition 2.6(b) the algebra KA h has exactly h right abutments, namely is both a left and a right abutment.
By the same proposition it follows that if an algebra Λ admits a module M that is both a left and a right abutment, then Λ ∼ = KA h and M is the unique indecomposable projective-injective KA h -module. In particular, M has the same height h as a left and a right abutment.
We have the following important Corollary.
Corollary 2.10. Let U be a left abutment realized by (e i , f i ) h i=1 (respectively a right abutment realized by (e i , g i−1 ) h i=1 ) and let Φ : KQ Λ /R ∼ −→ Λ be as in Proposition 2.6(a) (respectively as in Proposition 2.6(b)). Let π be the epimorphism π : KQ Λ /R −→ KA h given by identifying the full subquiver with vertices Q e · with A h . Then the morphism π • Φ −1 is independent of the choice of Φ and it satisfies Proof. Let us assume that U is a left abutment and Φ is as in Proposition 2.6(a); the other case is similar. Notice that we have a short exact sequence In particular, π h i=1 a i = 1 KA h . Let Φ, Ψ : KQ Λ /R → Λ be isomorphisms satisfying Φ(a i ) = e i = Ψ(a i ) and Φ(α i ) = f i = Ψ(α i ). By the description in Proposition 2.6(a) we have that Φ −1 (e · a) = Ψ −1 (e · a) for all a ∈ Λ. It follows that Corollary 2.10 justifies the following definition.
Definition 2.11. For a left abutment P realized by (e i , f i ) h i=1 (respectively a right abutment I realized by (e i , g i−1 ) h i=1 ) we denote the epimorphism π • Φ −1 : Λ ։ KA h by f P (respectively g I ) and we call it the footing at P (respectively I).
An easy consequence of Definition 2.11 is the following.
(a) If P is a left abutment realized by (e i , f i ) h i=1 , then f P (e · λ) = 0 implies e · λ = 0.
The following Lemma describes abutments in terms of the Auslander-Reiten quiver Γ(Λ) of Λ.
(a) P = e 1 Λ is a left abutment realized by (e i , f i ) h i=1 , if and only if P △ : is a full subquiver of Γ(Λ), there are no other arrows in Γ(Λ) going into P △ and, moreover, all northeast arrows are monomorphisms, all southeast arrows are epimorphisms and all modules in the same row have the same dimension. In particular, τ −i (e h Λ) is the simple top of e h−i Λ for 1 ≤ i ≤ h − 1. We call P △ the foundation of P .

if and only if
is a full subquiver of Γ(Λ), there are no other arrows in Γ(Λ) leaving △ I and, moreover, all northeast arrows are monomorphisms, all southeast arrows are epimorphisms and all modules in the same row have the same dimension. In particular, τ i D(Λe 1 ) is the simple socle of D(Λe i+1 ) for 1 ≤ i ≤ h − 1. We call △ I the foundation of I.
Proof. We only prove (a); (b) is similar. Assume first that P △ is a full subquiver of Γ(Λ) satisfying the required properties. Since all northeast arrows are monomorphisms and there are no other arrows going into P △, it follows that e 1 Λ is uniserial. For the factorization property, let P ′ be an indecomposable projective module such that there exists a nonzero homomorphism φ : e i Λ → P ′ and P ′ ⊆ e 1 Λ for , that is J is the additive closure of all indecomposable modules X such that [X] appears in the rightmost southeast diagonal of P △. Since the only indecomposable projective modules Y with [Y ] ∈ P △ are isomorphic to submodules of e 1 Λ, it follows that [P ′ ] ∈ P △. Since there are no arrows going into P △, the only arrows going out of P △ have one of the vertices {[J 1 ], . . . , [J h ]} as a source. Hence φ factors through J so that φ = g 1 • g 2 with g 2 : e i Λ → N and g 1 : N → P ′ for some N ∈ J . Moreover, since there are no other arrows going out of P △ \ {[J 1 ], . . . , [J h ]}, all squares in P △ correspond to almost split sequences and hence they are commutative. It follows that any morphism from e i Λ to J factors through e 1 Λ. Hence the morphism g 2 factors through e 1 Λ which shows that φ : e i Λ → P ′ factors through e 1 Λ, as required.
For the other direction we use induction on h ≥ 1. If h = 1 then e 1 Λ is a simple projective module and so there are no irreducible morphisms in Γ(Λ) into e 1 Λ. We assume the result is true for h = k and will prove it for h = k + 1. By induction hypothesis, and since by Remark 2.7 we have that e 2 Λ is also a left abutment of height h − 1, it follows that e2Λ △ : is also a full subquiver of Γ(Λ) and there are no other arrows in Γ(Λ) going into e2Λ △. Since e 1 Λ is uniserial, we have e 2 Λ ∼ = rad(e 1 Λ) and so there is an arrow [e 2 Λ] → [e 1 Λ] in Γ(Λ). We claim that this and the arrow [e 2 Λ] → [τ − (e 3 Λ)] are the only arrows in Γ(Λ) starting from [e 2 Λ]. To see this, note that any other arrow starting from [e 2 Λ] corresponds to the inclusion of e 2 Λ into an indecomposable projective module P ′ , since there are no other arrows going into [e 2 Λ]. But then this would correspond to some irreducible homomorphism that would not factor through e 1 Λ, contradicting the fact that e 1 Λ is a left abutment. Hence there is an almost split sequence Then a similar argument shows that there are exactly two arrows from [τ −(j−2) (e j Λ)] for 3 ≤ j ≤ h, exactly as required.
Since e 1 Λ is uniserial, we know that dim K (e h−i Λ) = i + 1. Since almost split sequences are exact sequences, it easily follows from simple dimension arguments that northeast arrows are monomorphisms, southeast arrows are epimorphisms and along the same row the dimensions remain the same. In particular, the last row has only simple modules, and since there is always an epimorphism e h−i Λ ։ τ −i (e h Λ) in P △, the result follows.
If P is a left abutment of Λ we set Similarly, if I is a right abutment of Λ we set Using Proposition 2.13 it can be shown that The following corollary shows that every abutment gives rise to an example of glued subcategories.
(a) Let P be a left abutment of Λ. Then mod Λ = F P ⊲ (mod Λ). (b) Let I be a right abutment of Λ. Then mod Λ = (mod Λ) ⊲ G I .
The following corollaries will be used later.
(a) Let P be a left abutment of Λ and M ∈ F P . Then proj. dim(M ) ≤ 1.
(b) Let I be a right abutment of Λ and N ∈ G I . Then inj. dim(N ) ≤ 1.
be a realization of P . Without loss of generality, we may assume that M is indecomposable. Then M corresponds to one of the vertices in P △, where P △ is as in Proposition 2.13(a). Since no arrows of Γ(Λ) have target in P △ but source outside of P △, it follows that the projective cover of M is e i Λ for some i ∈ {1, . . . , h}. Since Ω(M ) is a submodule of e i Λ, it follows that Ω(M ) = e j Λ for some j ≥ i or Ω(M ) = 0. In both cases we have proj. dim(M ) ≤ 1 as required.
(a) Let P be a left abutment realized by ( . Then for every λ ∈ Λ we have e · λ = e · λe · . where the last equality follows from Proposition 2.13 since there is no arrow going into P △ in Γ(Λ).
(a) Let P be a left abutment realized by (e i , f i ) h i=1 and let M ∈ F P . Then for every m ∈ M we have me · = m.
(b) Let I be a right abutment realized by (e i , g i−1 ) h i=1 and let N ∈ G I . Then for every n ∈ N we have ne · = n. Proof. We only prove (a); (b) is similar. If for a module X we have that xe · = x holds for all x ∈ X then it clearly holds for all submodules and epimorphic images of X, so by Proposition 2.13 it is enough to show (a) for M = e 1 Λ. But this follows immediately by Corollary 2.17.

Gluing via pullbacks.
The following definition is the the main concept of this section.
Definition 2.19. We call the pullback Λ of A fP −→ KA h gI ←− B the gluing of A and B along P and I and we denote it by Λ := B P ⊲I A.
In the following and when I and P are clear from context we will simply call Λ the gluing of A and B and denote it by Λ := B ⊲ A. Notice that since pullbacks are associative, the operation of gluing is associative too. Moreover, the indecomposable projective and the indecomposable injective modules over Λ can be related to the indecomposable projective and the indecomposable injective modules over A and B by considering certain idempotents. This is discussed in [IPTZ87, Construction 3.2]. We include the details here, adapted to our conventions, for the convenience of the reader.
We start with a simple example of gluing.
Example 2.20. Let A be a representation-directed algebra and P be a left abutment of A of height h. Let B = KA h and let I = I(h) be the unique indecomposable injective-projective B-module. By Example 2.9 we have that I is a right abutment of KA h of height h. The identity map Id KA h : KA h → KA h is the unique K-algebra morphism that satisfies the conditions of Corollary 2.10 and so the footing at I is The following lemma describes gluing of algebras given by quivers with relations.
Lemma 2.21. Let A = KQ A /R A be a representation-directed algebra given by a quiver with relations of the form where no path of the form α i · · · α i+k is in R A , and B = KQ B /R B be a representation-directed algebra given by a quiver with relations of the form Then the gluing of A and B over P and I is given by the bound quiver algebra Λ = KQ Λ /R Λ where Q Λ is the quiver and R Λ is generated by all elements in R A and R B and all paths starting from Q ′ A and ending in Q ′ B , under the identifications α i = β i = λ i .
Proof. That P and I are left and right abutments of height h follows by Proposition 2.6. The description of Λ as a quiver with relations is a straightforward calculation which is discussed after Lemma 3.4 in [IPTZ87].
Let us fix our setting for the rest of this section. First we fix two representation-directed algebras A and B, such that A admits a left abutment P realized by (e i , f i ) h i=1 and B admits a right abutment I realized by (ǫ i , g i−1 ) h i=1 . Notice that P and I have the same height. Accordingly, we have footing maps f P : A → KA h and g I : B → KA h . With this setting, the gluing Λ := B P ⊲I A is defined. That is, we have the following pullback diagram Suggestively for what follows, we write where all e i 's and ǫ i 's are primitive orthogonal idempotents. Furthermore, when clear from context, we will use the notation 1 C for both e · = h i=1 e i and ǫ · = h i=1 ǫ i . We also fix presentations of A and B as in Proposition 2.6. Specifically we have an isomorphism Φ A : is as in (2.1) and such that no path of the form α i · · · α i+k is in R A . Without loss of generality, and by Proposition 2.6(a), we denote the vertices of Q A by {h − l + 1, . . . , 0, 1, . . . , h} and the idempotent of Similarly we have an isomorphism .2) and such that no path of the form β i · · · β i+k is in R B . Without loss of generality, and by Proposition 2.6(b), we denote the vertices of Q B by {1, . . . , h, h + 1, . . . , m} and the idempotent of With this setting, the gluing That is, we have the following pullback diagram where Q Λ and R Λ are as in Lemma 2.21. In particular, the vertices of Q Λ are Proof. By the pullback diagrams (2.4) and (2.5) we get a commutative diagram where Φ Λ is the morphism induced by the universal property of the pullback. That Φ Λ is an isomorphism follows by the uniqueness of the pullback up to isomorphism. It remains to compute Φ Λ (l i ). If l i is the idempotent of KQ Λ /R Λ corresponding to the vertex i for some h − l + 1 ≤ i ≤ m, then by Lemma 2.21 and our assumptions on Φ A and Φ B we have By the fact that Λ is a pullback, it follows that In the following we set ε i := Φ Λ (l i ).
We further set We also introduce some notation to simplify expressions later in this section. We will denote [A ′ , B ′ ] := {A ′ , 1, . . . , h, B ′ } and we order the set For convenience, let us recall the functors defined by φ and ψ on the corresponding module categories. The epimorphism φ : Λ → A induces the restriction of scalars functor φ * : mod A → mod Λ, which has a left adjoint φ * (−) = − ⊗ Λ A and a right adjoint φ ! (−) = Hom Λ (A, −); similarly for ψ. Collectively, we have the functors Since φ and ψ are epimorphisms, it follows that φ * and ψ * are full and faithful. Our aim in this section is to show that if Λ = B ⊲ A then mod Λ = (mod B) ⊲ (mod A), where we identify mod A and mod B with their images under φ * and ψ * respectively. To this end, we need to verify that mod A and mod B satisfy the conditions of Definition 2.1.
Definition 2.24. Let X ∈ mod Λ. We will say that X is supported in Recall that the module category of a bound quiver algebra mod(KQ/R) is equivalent to the category rep(Q, R) of finite-dimensional representations of Q bound by R. We consider this equivalence as an identification. The support of a representation M = (M i , φ α ) of Q bound by R is the set We have the following equivalent characterizations of a module being supported in A or B.
Proof. Let us prove the result for A; the result for B is similar. Assume that (a) holds and we will show that (b) holds. Then X ∼ = φ * (M ) for some M ∈ mod A. Since (φ * , φ ! ) form an adjoint pair where the left adjoint is full and faithful, the unit of the adjunction is a natural isomorphism (see [Mac98, page 90]). Hence as required.
Assume that (b) holds and we will show that (c) holds. Using diagram (2.6) we have and it is well known that, viewed as a representation of Q A bound by R A , the restriction of scalars functor φ ′ * : mod(KQ A /R A ) → mod(KQ Λ /R Λ ) maps it to the representation obtained by putting zero in all vertices and arrows not in Q A . Hence Assume that (c) holds and we will show that (d) holds. For every x ∈ X we have by definition of the restriction of scalars functor and since Φ A is an isomorphism that where the last equality follows since by the assumption we have that the representation Φ Assume that (d) holds and we will show that (a) holds. By assumption we have that Then (a) follows by the commutativity of the left square in (2.6) and the fact that Φ Λ and Φ A are isomorphisms.
We have the following immediate corollary. Proof. Immediate by Lemma 2.25(d).
We can now identify the indecomposable projective and injective Λ-modules.
which follows immediately by Lemma 2.21.
Corollary 2.28. Let s A be the number of simple projective A-modules up to isomorphism and t A be the number of simple injective A-modules up to isomorphism. Similarly define s B , t B , s Λ and t Λ .
Finally, since {ε i } h−l+1≤i≤m is a complete set of primitive orthogonal idempotents for Λ by Lemma 2.23, it follows that there are exactly s B + (s A − 1) simple projective Λ-modules, as required.
Proof. By Lemma 2.21 it follows that for every The following lemma contains important information about the directedness that is required to Proof. Immediate from Lemma 2.21 by using the isomorphism Φ Λ to transfer computations to the bound quiver algebra KQ Λ /R Λ and using the fact that The following proposition is the most important step in showing the main result in this section.
Proof. Let us pick a sequence of nonzero morphisms where p h corresponds to the inclusion of the radical of ε h Λ. By applying Hom Λ (−, M ), and since Then clearly M ∼ = X ⊕ Y as vector spaces and it remains to show that both X and Y are submodules of M since by construction it is clear that X is supported in A and Y is supported in B.
Let us start by showing that We have s y (mε A ′ λε y ) = p mε A ′ λ and for any n ∈ rad(ε h Λ) we have n = nε B ′ by Corollary 2.29. Hence for any n ∈ rad(ε h Λ) we have where the last equality comes from Lemma 2.30(i).
Next let uε i ∈ U i and λ ∈ Λ. Again by Lemma 2.30(iv) we have λ = ε x λε y with x ≤ y ≤ B ′ and it is enough to show that uε i λ = uε i ε x λε y ∈ X. We can assume that x = i since otherwise u(ε i ε x )λε y = u0λε y = 0. If y < B ′ , then uε i λε y ∈ M ε y and it is enough to show uε i λε y ∈ U y . Since Since the left hand side is in U y by construction, we have k(uε i λε y ) ∈ U y as required. If y = B ′ then by Lemma 2.30(vi) we have uε i λε B ′ = uε i λ 1 ε h λ 2 ε B ′ . Using the same argument as before, we can show that uε i λ 1 ε h ∈ U h . We claim that then uε i λ 1 ε h λ 2 ε B ′ = 0 ∈ X, which is enough to show that X is a submodule of M . To show this claim it is enough to show that U h satisfies U h Λε B ′ = 0. To show this, let uε h ∈ U h and λ ∈ Λ. If λ ∈ rad(Λ) then by construction we have , since ε i are the only elements in Λ that act nontrivially on simple Λ-modules. Then As in the previous cases, we can assume that λ = ε i λε y with i ≤ y ≤ B ′ . If y = B ′ then vε i λε B ′ ∈ M ε B ′ and so vε i λ ∈ Y . If y ≤ h, using the same argument as in the previous case we can show that This shows that Y is a submodule of M and concludes the proof.
Lemma 2.34. Let M ∈ mod Λ be indecomposable. Then the following are equivalent.
Proof. By Proposition 2.13 it easily follows that for an A-module (respectively B-module) X we have supp(X) ⊆ {1, . . . , h} if and only if X ∈ F P (respectively X ∈ G I ). Then by Lemma 2.25 we have that M is supported in both A and B if and only if supp The result follows by the commutativity of the diagram (2.6) and the fact that Φ A , Φ B and Φ Λ are isomorphisms.
To simplify notation in the rest of this section, let us denote the subcategories φ * (mod A) ⊆ mod Λ and ψ * (mod B) ⊆ mod Λ by (mod A) * and (mod B) * respectively. Now we are ready to show the main result for this section. For condition (iii) let M ∈ (mod A) * \ (mod B) * be indecomposable and assume by way of contradiction that for some N ∈ (mod B) * there exists a nonzero morphism g : M → N . In particular, we have that M ։ Im g ֒→ N and so Im g ∈ (mod A) * ∩ (mod B) * = φ * (F P ), where the last equality follows by Lemma 2.34. Since both M and Im g are in the image of φ * and since φ * is full and reflects epimorphisms (as φ * is faithful), it follows that M ։ Im g is the image of an epimorphism in mod A. By the (functorial) isomorphism of Lemma 2.25(b), applying φ ! to M ։ Im g recovers this epimorphism. Hence there exists an epimorphism φ ! (M ) ։ φ ! (Im g) in mod A with φ ! (Im g) ∈ F P . By Proposition 2.13, this means that φ ! (M ) is in F P . But by Lemma 2.34 this implies that For condition (iv) notice that any g : N → M with N ∈ (mod B) * and M ∈ (mod A) * factors as N ։ Im g ֒→ M and Im g is in (mod A) * ∩ (mod B) * by Corollary 2.26.
The following corollaries describe the representation theory of Λ in terms of the representation theory of A and B and will be particularly useful in the following section.
Proof. Let f 0 : Y 0 → Y 1 be a nonzero morphism between indecomposable modules Y 0 , Y 1 ∈ mod Λ. We need to show that there exists no chain of nonzero nonisomorphisms f i : Assume by way of contradiction that such a chain exists. If all Y i are supported in B, then this gives rise to a chain of indecomposable B-module nonzero nonisomorphisms ψ Hence there exists some minimal j such that Y j is not supported in B. Then Y j is supported in A by Corollary 2.32. Since Y j is supported in A and not in B, and since mod Λ = (mod B) * ⊲ (mod A) * , it follows that Y i is supported in A and not in B for all i ≥ j by Definition 2.1(iii). Since Y k+1 ∼ = Y 0 and j was minimal, it follows that j = 0 and that all Y i are supported in A. Then this gives rise to a chain of indecomposable A-module nonzero nonisomorphisms Proof. We only prove (a1) Proof. Let M ∈ mod Λ and set U k = φ * Ω k (mod A) . Since gl. dim(A) = d 1 , we have that U k = 0 for k > d 1 . We claim that Ω j (M ) ⊆ add(U j , (mod B) * ) for any j ≥ 0. (2.7) We prove (2.7) by induction. The base case j = 0 follows immediately by Proposition 2.35. For the induction step, assume that (2.7) holds for j = k and we will show that it holds for j = k + 1. Then we have that with X ∈ U k and Y ∈ (mod B) * . By Proposition 2.35 we can write X ∼ = X 1 ⊕ X 2 with X 1 ∈ (mod A) * \ (mod B) * and X 2 ∈ (mod B) * . Then by Corollary 2.37(a1) we have that Ω(X 1 ) ∈ U k+1 and by Corollary 2.37(a2) we have that Ω(X 2 ), Ω(Y ) ∈ (mod B) * . Hence and the induction step is proved.
Next, let us now show that d 2 ≤ d. Let N be a B-module with proj. dim(N ) = d 2 . Then ψ * (N ) is a Λ-module and by Corollary 2.37(a2) we have that proj. dim(ψ * (N )) = d 2 . Hence d 2 ≤ d. Finally, let us show that d 1 ≤ d. Similarly to before, let L be an A-module with inj. dim(L) = d 1 . Then φ * (L) is a Λ-module and by Corollary 2.37(b2) we have that inj. dim(φ * (L)) = d 1 , which completes the proof.
under the identification (Id A ) * ( P △) = (f P ) * (△(h)). Hence we can view any A-module T ∈ F P as a KA h -module via the functor f ! P . Similarly, if I is a right abutment of A of height h, any A-module X ∈ G I can be viewed as a KA h -module through the identification A = A P (h) ⊲I KA h and the corresponding functor g * I . We finish this section with a corollary that describes the connection between abutments of Λ and abutments of A and B.
Proof. Let us indicatively show (a2) and (c1); the rest are similar. For (a2) notice that since h− l + 1 ≤ i ≤ 0, we have that ε i Λ is supported in A by Proposition 2.27. In particular, by the definition of φ, we have φ ! (ε i Λ) = e i A. Since F eiA ∩ F P = 0, it follows from Proposition 2.13 that the two subquivers

Part III: Fractures
In this section we will show how to use gluing to construct many examples of representation-directed algebras admitting n-cluster tilting subcategories. In subsection 3.1 we introduce the building blocks of our construction. In subsection 3.2 we show how the construction works. In subsection 3.3 we are interested in a special case of our construction which we can describe completely.
3.1. Fractured subcategories. First, let us introduce some notation. Let Λ be a representation-directed algebra. We set  is a partial order. Similarly we define ≤ on I ab . We will refer to elements of those sets as maximal or minimal with respect to these partial orders. We set P mab Λ = P mab := add P ∈ P ab | P is maximal , I mab Λ = I mab := add I ∈ I ab | I is maximal .
Definition 3.1. We call a subcategory C of mod Λ an n-cluster tilting subcategory if It is clear from the definition that mod Λ is the unique 1-cluster tilting subcategory of Λ. Observe that since Λ is representation-finite, then any additive subcategory of mod Λ is of the form add(M ) for some M ∈ mod Λ. When add(M ) is n-cluster tilting we call M an n-cluster tilting module.
Note that n-cluster tilting subcategories are usually defined in more general settings by adding the requirement of functorial finiteness, but since add(M ) is always functorially finite we can use the above definition.
Before we proceed, let us introduce one more piece of notation. Let C, V be subcategories of mod Λ. We set C \V to be the additive closure of all indecomposable modules X ∈ C such that X ∈ V. With this in mind we recall the following characterization of n-cluster tilting subcategories for representation-directed algebras.

Theorem 3.2. [Vas18, Theorem 1] Assume that Λ is a representation-directed algebra and let C be a subcategory of mod Λ. Then C is an n-cluster tilting subcategory if and only if the following conditions hold:
(1) P ⊆ C, (2) τ n and τ − n induce mutually inverse bijections is indecomposable for all indecomposable M ∈ C \P and 0 < i < n, (4) Ω −i (N ) is indecomposable for all indecomposable N ∈ C \I and 0 < i < n.
Then P = add{P (1) , . . . , P (k) , Q}. To generalize the definition of an n-cluster tilting subcategory for representation-directed algebras using Theorem 3.2, we first replace the basic module P (j) by a suitable basic module Then we set P L := add{T (1) , . . . , T (k) , Q}. Dually, we replace right abutments and define a subcategory I R in a similar manner. Then we replace all instances of P and I in Theorem 3.2 with P L and I R respectively. Since we want to generalize the definition of n-cluster tilting, we also want to have Ext i Λ (T (j) , T (j) ) = 0 for 0 < i < n and any 1 ≤ j ≤ k. Since by Corollary 2.16 we have that proj. dim(T ) ≤ 1, this simplifies to Ext 1 Λ (T (j) , T (j) ) = 0. Since T (j) ∈ F Pj , if we view T as a KA hmodule via f ! Pj , we conclude that f ! Pj (T (j) ) should be a tilting KA h -module. Tilting modules of KA h were classified in [HR81]. The following Proposition asserts that a basic tilting module of KA h has the correct number of indecomposable summands, which is necessary for our construction to work. Let P be a left abutment of Λ. Recall that by Example 2.41 we can view a Λ-module T in F P as a KA h -module via f ! P and dually for right abutments. With this in mind, we give the following definition.
Definition 3.4. Let Λ be a representation-directed algebra.
(a) Let P be a maximal left abutment of Λ realized by (e Notice that in particular a fracture is a basic module. The following lemma collects some basic information about fractures.
Lemma 3.5. Let Λ be a representation-directed algebra.
(a) Let T be a fracture of a maximal left abutment P , realized by (e i , f i ) h i=1 . Then T has h indecomposable summands and proj. dim(T ) ≤ 1. (b) Let T be a fracture of a maximal right abutment I, realized by (e i , g i−1 ) h i=1 . Then T has h indecomposable summands and inj. dim(T ) ≤ 1.

Proof. Follows immediately by Corollary 2.16 and Proposition 3.3.
Example 3.6. For a maximal left abutment P realized by (e i , f i ) h i=1 , there exists a unique (up to isomorphism) fracture of P that is projective, namely T = h i=1 e i Λ. To see that this is a fracture, notice that is a basic tilting KA h -module. The fact that T is the unique projective fracture of P follows by Lemma 3.5. Similarly, if I is a right abutment realized by (e i , g i−1 ) h i=1 , then T = h i=1 D(Λe i ) is the unique fracture of I that is injective.
Definition 3.7. Let Λ be a representation-directed algebra.
(a) A left fracturing T L of Λ is a module where T (P ) is a fracture of P . We set P L := add P \P ab , T L .
where T (I) is a fracture of I. We set I R := add I \I ab , T R .
(c) A fracturing of Λ is a pair (T L , T R ) where T L is a left fracturing of Λ and T R is a right fracturing of Λ.
It follows easily by the definition of a fracturing and of maximal left and right abutments that a fracturing is a basic module. Hence if (T L , T R ) is a fracturing of Λ, then we have by Proposition 3.3 that |P L | = |P| and |I R | = |I|. In particular, we always have |P L | = |I R |. Proof. We only prove (a); (b) is similar. First we show (a1) implies (a2). If P L = P then every module in P L is projective. In particular, T L is projective. To see that (a2) implies (a3), first notice that if is projective for every maximal left abutment Q of Λ. By Example 3.6 this implies that every indecomposable submodule of Q is isomorphic to a summand of T (Q) . Since an abutment is either maximal or isomorphic to a submodule of a maximal abutment, it follows that a representative of each isomorphism class of each abutment P of Λ appears exactly once as a direct summand of T L . Finally, (a3) implies (a1) immediately from the definition.
If Λ is an algebra, we will denote by P ab a left fracturing of Λ which is projective as a module and by I ab a right fracturing of Λ which is injective as a module. By Lemma 3.8, it follows that P ab and I ab are unique up to isomorphism. and T (I(3 ′ )) = 2 ′ ⊕ 2 ′ 3 ′ ⊕ 1 ′ 2 ′ 3 ′ . By construction, T (P (5)) is the unique (up to isomorphism) projective fracture of P (5) and the modules T (I(3)) and T (I(3 ′ )) are fractures of I(3) and I(3 ′ ) respectively. Then (T L B , T R B ) is a fracturing of B, where T L B = T (P (5)) and T R B = T (I(3)) ⊕ T (I(3 ′ )) . Since T (P (5)) is projective, we have P L = add(B) by Lemma 3.8. Following the definition, we also have Definition 3.10. Let n ≥ 2. Assume that Λ is a representation-directed algebra with a fracturing (T L , T R ) and let C be a subcategory of mod Λ. Then C is called a (T L , T R , n)-fractured subcategory if (1) P L ⊆ C, (2) τ n and τ − n induce mutually inverse bijections is indecomposable for all indecomposable M ∈ C \P L and 0 < i < n, (4) Ω −i (N ) is indecomposable for all indecomposable N ∈ C \I R and 0 < i < n.
Let us first note that Definition 3.10 makes sense for n = 1 as well. However, the case Λ = KA h behaves in a special way in that case for our purposes. Since the unique 1-cluster tilting subcategory for a representation-directed algebra Λ is mod Λ itself, and to avoid needlessly complicating the results and proofs of this section we opt to assume that n ≥ 2 when considering n-fractured subcategories.
Notice that conditions (1) and (2) in Definition 3.10 imply that I R ⊆ C, since |P L | = |I R |. In particular, we have that T L ∈ C and T R ∈ C. This definition generalizes the definition of an n-cluster tilting subcategory for a representation-directed algebra in the sense of the following proposition.
Proposition 3.11. Let n ≥ 2. Let Λ be a representation-directed algebra and (T L , T R ) be a fracturing of Λ. Let C be a (T L , T R , n)-fractured subcategory of mod Λ for n ≥ 2. Then C is an n-cluster tilting subcategory if and only if T L ∼ = P ab and T R ∼ = I ab .
Proof. If T L ∼ = P ab and T R ∼ = I ab , then Lemma 3.8 implies that P L = P and I R = I. Then Theorem 3.2 implies that C is an n-cluster tilting subcategory. Assume now that C is an n-cluster tilting subcategory of Λ. We will show that T L is projective (the proof that T R is injective is similar). Assume by way of contradiction that T L is not projective. Then proj. dim(T L ) = 1 by Lemma 3.5. Hence Ext 1 Λ (T L , Λ) = 0 which contradicts T L ∈ C.
Proposition 3.11 motivates the following definition.
Definition 3.12. Let n ≥ 2. Let Λ be a representation-directed algebra with a fracturing (T L , T R ) and C be a (T L , T R , n)-fractured subcategory. Then C will be called a left n-cluster tilting subcategory if T L ∼ = P ab and a right n-cluster tilting subcategory if T R ∼ = I ab .
Example 3.13. Let n ≥ 2 and Λ = KA h . Let (T L , T R ) be a fracturing of Λ. Since an indecomposable projective Λ-module is a submodule of the maximal left abutment P (1), it follows that P \P ab is empty and so P L = add(T L ). Similarly we have I R = add(T R ). Since τ n (M ) = 0 = τ − n (M ) for any M ∈ mod Λ, it is immediate from the definition that C ⊆ mod Λ is a (T L , T R , n)-fractured subcategory of Λ if and only if T L = T R and C = add(T L ).
Example 3.14. Let B and (T L B , T R B ) be as in Example 3.9. Let By computing τ − 2 , we find that A simple calculation verifies that C B is a (T L B , T R B , 2)-fractured subcategory. Since T L B is projective, it is a left n-cluster tilting subcategory. For the convenience of the reader who might want to verify those claims, we give the Auslander-Reiten quiver of B where we encircle the indecomposable modules which are in C B :

Main construction.
Our aim is to glue algebras admitting fractured subcategories in such a way that the resulting algebra also admits a corresponding fractured subcategory. To this end we need to first describe how to glue algebras with a fracturing. So let us fix two representation-directed algebras A and B with fracturings (T L A , T R A ), respectively (T L B , T R B ), and set Λ := B Q ⊲J A where Q is a left abutment of A and J is a right abutment of B, both of the same height h.
Let P be a maximal left abutment of Λ. Then either P is supported in B or P is not supported in B, in which case it is supported in A by Proposition 2.31. By Corollary 2.42(c1), in the first case ψ * (P ) is a left abutment of B and in the second case φ ! (P ) is a left abutment of A. Moreover, in either case it is a maximal left abutment by Corollary 2.39. Set Observe that if P is supported in B, then by construction the composition mod Λ to a basic tilting KA h -module and similarly if P is supported in A. Dually, maximal right abutments of Λ correspond to maximal right abutments of A or of B and so we set and T R Λ := Then, by the above considerations it follows that (T L Λ , T R Λ ) is a fracturing of Λ, which we call the gluing of the fracturings (T L A , T R A ) and (T L B , T R B ) at Q and J and we denote it by ( . Although trivial gluing is a very special case of gluing which is not of much interest in this paper, it behaves somewhat unexpectedly with respect to gluing of fracturings. We illustrate the situation with the following example. A and so T R Λ = T R A . Moreover, by Corollary 2.39 it follows that no maximal left abutment of Λ is supported in B except for Q when h = h Q . We consider the cases h < h Q and h = h Q separately. Assume that h < h Q . Then Q is a maximal left abutment of Λ not supported in B. Hence T ) and that We need to prove conditions (1)-(4) of Definition 3.10. We pick idempotents of A, B and Λ as in Section 2.2.2. For condition (1) we need to show that P L Λ ⊆ C Λ , or equivalently By the construction of T L Λ , and since T L A ∈ add{C A } and T L B ∈ add{C B }, it follows that T L Λ ∈ C Λ . Then, if ε i Λ is an indecomposable projective Λ-module, it is enough to show that if ε i Λ is not a left abutment, then ε i Λ ∈ C Λ . If 1 ≤ i ≤ m, then ε i Λ is supported in B by Proposition 2.27 and so ψ * (ε i Λ) = ǫ i B. By Corollary 2.42(b1) it follows that ǫ i B is not a left abutment of B and so ǫ i B ∈ C B . Hence If e i A is not a left abutment of A, then a similar argument as before shows that ε i Λ ∈ C Λ . If e i A is a left abutment of A, then we must have that φ * (F eiA ∩ F P ) = 0, since this intersection being zero implies via Corollary 2.42(a2) that ε i Λ is a left abutment, contradicting our assumption. In particular, we have F eiA ∩ F e1A = 0, since by assumption P ∼ = e 1 A. Since i < 1, by Proposition 2.13 we have that F e1A F eiA and that e1A △ is a full subquiver of eiA △. In particular, e 1 A and e i A are both abutments appearing in the same radical series of the maximal abutment Q and the height of e i A is greater than h. Since h ≥ lvl(T A ) ⊆ C A , it follows that ε i Λ ∈ C Λ , as required.
Hence, condition (1) is satisfied. Conditions (3) and (4) follow immediately by Corollary 2.37 and the corresponding conditions being true for C A and C B .
It remains to show that condition (2) holds for C Λ . Let M ∈ C \P L Λ . We will show that τ n (M ) ∈ C \I R Λ and τ − n τ n (M ) ∼ = M ; the dual fact that if N ∈ C \I R Λ then τ − n (N ) ∈ C \P L Λ and τ n τ − n (N ) ∼ = N can be shown similarly.
Assume first that ψ * (M ) ∈ C B . We claim that ψ * (M ) ∈ (C B ) \P A similar argument as before shows that Moreover, by Corollary 2.37, it follows that τ n (M ) ∼ = ψ * τ n ψ * (M ) and so τ n (M ) ∈ (C Λ ) \I R Λ . The previous argument shows also that we can compute The following corollary of Theorem 3.16 is of particular interest.
Corollary 3.17. Let A be a strongly (n, d 1 )-representation-directed algebra and B be a strongly (n, d 2 )-representation-directed algebra. Let P be a simple projective A-module and I be a simple injective B-module. Then Λ = B P ⊲I A is a strongly (n, d)-representation-directed algebra for some d Proof. First we have that Λ is representation-directed by Corollary 2.36 and that max{d 1 , d 2 } ≤ d ≤ d 1 + d 2 by Corollary 2.38. It remains to show that Λ admits an n-cluster tilting subcategory C Λ . If n = 1, then C Λ = mod Λ. Assume that n ≥ 2. By Proposition 3.11, we have that there exists a (P Since P and I are simple, we have that both P and I have height 1. In particular, if Q is a maximal left A-abutment with F P ⊆ F Q and J is a maximal right B-abutment with G I ⊆ G J , it follows that We have is a projective B-module by assumption, and so its image under ψ * is a projective Λ-module by Proposition 2.27. If R is supported in A, then there is no arrow going into the triangle R △. Then by Corollary 2.39 there is no arrow going into the triangle φ ! (R) △. Let h ′ be the height of R. Since T are all the different leftmost vertices in the triangle φ ! (R) △. Hence lifting this through φ * , we again get a direct sum corresponding to the leftmost vertices of the triangle R △, which is a projective Λ-module, completing the proof.
Since representation-directed algebras always have simple projective and injective modules, Corollary 3.17 can be used to construct arbitrarily many n-cluster tilting subcategories from known n-cluster tilting subcategories of representation-directed algebras.
We describe the next simplest case of using Theorem 3.16. First we need to have T R A = I A and T L A to have exactly one nonprojective fracture T corresponding to a maximal right abutment J. Then, after gluing the resulting subcategory will be a (P Λ , I Λ , n)-fractured subcategory or equivalently an n-cluster tilting subcategory.
Even if there are more nonprojective fractures chosen for the left fracturing of A (or similarly noninjective fractures chosen for the right fracturing of B), by the construction of gluing one can glue at each fracture independently. Say we have an algebra Λ and that at each nonprojective fracture we glue by a left n-cluster tilting subcategory, while at each noninjective fracture we glue by a right n-cluster tilting subcategory and each gluing is compatible as per the requirements of Theorem 3.16. Then the result will be an algebra such that the gluing of all the fractured subcategories is an n-cluster tilting subcategory. We illustrate with a detailed example.
Example 3.18. Let B, (T L B , T R B ) and C B be as in Example 3.14. Recall that C B is a left n-cluster tilting subcategory obtained by repeatedly applying τ − 2 starting from B and there are two noninjective fractures in T R B , namely We want to glue two appropriate algebras with B, one alongside . Consider the algebra A as in Example 2.40. It is easy to see that A admits a 2-cluster tilting subcategory, given by C A = add(A ⊕ D(A)). Then by Proposition 3.11 we have that C A is (T L A , T R A , 2)-fractured subcategory where Hence, viewing T (I(3) B ) and T (P (1) A ) as KA 3 -modules via the respective functors, we have that they coincide since In particular, by Theorem 3.16, the algebra . Viewing the Auslander-Reiten quivers of B and A embedded in the Auslander-Reiten quiver of Λ 1 we can find an additive generator of C Λ1 . If we denote the indecomposable modules in the 2-fractured subcategories by encircling the corresponding vertices we have: In particular, C Λ1 is a 2-left cluster tilting subcategory, as expected. Moreover, Λ 1 has two maximal right abutments, namely I(2) Λ1 and I(3 ′ ) Λ1 . The fracture corresponding to the first one is injective, while the fracture of the second one is , which is noninjective. Hence we want to glue at I(3 ′ ) Λ1 . Let C be the algebra given by the quiver with relations Then the Auslander-Reiten quiver Γ(C) of C is Hence by Proposition 2.13 there is a unique maximal left abutment, namely P ( It is easy to see that C has a (T L C , T R C , 2)-fractured subcategory such that the gluing is compatible according to Theorem 3.16. Hence the gluing of the subcategories C Λ1 and C C is a 2-cluster tilting subcategory. Concretely, the Auslander-Reiten quivers of Λ 1 and C along with their 2-fractured subcategories are Γ(Λ 1 ) and indecomposables in C Λ1 Γ(C) and indecomposables in C C we glue here the algebra Λ 2 is given by the quiver with relations 1 2 3 4 5 6 7 , and the Auslander-Reiten quiver Γ(Λ 2 ) of Λ 2 with the 2-cluster tilting subcategory C Λ2 is Γ(Λ 2 ) and indecomposables in C Λ2 .
Remark 3.19. The algebras of Example 3.18 and Corollary 3.17 give rise to algebras with many interesting properties. For instance let us consider the number of sinks and sources in the quiver of an algebra. Following the notation of Corollary 2.28, and since the number of sinks (respectively sources) in the quiver of an algebra is equal to the number of simple projective (respectively injective) modules, we denote for an algebra Λ by s Λ the number of sources in its quiver and by t Λ the number of sinks in its quiver. Then let A be a strongly (2, d A )-representation-directed algebra with s A = 2 and t A = 1 (for example, we may take A to be the algebra Λ 2 as in Example 3.18). Let B be a (2, d B )-representation-directed algebra which admits a 2-cluster tilting subcategory. By gluing at the simple projective A-module and any simple injective B-module we get the algebra B (1) = B ⊲ A. By Corollary 3.17, we have that B (1) admits a 2-cluster tilting subcategory. By Corollary 2.28 we have that (s B (1) , t B (1) ) = (s B , t B + 1). Continuing inductively, let B (i) be a sequence of algebras defined by B (i) = B (i−1) ⊲ A where the gluing is done over any simple projective A-module and any simple injective B (i−1) -module. Then we get that B (i) admits a 2-cluster tilting subcategory and (s B (i) , t B (i) ) = (s B , t B + i).
A similar argument shows that if we let B (j) be a sequence of algebras defined by , where all gluings are done over simple modules, then again B (j) admits a 2-cluster tilting subcategory and (s B (j) , t B (j) ) = (s B + j, t B ). More generally, we have that In particular, by choosing (s B , t B ) = (1, 1) (for example, we may take B = KA h / rad(KA h ) h−1 for some h ≥ 3), we have that for any pair (s, t) with s, t ≥ 1 there exists an algebra Λ such that Λ admits a 2-cluster tilting subcategory and (s Λ , t Λ ) = (s, t). It follows that for any given pair of numbers (s, t), there exists a quiver Q with s sinks and t sources and a bound quiver algebra Λ = KQ/R such that Λ admits a 2-cluster tilting subcategory. Note that by construction the number of vertices of the quiver of Λ is of the order of s + t but can be made arbitrarily large.
In Example 3.18 it was not clear how one should find the algebras A and C. They depended on the type of fractures that the algebra B had and clearly they are not unique since we can always glue at simple modules via Corollary 3.17. The fractures in this example corresponded to slice modules of KA 3 and we will see in section 3.3 how we can find appropriate algebras to glue in this case. More generally we have the following question. If we can answer Question 1(a) (respectively 1(b)) affirmatively we will say that we can complete T on the left (respectively right). Notice that by symmetry we can complete T on the left if and only if we can complete D(T ) on the right by taking A = B op .
As a special case, . This condition is equivalent to saying that viewing the indecomposable summands of T as vertices in △(h), they are symmetric along the perpendicular bisector of the bottom line of the triangle. It follows that if we can answer Question 1(a) affirmatively in this case, then the algebra B D(I) ⊲I B op admits an n-cluster tilting subcategory by Theorem 3.16. A similar result holds if we can answer Question 1(b) affirmatively. We illustrate this situation with an example.
Example 3.20. Let A be given by the quiver with relations . If we set T R A = I ab to be an injective right fracturing of A and  In particular, C A is a right 3-cluster tilting subcategory. If we view the fracture appearing in the foundation of P (1) A as a KA 3 -module, we have Then the algebra B = A op is given by the quiver with relations we glue here and the Auslander-Reiten quiver of Λ with its 3-cluster tilting subcategory is Γ(Λ) and indecomposables in C Λ . 3.3. The case of slice modules. In this section, we answer Question 1 positively in the case of T being a slice module. We begin with the definition of slice modules for KA h ; for the general definition of slice modules we refer to [HR81]. Since the possible lengths of T i are 1 to h, we can assume without loss of generality that for a slice of KA h , we have l(T i ) = i. If we denote the indecomposable KA h -modules by M (i, j) as in (1.1), it follows that a slice of KA h is a set of modules {M (i 1 , 1), M (i 2 , 2), . . . , M (i h , h)} such that i k = i k−1 or i k = i k−1 − 1. In particular, i h = 1 and i h−1 = 1 or i h−1 = 2.
Definition 3.22. Let Λ be a representation-directed algebra and let T be a fracture corresponding to a maximal abutment of Λ. We will say that T is a slice fracture if T viewed as a KA h -module is a slice module.
Our aim is to answer Question 1 affirmatively when T is a slice module. Notice that if T is a slice module, then D(T ) is also a slice module. Hence by symmetry it is enough to answer Question 1(a). The following computational lemma will be used.
Before we proceed with the main result of this section, let us explain how Lemma 3.23 will be used. By Proposition 2.6 there is a unique maximal left abutment of Λ m,h , namely P (m − h + 1) = M (1, h). Moreover, the Auslander-Reiten quiver of Λ m,h is a subquiver of △(m) where we remove all the vertices corresponding to indecomposable modules of length at least h + 1. Described otherwise, it is the same as the quiver △(h) with the addition of more diagonals on the right hand side of the same height h.
Let T be a slice fracture of P (m − h) and letT ⊕ P (m − h + 1) ∼ = T . Then Lemma 3.23 implies that the action of τ − n translatesT through the Auslander-Reiten quiver of Γ(Λ m,h ) without changing its shape. In other words, for m large enough we have the following pictures: In each of the above pictures the leftmost thick line represents the indecomposable summands ofT in the foundation of P (m − h + 1), the middle thick line represents the indecomposable summands of τ − n (T ) and the rightmost thick line represent the indecomposable summands of τ −2 n (T ). Notice that in the case of n being evenT is reflected horizontally at every application of τ − n . Moreover, the module P (m−h), which would be at the top of the slice, is injective and so τ − n (P (m−h+1)) = 0. Additionally, the above applications of τ − n are invertible by τ n . The idea of the proof is that by choosing m correctly we can stop precisely at the point where the thick diagonal aligns with the end of Λ m,h . Then we can remove these aligned modules from our slice and consider the leftover piece as a slice of smaller height.
where now the encircled modules form a 4-cluster tilting subcategory. As a consequence, the algebra Λ 12,5 P (6)
Proposition 3.25. Let {M (i 1 , 1), . . . , M (i h , h)} be a slice of KA h and T = h k=1 M (i k , k). Then we can complete T on the left and on the right.
Proof. Let n ≥ 2. As mentioned before, by symmetry, it is enough to show that we can complete T on the right. We will use induction on h. For h = 1 we have only one indecomposable KA 1 -module, say N , and so add(T ) = add(N ) is an n-cluster tilting subcategory for any n. For the induction step, assume that we can complete any slice of KA h−1 on the right and we will show that we can complete any slice of KA h on the right.
We  But this shows that A ′ completes T on the right, as required.
For the case i h−1 = 1 we consider the cases n being odd and n being even separately. For the case n being odd, set m := n+1 2 h. First we glue KA h Since this is a trivial gluing as in Example 2.20, the resulting algebra is isomorphic to Λ m,h again. It is clear that viewing the modules M (i k , k) as Λ m,h -modules we have Computing τ − n by using Lemma 3.23 gives In particular we have which is a right abutment of Λ h+ n−1 2 h,h of height h − 1. Moreover, the set {τ − n (M (i k , k)} h−1 k=1 ⊆ F M(1,h−1) is a fracture which is a slice, viewed as a KA h−1 -module. By induction hypothesis we can complete this slice on the right. Hence there exists an algebra A and a maximal left abutment P of A such that A admits a (T L A , I ab A , n)-fractured subcategory and the only nonprojective fracture of T L A is . It follows that the algebra A ′ = Λ m,h P ⊲I (h − 1) A completes the fracture T on the right.
Finally, for the case n being even, set m := i 1 + n 2 h. A similar computation shows that In this section we construct examples of (n, d)-representation-finite algebras. Specifically, if n is odd we will construct an (n, d)-representation-finite algebra for any d ≥ n and if n is even we will construct an (n, d)-representation-finite algebra for any d odd or d ≥ 2n.
In our constructions we again use acyclic Nakayama algebras. Recall that the Auslander-Reiten quiver Γ(KA h /R) of a quotient of KA h by an admissible ideal is a full subquiver of △(h) with the property that if a vertex (i, j) is not in Γ(KA h /R), then the vertices (i − 1, j + 1) and (i, j + 1) are not in Γ(KA h /R). Recall also that acyclic Nakayama algebras can be classified by Kupisch series, first introduced in [Kup58].
Then the correspondence between m-Kupisch series and acyclic Nakayama algebras is given by where in the above description of R paths not belonging to A m should just be ignored. Using this identification, we will identify a Kupisch series with the corresponding acyclic Nakayama algebra.
Moreover, given an m-Kupisch series we explain how to describe the Auslander-Reiten quiver of the corresponding acyclic Nakayama algebra. We will also use the following lemma for computing syzygies, cosyzygies and Auslander-Reiten translations of modules over acyclic Nakayama algebras.
Lemma 4.2. Let Λ be an acyclic Nakayama algebra and M (i, j) = 0 be a Λ-module.
Proof. The claims about the Auslander-Reiten translations are [Vas18,Lemma 4.7]. The claim about syzygy in (a) follows by noticing that M (i + j − u i+j , u i+j ) is a projective cover of M (i, j) and the claim about cosyzygy in (b) follows by noticing that M (i, v i ) is an injective envelope of M (i, j).
In the rest of this and the next subsection we will only draw Auslander-Reiten quivers of Nakayama algebras. When drawing such an Auslander-Reiten quiver we will only draw the vertices as the arrows can be inferred by our conventions. We write some vertices in bold to denote that the additive closure of the indecomposable modules corresponding to the bold vertices is an n-cluster tilting subcategory for some n. Moreover, we will denote by h (k) a sequence h, h, . . . , h where h appears k times in a Kupisch series. We give an example using this notation. where the bold vertices correspond to the indecomposable modules in C. Using Theorem 3.2 we can see that C Λ is a 2-cluster tilting subcategory.
The algebras Λ m,2 are of special interest. Specifically, we will use the following Proposition. The following theorem will be our main tool in constructing examples of (n, d)-representation-finite algebras.
(a) Let A be a strongly (n, d)-representation-directed algebra and assume that there exists a simple projective A-module P with injective dimension d. Then Λ = Λ n+1,2 P

⊲I
(1) A is strongly (n, d + n)-representation-directed and there exists a simple projective Λ-module P ′ with injective dimension d + n.
(b) Let B be a strongly (n, d)-representation-directed algebra and assume that there exists a simple injective A-module I with projective dimension d. Then Λ = B P (n + 1) ⊲I Λ n+1,2 is strongly (n, d+ n)-representation-directed and there exists a simple injective Λ-module I ′ with projective dimension d + n.

⊲I
(1) A admits an n-cluster tilting subcategory and has global dimension at most d + n. To finish the proof it is enough to show that there exists a simple projective Λ-module P ′ with injective dimension d + n.
Since ψ * (I(1)) ∼ = φ * (P ) is supported in A, it follows from Corollary 2.37(b2) that the injective dimension of φ * (P ) is the same as the injective dimension of P , which is d by assumption. Hence the injective dimension of P ′ is d + n which completes the proof.
In particular we have the following corollary.
(a) Let A be a strongly (n, d)-representation-directed algebra and assume that there exists a simple projective A-module P with injective dimension d. Let Λ be the algebra Λ = Λ n+1,2 P (n + 1)

⊲I
(1) · · · P (n + 1) where there are k − 1 terms Λ n+1,2 on the right-hand side of the above expression. Then Λ is strongly (n, kn + d)-representation-directed. (b) Let B be a strongly (n, d)-representation-directed algebra and assume that there exists a simple injective B-module I with projective dimension d. Let Λ be the algebra Λ = B P (n + 1) ⊲I Λ n+1,2 P (n + 1)
Proof. Follows immediately by applying Theorem 4.5 k times.
Hence we have examples of (n, kn)-representation-finite algebras for any k. Our construction of examples of (n, d)-representation-finite algebras for d = kn will follow the same spirit.
4.1. The case of n being odd. Let us start with the case of n being odd. Given n, we will construct a strongly (n, d)-representation-directed algebra for any n ≤ d ≤ 2n − 1. Moreover, each such algebra will admit a simple projective module of injective dimension d. Then by applying Corollary 4.6 we obtain an example of an (n, d)-representation-directed algebra for any d.
Consider the following motivating example.

Similar computations show that
is an n-cluster tilting subcategory. Moreover, since proj. dim(J 1 ) < d and proj. dim(J 2 ) = d and J 1 and J 2 are the only indecomposable injective nonprojective Λ-modules, we have gl. dim(Λ) = d. For the final part, we have that J 2 ∼ = M n + d 2 + 1, 1 ∼ = I(1) and hence proj. dim(I(1)) = d. (b) The two cases are similar; let us only prove the case h > 2. The Auslander-Reiten quiver Γ(Λ) in this case has the form • N ∼ = M (2n−d+3)(d+1) 8 + 1, 1 , • J h ∼ = M 1 4 (n(2n − d + 5) + d + 5), 1 . Moreover the bold vertices correspond to the indecomposable projective-injective Λ-modules, the vertices Q i correspond to the indecomposable projective noninjective Λ-modules and the vertices J i correspond to the indecomposable injective nonprojective Λ-modules. Using Lemma 4.2 we compute Let us indicatively show that τ n (J i ) ∼ = Q i+1 for 1 ≤ i ≤ h − 1; the other computations are similar. Looking at the Auslander-Reiten quiver Γ(Λ) it is clear that the projective cover of J i corresponds to the vertex exactly to the left and above of J i . In other words, we have Then by Lemma 4.2 we have (for simplicity, we write (x, y) instead of M (x, y)) Assume that i > 1. Again it is easy to see that by noticing that J i−1 is a vertex in the Auslander-Reiten quiver Γ(Λ) and the sum of its coordinates is equal to the sum of the coordinates of Ω(J i ). Then It follows by a simple induction that for every which again satisfies (4.1). Hence Since s = d−n 2 (h + 1) + 2h, and since 1 ≤ i ≤ h − 1, it follows again by a simple induction that for every 1 ≤ i ≤ h − 1 and 1 ≤ k ≤ d−n 2 + 1 we have Ω 2k Ω 2i−2 (J i ) ∼ = τ k(h+1) τ i−1 (J 1 ). In particular, for k = d−n 2 + 1 we have Rewriting the above, we have shown that for 1 ≤ i ≤ h − 1 we have Clearly Ω d−n+2i (J i ) also satisfies (4.1) and so Hence It follows again by a simple induction that for every 1 ≤ i ≤ h − 1 and 0 ≤ k ≤ h − i we have In particular, for k = h − i − 1 we have which shows that τ n (J i ) ∼ = Q i+1 . Next, it follows from Theorem 3.2 that ) is an n-cluster tilting subcategory. For the computation of the global dimension, notice that again using Lemma 4.2 as well as the previous computations, for 1 ≤ i ≤ h − 1 we have A similar computation shows that Ω d (J h ) ∼ = Q h . Since n and d are both odd and since Q i is projective for any i it follows that gl. dim(Λ) = max{proj. dim(J i ) | 1 ≤ i ≤ h} = d.
Corollary 4.10. Let n be odd and d ≥ n. There exists an (n, d)-representation-finite algebra Λ.
4.2. The case of n being even. In this case we have the following families of (n, d)-representationfinite algebras.
Proposition 4.11. Let n be even and 0 < k < n.
Proof. The proof is similar to the proof of Proposition 4.9. Computations are done using Lemma 4.2.
Corollary 4.12. Let n be even and d ≥ 2n. There exists an (n, d)-representation-finite algebra Λ.
Let us give an example in this case as well.
Example 4.13. Let n = 6. Using Proposition 4.11 and Example 4.7 we have in Table 2 a list of (6, d)-representation-finite algebras Λ where the 6-cluster tilting subcategories are denoted by the bold vertices in the Auslander-Reiten quivers. Using Corollary 4.6 and the list in Table 2, we obtain a (6, d)-representation-finite algebra for any d ≥ 12.   Remark 4.15. Let us note that Theorem 4.14 is not sharp in the sense that there exist alebras that are (n, d)-representation-finite where n is even, while n < d < 2n and d is odd. For example, in [Vas18, Example 3.8] it was shown that the path algebra of the quiver with relations

Summary of notation
Due to the length and the technical nature of the paper, we include a list of terminology with references to the corresponding numbered definitions as well as a list of symbols with a short description and a reference to where they are first encountered.

C
The set of isomorphism classes of indecomposable modules in C ⊆ mod Λ. Section 1.1.

|C|
The cardinality of C. Section 1.1. add(C) The additive closure of C. Section 1.1.
Sub(C) (resp. Fac(C)) The subcategory of mod Λ containing all submodules (resp. factor modules) of modules in C. Section 1.1.

D
The standard duality D = Hom K (−, K) between mod Λ and mod Λ op . Section 1.1.
The category L is the gluing of its subcategories B and A. Definition 2.1.
The realization of a left abutment P = e 1 Λ and of a right abutment I = D(Λe h ). Section 2.2.1.

Q Λ
The ordinary quiver of an algebra Λ. Section 2.2.1.
f P (resp. g I ) The footing f P : Λ ։ KA h for a left abutment P and g I : Λ ։ KA h for a right abutment I. Definition 2.11. P △ (resp. △ I ) The foundation of a left abutment P and of a right abutment I. Proposition 2.13.
F P (resp. G I ) The smallest additive subcategory of mod Λ containing all modules in the foundation of a left abutment P (resp. a right abutment I).
The gluing of A and B along P and I. Definition 2.19.

supp(M )
The support of a representation M of a bound quiver algebra. Section 2.2.2.
The additive closure of the category of maximal left (resp. right) abutments of Λ. Section 3.1.

C \V
The additive closure of all indecomposable modules X ∈ C such that X ∈ V. Section 3.1.

lvl(T )
The level of a fracture T . Definition 3.4.
T L (resp. T R ) A left (resp. right) fracturing, that is a direct sum of fractures of maximal left (resp. right) abutments. Definition 3.7.
P ab (resp. I ab ) A left (resp. right) fracturing which is projective (resp. injective) as a module. Section 3.1.
h (k) A sequence h, h, . . . , h where h appears k times. Section 4.