n-APR tilts of a class of Koszul algebra realized by τ_[n]-mutations

Abstract

In this paper, we introduce the τ_[n]-mutations of a class of Koszul algebra and prove that for the Koszul algebra with the global dimension ≤ n, if its Koszul dual is an admissible (n-1)-translation algebra, then the quiver of endomorphism algebra of n-APR tilting module can be realized by τ_[n]-mutation.

Graphical abstract

Keywords

n-APR tilting module / n-translation algebra / τ_[n]-mutation

1 Introduction

The tilting theory plays an important role in the study of representation theory. In particular, the tilting modules induced by simple modules are of crucial significance in tilting theory. For example, the endomorphism algebra of the APR tilting module realizes the action of the BGP reflection on the quiver [1]. The BGP reflection clearly describes the mutation of the quiver and plays a vital role in the study of classical representation theory. The n-APR tilting module was recently introduced by Iyama and Oppermann, which generalized the classical APR tilting module to higher dimensions. It plays an important role in the higher-dimensional Auslander-Reiten theory [12]. For instance, the n-APR tilting modules preserve many important invariants such as n- for finiteness and n- for infiniteness [14-16]. Inspired by classical APR-tilting modules [4], we are highly interested in the following question:

Can the endomorphism algebra of the n-APR tilting module realize an action similar to that of the BGP reflection on the quiver?

Given a finite quiver Q, the translation quiver ZQ with translation τ plays a crucial role in realizing the BGP reflection. The quiver of the endomorphism algebra of the APR tilting module induced by a sink i of Q (i.e., the BGP reflection) can be realized in the translation quiver ZQ: consider Q as Q\times \left\{1\right\}, and then replace (i,1) with its translation \tau (i,1)=(i,0); the full sub-quiver of ZQ obtained in this way is the quiver of the endomorphism algebra of the APR tilting module.

Recently, Guo [6-10] generalized the classical translation quiver ZQ with translation τ to the admissible n-translation quiver with translation τ_[n], and defined the stable n-translation quiver \mathbb{Z}{|}_{n}Q of the admissible n-translation quiver through the smash product. The classical translation quiver ZQ with translation τ can be regarded as the case when n=1. Guo [8] observed a profound connection between the admissible n-translation algebras and Iyama's higher-dimensional representation theory. In addition, it is noted that Mizuno [15] has recently characterized the quiver of the endomorphism algebra of the n-APR tilting module. In this paper, considering the admissible n-translation algebras and Mizuno's characterization of the quiver of the endomorphism algebra in the n-APR tilting module, we investigate the action of the “higher-dimensional” BGP reflection on the quiver. The following shows the main results of this paper.

Theorem 1.1　 Let \mathrm{\Lambda }=kQ/\left(\rho \right) be a finite-dimensional basic Koszul algebra with global dimension n ≥ 2. Let a ∈ Q₀ be a sink of the Q and it induces the n-APR tilting module T_a.

If the Koszul dual of \mathrm{\Lambda } is an admissible (n–1)-translation algebra, then the n-APR tilt {\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathrm{\Lambda }}(T_a{)}^{\mathrm{o}\mathrm{p}} of \mathrm{\Lambda } with respect to the point a is the τ_[n]-mutation of \mathrm{\Lambda }with respect to a.

In Section 2, some basic knowledge is introduced. In Section 3, the connection between the n-APR tilting module and τ_[n]-mutation is discussed, and the main theorems of this paper are proved. In Section 4, the n-cubic pyramid algebra is used to illustrate Theorem 1.1 of this paper. The n-cubic pyramid algebra was recently introduced by Guo Jinyun and the first author of this paper. For relevant definitions and more information, please refer to [9].

2 Preliminaries

In this paper, unless otherwise emphasized, it is always assumed that the algebra \mathrm{\Lambda }is a finite-dimensional algebra over an algebraically closed field k, and \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda } is the category of finitely-generated left \mathrm{\Lambda }-modules. For a finitely-generated left \mathrm{\Lambda }-module M, it is denoted to add M as a full sub-category of mod \mathrm{\Lambda }, whose objects are all the direct summands of finite direct sums of M. Denote (-)* as the functor {\text{Hom}}_{\mathrm{\Lambda }}(-,\mathrm{\Lambda }), and the standard dual functor {\text{Hom}}_k(-,k) as D. For M ∈ \mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }, take its minimal projective resolution P_1\stackrel{f_1}{\to }{P}_0\stackrel{f_0}{\to }M\to 0. Denote the syzygy ΩM=Ker f₀ and \mathrm{T}\mathrm{r}M:=\mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}{f}_1^{\ast }. For more knowledge, please refer to [2].

In 2011, Iyama and Oppermann [12] introduced the n-APR tilting module, which generalizes the classical APR tilting module [1] to higher dimensions.

Definition 2.1 [11, Definition 3.1]　Let \mathrm{\Lambda }=kQ/\left(\rho \right) be a finite-dimensional k-algebra, and a be a sink of Q, and P(a) be the simple projective \mathrm{\Lambda }-module corresponding to the vertex a, \mathrm{\Lambda }=P\left(a\right)\oplus M. If id P(a) = n and {\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathrm{\Lambda }}^i(D\mathrm{\Lambda },P(a\left)\right)=0, , then T_a:={\tau }_n^{-}P\left(a\right)\oplus M is called an n-APR tilting module of \mathrm{\Lambda }, and {\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}\left(T_a\right) is called the n-APR tilt of \mathrm{\Lambda }. Here, {\tau }_n^{-}=\mathrm{T}\mathrm{r}{\mathrm{\Omega }}^{n-1}D.

The n-APR tilting module is a tilting module with projective dimension n [11, Theorem 3.2]. In particular, the 1-APR tilting module is the APR tilting module. Mizuno characterizes the quiver of the endomorphism algebra of the n-APR tilting module of an algebra with global dimension n [14, Proposition 6.55]. The method is as follows: Take the minimal injective resolution of P(a)

0\to P\left(a\right)\to I^0\to I^1\to \cdots \to I^n\to 0.

According to [11, Theorem 3.2], there is a minimal projective resolution of {\tau }_n^{-}P\left(a\right)

0\to P^0\to P^1\to \cdots \to P^n\to {\tau }_n^{-}P\left(a\right)\to 0,

where P^i={\nu }^{-}\left(I^i\right), 0\le i\le n, \nu =D{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathrm{\Lambda }}(-,\mathrm{\Lambda }), P^i\in \mathrm{a}\mathrm{d}\mathrm{d}\left(\mathrm{\Lambda }/P\left(a\right)\right), and since P(a) is a simple projective \mathrm{\Lambda }-module, there is P^0=P\left(a\right). Decompose P^n-1 and Pⁿ into direct sums of indecomposable projective \mathrm{\Lambda }-modules, and denote them as {\oplus }_{b\in B}{P}_{i_b} and {\oplus }_{c\in C}{P}_{j_c}, respectively. Denote the homomorphism α as {\left(a_{b}c\right)}_{b}c:{\oplus }_{b\in B}{P}_{i_b}\to {\oplus }_{c\in C}{P}_{j_c}, where a_{b}c\in e_{i_b}\mathrm{\Lambda }{e}_{j_c} is the right-multiplication a_{bc} map from P_{i_b} toP_{j_c}. Define a new set of arrow directions a_c^{\ast }:k\to j_c,c\in C. Define a new bound quiver \left(Q^{\prime },{\rho }^{\prime }\right)={\sigma }_a\left(Q,\rho \right):

• Vertex set: Q_0^{\prime }=Q_0;

• Arrow direction set: Q_1^{\prime }=\left\{\beta \in Q_1|t\left(\beta \right)\ne a\right\}\cup \left\{a_c^{\ast }:a\to j_c|c\in C\right\};

• Relation set {\rho }^{\prime }=\{\begin{array}{ll}0,& \mathrm{i}\mathrm{f}n=1;\\ \{r\in \rho |t\left(r\right)\ne a\}\cup \{\sum _{c\in C} a_{bc}{a}_c^{\ast }|b\in B\},& \mathrm{i}\mathrm{f}n>1.\end{array}

Proposition 2.1 [14, Proposition 6.55]　 Let \mathrm{\Lambda }=kQ/\left(\rho \right) be a finite-dimensional k-algebra, where ρ is its minimal relation set. Suppose a\in Q_0 is a sink such that T_a={\tau }_n^{-1}P\left(a\right)\oplus \Lambda /\Lambda e_a\Lambda is an n-APR tilting module of \mathrm{\Lambda }. Let \left(Q^{\prime },{\rho }^{\prime }\right):={\sigma }_a\left(Q,\rho \right). Then there exists an isomorphism {End}_{\Lambda }^{op}\left(T_a\right)\cong k{Q}^{\prime }/\left({\rho }^{\prime }\right), and here ρ′ is the minimal relation set of the bound quiver of {End}_{\Lambda }^{op}\left(T_a\right).

3 τ_[n]-mutation and n-APR tilting

This section will present the proof of the main Theorem 1.1.

First, some necessary concepts are provided.

For a given finite-dimensional basic algebra \mathrm{\Lambda }, by Gabriel's theorem, there exists a quiver Q and a set of relations ρ such that \Lambda \cong kQ/\left(\rho \right). (Q, ρ) is called the bound quiver of \mathrm{\Lambda }. Let \mathrm{\Lambda }={\mathrm{\Lambda }}_0\oplus {\mathrm{\Lambda }}_1\oplus \cdots \oplus {\mathrm{\Lambda }}_l, such that {\mathrm{\Lambda }}_i{\mathrm{\Lambda }}_j={\mathrm{\Lambda }}_{i+j}, then it is called a graded algebra. In the following context, unless otherwise emphasized, \mathrm{\Lambda } always refers to a finite-dimensional graded algebra whose bound quiver is (Q, ρ). Suppose there is a one-to-one correspondence {\tau }_{\left[n\right]}:Q_P\to Q_I defined on two subsets Q_P,Q_I\subseteq Q_0 of Q₀, and the bound quiver (Q, ρ) satisfies the following four conditions, then Q is called an n-translation quiver, denoted by \left(Q,\rho ,{\tau }_{\left[n\right]}\right), and {\tau }_{\left[n\right]} is called the n-translation of Q [8]:

(1) For any vertex i in Q_P, there exists a bound path of length n + 1 from {\tau }_{\left[n\right]}i to i; the length of the longest bound path is n + 1, and for any bound path p of length n + 1, there exists i\in Q_P, such that s\left(p\right)={\tau }_{\left[n\right]}i,t\left(p\right)=i.

(2) For any i\in Q_P, any two bound paths of length n + 1 from {\tau }_{\left[n\right]}i to i are linearly dependent.

(3) For any i\in Q_P,j\in Q_0, let u=\sum _{s=1}^m a_s{p}_s be a linear combination of bound paths p_s of length from j to i. Then there exists a path q of length n+1-t from {\tau }_{\left[n\right]}i to j, such that uq is nonzero in kQ/\left(\rho \right).

(4) For any i\in Q_I,j\in Q_0, let u=\sum _{s=1}^m a_s{p}_s be a linear combination of bound paths p_s of length from i to j. Then there exists a path q of length n+1-t from j to {\tau }_{\left[n\right]}^{-1}i, such that qu is nonzero in kQ/\left(\rho \right).

The vertices in Q₀\Q_P and Q₀\Q_I are called projective vertices and injective vertices respectively. A graded algebra is called an n-pretranslation algebra if its bound quiver is an n-translation quiver. An n-translation quiver is called a stable n-translation quiver if Q_P=Q_0=Q_I, and the corresponding bound path algebra is called a stable n-pretranslation algebra.

Let \left(Q,\rho ,{\tau }_{\left[n\right]}\right) be an n-translation quiver and p be a bound path from i to j. If any bound path of length l(p) from i to j is linearly related to p, then p is called:

(1) t-th left stark with respect to the vertex i′, and if for any bound path w of length from i′ to i, pw is a bound path.

(2) t-th right stark with respect to the vertex j′, and if for any bound path u of length from j to j′, up is a bound path.

Let \left(Q,\rho ,{\tau }_{\left[n\right]}\right)be an n-translation quiver, bound path p is called:

(1) Right (left) translatable, and if p can be written as a linear combination of bound paths inkQ/\left(\rho \right) that do not pass through any projective (injective) vertices.

(2) Semi-translatable, and if p can be written as a linear combination of paths in the form p′p′′ in kQ/\left(\rho \right), where p′′ is a bound path that does not pass through any injective vertices and p′ is a bound path that does not pass through any projective vertices.

(3) Translatable, and if p is either left translatable, or right translatable, or semi-translatable.

An n-translation quiver Q is called admissible if it also satisfies the following three conditions:

(1) For each bound path g, there exist paths q′ and q, such that q′gq is a bound path of length n+1;

(2) Any bound path from i\in Q_I to j\in Q_P is linearly related to a set of translatable bound paths;

(3) Let i\in Q_P, p be a bound path ending at i, and q be a bound path starting from {\tau }_{\left[n\right]}i, such that l\left(p\right)+l\left(q\right)\le n. If p passes through a projective vertex and q passes through an injective vertex, then either p is a n+1-\left(l\left(p\right)+l\left(q\right)\right)th left stark with respect to t(q) or q is a n+1-\left(l\left(p\right)+l\left(q\right)\right)th right stark with respect to s(p).

Let \Lambda =kQ/\left(\rho \right) be a Koszul algebra and the bound quiver of its Koszul dual \mathrm{\Gamma }:={\Lambda }^{!,\mathrm{o}\mathrm{p}}=kQ/\left({\rho }^{!,\mathrm{o}\mathrm{p}}\right)be an admissible (n-1)-translation quiver. Denote its (n-1)-translation by {\tau }_{\left[n-1\right]}, the set of projective vertices of (Q,{\rho }^{!,\mathrm{o}\mathrm{p}}) as P, and the set of injective vertices as I. Denote the trivial extension of {\mathrm{\Lambda }}^{!,\mathrm{o}\mathrm{p}} and the bound quiver of the trivial extension as \tilde{\mathrm{\Gamma }} and (\tilde{Q},{\rho }_{\tilde{\mathrm{\Gamma }}}), respectively. According to [8, Proposition 4.2], there are

(1) {\tilde{Q}}_0=Q_0 and {\tilde{Q}}_1=Q_1\cup \{{\beta }_i:i\to {\tau }_{\left[n-1\right]}i\mid i\in Q_0\mathrm{\setminus }\mathcal{P}\};

(2) {\rho }_{\tilde{\mathrm{\Gamma }}}={\rho }^{!,\mathrm{o}\mathrm{p}}\cup \{{\beta }_{{\tau }_{[n-1]}i}{\beta }_i\mid i,{\tau }_{[n-1]}i\in Q_0\setminus \mathcal{P}\}\cup \left\{{\tau }_{[n-1]}\right(\alpha ){\beta }_i-{\beta }_j\alpha \mid \alpha :i\to j\in Q_1,i,j\in Q_0\setminus \mathcal{P}\};

(3) \tilde{\Gamma } is a stable n-pretranslation algebra and the n-translation is trivial.

Based on [8], the quiver of smash product \tilde{\mathrm{\Gamma }}\mathrm{♯}k{\mathbb{Z}}^{\mathbb{\ast }}is (\mathbb{Z}{|}_{n-1}Q,{\rho }_{\mathbb{Z}{|}_{n-1}Q}):

• Vertex set: \mathbb{Z}{|}_{n-1}{Q}_0=Q_0\times \mathbb{Z}=\left\{\left(a,t\right)|a\in Q_0,t\in \mathbb{Z}\right\};

• Arrow direction set:

\begin{array}{rl}Z{|}_{n-1}{Q}_1& =\left\{\right(\alpha ,t):(i,t)\to (j,t)\mid \alpha :i\to j\in Q_1,t\in Z\}\\ & \cup \left\{\right({\beta }_i,t):(i,t)\to ({\tau }_{[n-1]}i,t+1)\mid i\in Q_0\setminus P,t\in Z\};\end{array}

• Relation set: {\rho }_{\mathbb{Z}}{|}_{n-1}Q={\cup }_{t\in \mathbb{Z}}\left(\left({\rho }^{!,\mathbf{o}\mathbf{p}}\right)\left(t\right)\cup \left({\rho }^{!,\mathbf{o}\mathbf{p}}\right){\left(t\right)}^{\ast }\right).

For any path g={\alpha }_l\cdots {\alpha }_1 in Q, denote g\left(t\right):=\left({\alpha }_l,t\right)\cdots \left({\alpha }_1,t\right), and for a linear combination z=\sum _s a_s{g}_s of paths \{g_s\mid s=1,2,\dots ,r\} in Q, denote z\left(t\right):=\sum _s a_s{g}_s\left(t\right), and also denote

\begin{array}{rl}\left({\rho }^{!,\mathrm{o}\mathrm{p}}\right)\left(t\right)& =\left\{p\right(t)\mid p\in {\rho }^{!,\mathrm{o}\mathrm{p}}\},\\ \left({\rho }^{!,\mathrm{o}\mathrm{p}}\right)(t{)}^{\ast }& =\{\left({\tau }_{[n-1]}\right(\alpha ),t+1)({\beta }_i,t)-({\beta }_j,t)(\alpha ,t)|i,j\in Q_0\setminus P,\alpha :i\to j\in Q_1\}\\ & \cup \left\{\left({\beta }_{{\tau }_{\left[n-1\right]}i},t+1\right)\left({\beta }_i,t\right)|i,{\tau }_{\left[n-1\right]}i\in Q_0\setminus \mathcal{P}\right\}.\end{array}

If i is a sink of the quiver Q, define the τ_[n]-mutation \left(s_i^{-}Q,s_i^{-}\rho \right) of (Q, p) with respect to i as the bound full sub-quiver of the quiver {\left(\mathbb{Z}{|}_{n-1}Q,{\rho }_{\mathbb{Z}{|}_{n-1}Q}\right)}^{!,\mathrm{o}\mathrm{p}}of the Koszul dual of \tilde{\mathrm{\Gamma }}\mathrm{♯}k{\mathbb{Z}}^{\mathbb{\ast }}. Its vertex set is \left(Q_0\times \left\{1\right\}\mathrm{\setminus }\left(i,1\right)\right)\cup \{{\tau }_{\left[n\right]}\left(i,1\right)=\left(i,0\right)\} (that is, replace (i, l) with its n-translation {\tau }_{\left[n\right]}\left(i,1\right)=\left(i,0\right)). More precisely:

• Vertex set: {\left(s_i^{-}Q\right)}_0=Q_0\times \left\{1\right\}\mathrm{\setminus }\left(i,1\right)\cup \{{\tau }_{\left[n\right]}\left(i,1\right)=\left(i,0\right)\};

• Arrow direction set: {\left(s_i^{-}Q\right)}_1=\{\left(\alpha ,1\right)\mid \alpha \in Q_1,t\left(\alpha \right)\ne a\}\cup \{\left({\beta }_i,0\right):\left(i,0\right)\to \left({\tau }_{\left[n-1\right]}i,1\right)\};

• Relation Set:

\begin{array}{rl}& s_i^{-}\rho =\\ & \{\begin{array}{ll}0,& \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}n=1;\\ \left\{r\right(1)\in \rho (1\left)\right|t\left(r\right)\ne i,r\in \rho \}\cup \{(\gamma ,1)({\beta }_i,0)\left|s\right(\gamma )={\tau }_{[n-1]}i,\gamma \in Q_1\},& \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}n>1.\end{array}\end{array}

Here, the bound path algebra k{s}_i^{-}Q/\left(s_i^{-}\rho \right) of \left(s_i^{-}Q,s_i^{-}\rho \right) is called the τ_[n]-mutation with respect to i.

Dually, if j is a source of the quiver Q, define the co-τ_[n]-mutation\left(s_j^{+}Q,s_j^{+}\rho \right) of (Q, ρ) with respect to j as the bound sub-quiver of the quiver (\mathbb{Z}{|}_{n-1}Q,{\rho }_{\mathbb{Z}{|}_{n-1}Q}{)}^{!,\mathrm{o}\mathrm{p}} of the Koszul dual of \tilde{\Gamma }\mathrm{♯}k{\mathbb{Z}}^{\mathbb{\ast }}. Its vertex set is (Q_0\times \{1\}\setminus (j,1\left)\right)\cup \left\{{\tau }_{\left[n\right]}^{-1}(j,1)=(j,2)\right\} (replace (j, 1) with its inverse n-translation {\tau }_{\left[n\right]}^{-}1\left(j,1\right)=\left(j,2\right)). Here, the bound path algebra k{s}_j^{+}Q/\left(s_j^{+}\rho \right) of \left(s_j^{+}Q,s_j^{+}\rho \right) is called the co-τ_[n]-mutation of kQ/\left(\rho \right) with respect to j.

Proof of Theorem 1.1　According to [3], \mathrm{\Lambda }^op, \mathrm{\Lambda }^!, {\mathrm{\Lambda }}^{!,\mathrm{o}\mathrm{p}} are all Koszul algebras, \mathrm{g}\mathrm{l}.\mathrm{d}\mathrm{i}\mathrm{m}{\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}=\mathrm{g}\mathrm{l}.\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{\Lambda }=n (see [13]). According to the properties of the Koszul complex [3, p. 485], the left simple \mathrm{\Lambda }^op-module {\mathrm{\Lambda }}_0^{\mathrm{o}\mathrm{p}}{e}_a=S\left(a\right) corresponding to the source vertex a of Q^op has the following minimal projective resolution:

0\to {\Lambda }^{\mathrm{o}\mathrm{p}}\otimes D\left({\Lambda }_n^{!,\mathrm{o}\mathrm{p}}\right)e_a\to \cdots \to {\Lambda }^{\mathrm{o}\mathrm{p}}\otimes D\left({\Lambda }_0^{!,\mathrm{o}\mathrm{p}}\right)e_a\to {\Lambda }_0^{\mathrm{o}\mathrm{p}}{e}_a=S\left(a\right)\to 0.

Therefore, there is a minimal injective resolution of the left simple \Lambda-module {\Lambda }_0{e}_a=S\left(a\right)=P\left(a\right) corresponding to the sink a of Q.

0\to P\left(a\right)\cong D\left({\Lambda }_0^{\mathrm{o}\mathrm{p}}{e}_a\right)\to D\left({\Lambda }^{\mathrm{o}\mathrm{p}}\otimes D\left({\Lambda }_0^{!,\mathrm{o}\mathrm{p}}\right)e_a\right)\to \cdots \to D\left({\Lambda }^{\mathrm{o}\mathrm{p}}\otimes D\left({\Lambda }_n^{!,\mathrm{o}\mathrm{p}}\right)e_a\right)\to 0.

As a left \mathrm{\Lambda }-module, there is the following isomorphism:

\begin{array}{rl}D\left({\Lambda }^{\mathrm{o}\mathrm{p}}\otimes D\left({\Lambda }_n^{!,\mathrm{o}\mathrm{p}}\right)e_a\right)& \cong D\left({\Lambda }^{\mathrm{o}\mathrm{p}}\otimes D\left(e_a{\Lambda }_n^{!,\mathrm{o}\mathrm{p}}\right)\right)\\ & \cong e_a{\Lambda }_n^{!,\mathrm{o}\mathrm{p}}\otimes D\left({\Lambda }^{\mathrm{o}\mathrm{p}}\right)\\ & \cong e_{{\tau }_{\left[n-1\right]}a}\left(D{\Lambda }^{\mathrm{o}\mathrm{p}}\right)\\ & \cong D\left({\Lambda }^{\mathrm{o}\mathrm{p}}{e}_{{\tau }_{\left[n-1\right]}a}\right)\\ & \cong D\left(e_{{\tau }_{\left[n-1\right]}a}\Lambda \right)\\ & =I\left({\tau }_{\left[n-1\right]}\alpha \right),\end{array}

and

\begin{array}{rl}D({\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}\otimes D({\mathrm{\Lambda }}_{n-1}^{!,\mathrm{o}\mathrm{p}}\left)e_a\right)& \cong D({\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}\otimes D(e_a{\mathrm{\Lambda }}_{n-1}^{!,\mathrm{o}\mathrm{p}}\left)\right)\\ & \cong e_a{\mathrm{\Lambda }}_{n-1}^{!,\mathrm{o}\mathrm{p}}\otimes D\left({\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}\right)\\ & \cong \underset{\alpha :{\tau }_{[n-1]}a\to j}{\oplus } e_{j}D\left({\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}\right)\\ & \cong \underset{\alpha :{\tau }_{[n-1]}a\to j\in Q_1}{\oplus } D\left({\mathrm{\Lambda }}^{\mathrm{o}\mathrm{p}}{e}_j\right)\\ & \cong \underset{\alpha :{\tau }_{[n-1]}a\to j\in Q_1}{\oplus } D\left(e_j\mathrm{\Lambda }\right)\\ & =\underset{\alpha :{\tau }_{[n-1]}a\to j\in Q_1}{\oplus } I\left(j\right).\end{array}

Hence, the injective resolution mentioned above can be written as

0\to P\left(a\right)\to I\left(a\right)\to \cdots \to \underset{\alpha :{\tau }_{[n-1]}a\to j\in Q_1}{\oplus } I\left(j\right)\to I\left({\tau }_{[n-1]}a\right)\to 0.

With the proof [11, Theorem 3.2], minimal projective resolution of left \mathrm{\Lambda }-module {\tau }_n^{-}P\left(a\right) is

0\to P\left(a\right)\to \cdots \to \underset{\alpha :{\tau }_{[n-1]}a\to j\in Q_1}{\oplus } P\left(j\right)\stackrel{\left(\alpha \right)}{\to }P\left({\tau }_{[n-1]}a\right)\to {\tau }_n^{-}P\left(a\right)\to 0.

From Proposition 2.1, the quiver of {\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathrm{\Lambda }}(T_a{)}^{\mathrm{o}\mathrm{p}} is

• Vertex set: Q_0^{\prime }=Q_0;

• Arrow direction set: Q_1^{\prime }=\{\alpha \in Q_1\mid t\left(\alpha \right)\ne a\}\cup \{a^{\ast }:a\to {\tau }_{\left[n-1\right]}a\};

• Relation set: {\rho }^{\prime }=\{r\in \rho \mid t\left(r\right)\ne a\}\cup \{\alpha a^{\ast }\mid \alpha :{\tau }_{\left[n-1\right]}a\to j\in Q_1\}.

By Comparing \left(s_a^{-}Q,s_a^{-}\rho \right), it can be seen that it is the same bound quiver as \left(Q^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }}\right). □

Note　In fact, as can be seen from the proofs of the theorems mentioned above, the condition “admissible” in the theorem is only used in the last sentence. However, the characterization of its quiver does not use the “admissible” condition.

4 n-cubic pyramid algebra

Let m\ge 3,n\ge 2, and denote

{\mathbf{e}}_i:=(\stackrel{1}{0},\dots ,\stackrel{i-1}{0},\stackrel{i}{1},\stackrel{i+1}{0},\dots ,\stackrel{n}{0})\in {\mathbb{Z}}^n,\mathbf{e}\mathbf{:=}\left(\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\dots \mathbf{,}\mathbf{1}\right)\in {\mathbb{Z}}^{\mathbf{n}}\mathbf{,}{\mathbf{e}}_{\mathbf{0}}\mathbf{:=}\left(\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\dots \mathbf{,}\mathbf{0}\right)\in {\mathbb{Z}}^{\mathbf{n}}\mathbf{.}

Let \mathbf{i}\mathbf{=}\left({\mathbf{i}}_{\mathbf{1}}\mathbf{,}{\mathbf{i}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{i}}_{\mathbf{n}}\right)\in {\mathbb{Z}}^{\mathbf{n}}, and denote \mathbf{i}\left(\mathbf{1}\right)\mathbf{=}\mathbf{i}\mathbf{+}{\mathbf{e}}_{\mathbf{1}}\mathbf{,}\mathbf{i}\left(\mathbf{t}\right)\mathbf{=}\mathbf{i}\mathbf{-}{\mathbf{e}}_{\mathbf{t}\mathbf{-}\mathbf{1}}\mathbf{+}{\mathbf{e}}_{\mathbf{t}}, where 2\le t\le n, denote \mathbf{i}\left({\mathbf{t}}_{\mathbf{1}}\mathbf{,}{\mathbf{t}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{t}}_{\mathbf{s}}\right)\mathbf{=}\mathbf{i}\left({\mathbf{t}}_{\mathbf{1}}\right)\left({\mathbf{t}}_{\mathbf{2}}\right)\cdots \left({\mathbf{t}}_{\mathbf{s}}\right)\mathbf{.} Similarly, denote \left(1\right)\mathbf{i}\mathbf{=}\mathbf{i}\mathbf{-}{\mathbf{e}}_{\mathbf{1}}\mathbf{,}\left(\mathbf{t}\right)\mathbf{i}\mathbf{=}\mathbf{i}\mathbf{+}{\mathbf{e}}_{\mathbf{t}\mathbf{-}\mathbf{1}}\mathbf{-}{\mathbf{e}}_{\mathbf{t}}, where 2\le t\le n, and denote \left(t_1,t_2,\dots ,t_s\right)\mathbf{i}\mathbf{=}\left({\mathbf{t}}_{\mathbf{s}}\right)\left({\mathbf{t}}_{\mathbf{s}\mathbf{-}\mathbf{1}}\right)\cdots \left({\mathbf{t}}_{\mathbf{1}}\right)\mathbf{i}. Obviously, \left(t\right)\mathbf{i}\left(\mathbf{t}\right)\mathbf{=}\mathbf{i}.

In this section, we examine the n-cubic pyramid algebra \left(Q\left(n\right),\rho \left(n\right)\right)=\left(Q{\left(n\right)}_0,Q{\left(n\right)}_1,{\tau }_{\left[n-1\right]}\right), where

• Vertex set: Q{\left(n\right)}_0=\{\mathbf{i}\mathbf{=}\left({\mathbf{i}}_{\mathbf{1}}\mathbf{,}{\mathbf{i}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{i}}_{\mathbf{n}}\right)\mid \mathbf{1}\le {\mathbf{i}}_{\mathbf{t}}\mathbf{,}\sum _{\mathbf{t}\mathbf{=}\mathbf{1}}^{\mathbf{s}}{\mathbf{i}}_{\mathbf{t}}\le \mathbf{m}\mathbf{+}\mathbf{s}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{1}\le \mathbf{s}\le \mathbf{n}\}\mathbf{;}

• Arrow direction set: Q{\left(n\right)}_1=\{{\gamma }_{\mathrm{i}}^{\left(t\right)}:\mathbf{i}\to \mathbf{i}\left(\mathbf{t}\right)\mid \mathbf{1}\le \mathbf{t}\le \mathbf{n}\mathbf{,}\mathbf{i}\mathbf{,}\mathbf{i}\left(\mathbf{t}\right)\in \mathbf{Q}{\left(\mathbf{n}\right)}_{\mathbf{0}}\}\mathbf{;}

• Relation set:

• (n-1)-translation: {\tau }_{\left[n-1\right]}\left(\mathbf{i}\right)=\mathbf{i}\mathbf{-}{\mathbf{e}}_{\mathbf{n}}\mathbf{,}\mathbf{i}\in \{\left({\mathbf{i}}_{\mathbf{1}}\mathbf{,}{\mathbf{i}}_{\mathbf{2}}\mathbf{,}\dots \mathbf{,}{\mathbf{i}}_{\mathbf{n}}\right)\in \mathbf{Q}{\left(\mathbf{n}\right)}_{\mathbf{0}}\mid {\mathbf{i}}_{\mathbf{n}}>\mathbf{1}\}\mathbf{.}

As shown in [9], the n-representation finite (n-1)-Auslander algebra T_m^{\left(n\right)}\left(k\right) is the Koszul dual of the bound path algebra kQ(n)/(ρ(n)) corresponding to (Q(n),ρ(n)), and (Q(n),ρ(n)) is an admissible (n-1)-translation quiver. From the last sentence of the proof of [9, Theorem 4.5] and [5], it is known that T_m^{\left(n\right)}\left(k\right) is an almost Koszul algebra of type (m-1,n), but its global dimension of T_m^{\left(n\right)}\left(k\right) is n (see [11]). Thus, T_m^{\left(n\right)}\left(k\right) is a Koszul algebra, and its unique sink a=(1,1,\dots ,1,m) induces an n-APR tilting module [12]. Therefore, T_m^{\left(n\right)}\left(k\right) satisfies the conditions of Theorem 1.1, and there are:

Corollary 4.1　 For any m\ge 2,n\ge 1, let T_a be the n-APR tilting module induced by the unique sink a=(1,1,\dots ,1,m) of T_m^{\left(n\right)}\left(k\right). Then {End}_{T_m^{\left(n\right)}\left(k\right)}(T_a{)}^{op} is the τ_[n]-mutation of T_m^{\left(n\right)}\left(k\right) with respect to a.

Example 4.1　Take m = 4, n = 2, then \left(Q\right(2),\rho (2\left)\right)is the following quiver:

From [8, Proposition 4.2], the trivial extension quiver of T_4^{\left(2\right)}(k{)}^{!,\mathrm{o}\mathrm{p}} is \left(\tilde{Q}\right(2),\tilde{\rho }(2\left)\right).

And \tilde{\rho }\left(2\right)=\{{\gamma }_{\mathbf{i}\left(\mathbf{i}\right)}^{\left(i\right)}{\gamma }_{\mathbf{i}}^{\left(i\right)}\mid i=1,2,3\}\cup \{{\gamma }_{\mathbf{i}\left(\mathbf{j}\right)}^{\left(i\right)}{\gamma }_{\mathbf{i}}^{\left(j\right)}+{\gamma }_{\mathbf{i}\left(\mathbf{i}\right)}^{\left(j\right)}{\gamma }_{\mathbf{i}}^{\left(i\right)}\mid i\ne j,i,j\ne 3\}\cup \{{\gamma }_{\mathbf{i}\left(\mathbf{3}\right)}^{\left(i\right)}{\gamma }_{\mathbf{i}}^{\left(3\right)}-{\gamma }_{\mathbf{i}\left(\mathbf{i}\right)}^{\left(3\right)}{\gamma }_{\mathbf{i}}^{\left(i\right)}\mid i\ne 3\}.

Then \left(\mathbb{Z}{|}_{1}Q\right(2),{\rho }_{\mathbb{Z}{|}_{1}Q\left(2\right)}{)}^{!,\mathrm{o}\mathrm{p}}=(\mathbb{Z}{|}_{1}Q\left(2\right),{\rho }_{\mathbb{Z}{|}_{1}Q\left(2\right)}^{!,\mathrm{o}\mathrm{p}}) is

Here, {\rho }_{\mathbb{Z}{|}_{1}Q\left(2\right)}^{!,\mathrm{o}\mathrm{p}}=\{{\gamma }_{\mathbf{i}\left(\mathbf{j}\right)}^{\left(i\right)}{\gamma }_{\mathbf{i}}^{\left(j\right)}-{\gamma }_{\mathbf{i}\left(\mathbf{i}\right)}^{\left(j\right)}{\gamma }_{\mathbf{i}}^{\left(i\right)}\mid i\ne j,i,j\ne 3\}\cup \{{\gamma }_{\mathbf{i}\left(\mathbf{3}\right)}^{\left(i\right)}{\gamma }_{\mathbf{i}}^{\left(3\right)}+{\gamma }_{\mathbf{i}\left(\mathbf{i}\right)}^{\left(3\right)}{\gamma }_{\mathbf{i}}^{\left(i\right)}\mid i\ne 3\}. Then the 2-APR tilt of T_4^{\left(2\right)}\left(k\right) with respect to vertex (14) is a τ_[2]-mutation of T_4^{\left(2\right)}\left(k\right), and its bound quiver is

The relation set is \{{\gamma }_{\mathrm{i}\left(j\right)}^{\left(i\right)}{\gamma }_{\mathrm{i}}^{\left(j\right)}-{\gamma }_{\mathrm{i}\left(i\right)}^{\left(j\right)}{\gamma }_{\mathrm{i}}^{\left(i\right)}\mid i\ne j,i,j\ne 3\} (see Theorem 1.1). It is noticed that τ_[2](141) = (140). Therefore, it is the bound full sub-quiver obtained from {\left(\mathbb{Z}{|}_{1}Q\left(2\right),{\rho }_{\mathbb{Z}{|}_{1}Q\left(2\right)}\right)}^{!,\mathrm{o}\mathrm{p}} by replacing (141) in \left(Q,\rho \right)\times \left\{1\right\} with τ_[2] (141).

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