$\tau$-Tilting Finite Cluster-Tilted Algebras

TL;DR

This paper establishes a precise equivalence between $ au$-tilting finiteness and representation-finiteness for cluster-tilted algebras, deepening understanding of their structural properties.

Contribution

It proves that a cluster-tilted algebra is $ au$-tilting finite if and only if it is representation-finite, providing a complete characterization.

Findings

$ au$-tilting finiteness coincides with representation-finiteness for cluster-tilted algebras

The paper offers a new criterion for classifying cluster-tilted algebras

Enhances understanding of the relationship between $ au$-tilting theory and representation theory

Abstract

Let B be a cluster-tilted algebra. We prove that B is $\tau$-tilting finite if and only if B is representation-finite.

Equations

$$\{\text{tilted}\text{algebras}\}⟼\{\text{cluster-tilted}\text{algebras}\}$$

$$C⟼B=C⋉E$$

Full text

1 Introduction

The theory of $\tau$-tilting was introduced by Adachi, Iyama and Reiten in [1] as a far-reaching generalization of classical tilting theory for finite dimensional associative algebras. One of the main classes of objects in the theory is that of $\tau$-rigid modules: a module $M$ over an algebra $\Lambda$ is $\it{\tau_{\Lambda}}$-$\it{rigid}$ if $\mathop{\text{Hom}_{\Lambda}(M,\tau_{\Lambda}M)}=0$, where $\tau_{\Lambda}M$ denotes the Auslander-Reiten translation of $M$; such a module $M$ is called $\it{\tau_{\Lambda}}$-$\it{tilting}$ if the number $|M|$ of non-isomorphic indecomposable summands of $M$ equals the number of isomorphism classes of simple $\Lambda$-modules. Recently, a new class of algebras were introduced by Demonet, Iyama, Jasso in [10] called $\it{\tau_{\Lambda}}$-$\it{tilting~{}finite}$ algebras. They are defined as finite dimensional algebras with only a finite number of isomorphism classes of basic $\tau_{\Lambda}$-tilting modules.

An obvious sufficient condition for an algebra to be $\tau_{\Lambda}$-titling finite is for it to be representation-finite. In general, this condition is not necessary. The aim of this note is to prove for $\it{cluster}$-$\it{tilted~{}algebras}$, this condition is in fact necessary.

Tilted algebras are the endomorphism algebras of tilting modules over hereditary algebras, introduced by Happel and Ringel in [11]. Cluster-tilted algebras are the endomorphism algebras of cluster-tilting objects over cluster categories of hereditary algebras, introduced by Buan, Marsh and Reiten in [8]. The similarity in the two definitions lead to the following precise relation between tilted and cluster-tilted algebras, which was established in [2] by Assem, Br$\ddot{\text{u}}$stle, and Schiffler.

There is a surjective map

[TABLE]

[TABLE]

where $E$ denotes the $C$-$C$-bimodule $E=\text{Ext}_{C}^{2}(DC,C)$ and $C\ltimes E$ is the trivial extension.

This result allows one to define cluster-tilted algebras without using the cluster category. Using this construction, we show the following.

Theorem 1.1**.**

Let $B$ be a cluster-tilted algebra. Then $B$ is $\tau_{B}$-tilting finite if and only if $B$ is representation-finite.

2 Notation and Preliminaries

We now set the notation for the remainder of this paper. All algebras are assumed to be finite dimensional over an algebraically closed field $k$. If $\Lambda$ is a $k$-algebra then denote by $\mathop{\text{mod}}\Lambda$ the category of finitely generated right $\Lambda$-modules and by $\mathop{\text{ind}}\Lambda$ a set of representatives of each isomorphism class of indecomposable right $\Lambda$-modules. We denote by $\operatorname{add}M$ the smallest additive full subcategory of $\mathop{\text{mod}}\Lambda$ containing $M$, that is, the full subcategory of $\mathop{\text{mod}}\Lambda$ whose objects are the direct sums of direct summands of the module $M$. Given $M\in\mathop{\text{mod}}\Lambda$, the projective dimension of $M$ is denoted $\operatorname{pd}_{\Lambda}M$. We let $\tau_{\Lambda}$ and $\tau^{-1}_{\Lambda}$ be the Auslander-Reiten translations in $\mathop{\text{mod}}\Lambda$. We let $D$ be the standard duality functor $\operatorname{Hom}_{k}(-,k)$. Finally, $\Gamma(\mathop{\text{mod}}\Lambda)$ will denote the Auslander-Reiten quiver of $\Lambda$.

2.1 Tilted Algebras

Tilting theory is one of the main themes in the study of the representation theory of algebras. Given a $k$-algebra $A$, one can construct a new algebra $B$ in such a way that the corresponding module categories are closely related. The main idea is that of a tilting module.

Definition 2.1**.**

Let $A$ be an algebra. An $A$-module $T$ is a partial tilting module if the following two conditions are satisfied:

1. (1)

$\operatorname{pd}_{A}T\leq 1$. 2. (2)

$\operatorname{Ext}_{A}^{1}(T,T)=0$.

A partial tilting module $T$ is called a tilting module if it also satisfies the following additional condition:

1. (3)

There exists a short exact sequence $0\rightarrow A\rightarrow T^{\prime}\rightarrow T^{\prime\prime}\rightarrow 0$ in $\mathop{\text{mod}}A$ with $T^{\prime}$ and $T^{\prime\prime}$ $\in\operatorname{add}T$.

We now state the definition of a tilted algebra.

Definition 2.2**.**

Let $A$ be a hereditary algebra with $T$ a tilting $A$-module. Then the algebra $B=\operatorname{End}_{A}T$ is called a tilted algebra.

2.2 Cluster categories and cluster-tilted algebras

Let $C=kQ$ be the path algebra of the quiver $Q$ and let $\mathcal{D}^{b}(\mathop{\text{mod}}C)$ denote the derived category of bounded complexes of $C$-modules. The $cluster$ $category$ $\mathcal{C}_{C}$ is defined as the orbit category of the derived category with respect to the functor $\tau_{\mathcal{D}}^{-1}[1]$, where $\tau_{\mathcal{D}}$ is the Auslander-Reiten translation in the derived category and $[1]$ is the shift. Cluster categories were introduced in [7], and in [9] for type $\mathbb{A}$.

An object $T$ in $\mathcal{C}_{C}$ is called $cluster$-$tilting$ if $\text{Ext}_{\mathcal{C}_{C}}^{1}(T,T)=0$ and $T$ has $|Q_{0}|$ non-isomorphic indecomposable direct summands where $|Q_{0}|$ is the number of vertices of $Q$. The endomorphism algebra $\operatorname{End}_{\mathcal{C}_{C}}T$ of a cluster-tilting object is called a $cluster$-$tilted$ $algebra$ [8].

2.3 Relation extensions

Let $C$ be an algebra of global dimension at most 2 and let $E$ be the $C$-$C$-bimodule $E=\text{Ext}_{C}^{2}(DC,C)$.

Definition 2.3**.**

The relation extension of $C$ is the trivial extension $B=C\ltimes E$, whose underlying $C$-module structure is $C\oplus E$, and multiplication is given by $(c,e)(c^{\prime},e^{\prime})=(cc^{\prime},ce^{\prime}+ec^{\prime})$.

Relation extensions were introduced in [2]. In the special case where $C$ is a tilted algebra, we have the following result.

Theorem 2.4**.**

[2, Theorem 3.4]. Let C be a tilted algebra. Then $B=C\ltimes\emph{Ext}_{C}^{2}(DC,C)$ is a cluster-tilted algebra. Moreover all cluster-tilted algebras are of this form.

2.4 Slices and local slices

Definition 2.5**.**

A $\it{slice}$ $\Sigma$ in $\Gamma(\mathop{\text{mod}}A)$ is a set of indecomposable $A$-modules such that

1. (1)

$\Sigma$ is sincere. 2. (2)

Any path in $\mathop{\text{mod}}A$ with source and target in $\Sigma$ consists entirely of modules in $\Sigma$. 3. (3)

If $M$ is an indecomposable non-projective $A$-module then at most one of $M$ , $\tau_{A}M$ belongs to $A$. 4. (4)

If $M\rightarrow S$ is an irreducible morphism with $M,S\in\mathop{\text{ind}}A$ and $S\in\Sigma$, then either $M$ belongs to $\Sigma$ or $M$ is non-injective and $\tau_{A}^{-1}M$ belongs to $\Sigma$.

The existence of slices is used to characterize tilted algebras in the following way.

Theorem 2.6**.**

[12]*

Let $B=\emph{End}_{A}T$ be a tilted algebra. Then the class of $B$-modules $\emph{Hom}_{A}(T,DA)$ forms a slice in $\mathop{\text{mod}}B$. Conversely, any slice in any module category is obtained in this way.*

The following notion of local slices was introduced in [3] in the context of cluster-tilted algebras. We say a path $X=X_{0}\rightarrow X_{1}\rightarrow X_{2}\rightarrow\dots\rightarrow X_{s}=Y$ in $\Gamma(\mathop{\text{mod}}A)$ is $\it{sectional}$ if, for each $i$ with $0<i<s$, we have $\tau_{A}X_{i+1}\neq X_{i-1}$.

Definition 2.7**.**

A $\it{local~{}slice}$ $\Sigma$ in $\Gamma(\mathop{\text{mod}}A)$ is a set of indecomposable $A$-modules inducing a connected full subquiver of $\Gamma(\mathop{\text{mod}}A)$ such that

1. (1)

If $X\in\Sigma$ and $X\rightarrow Y$ is an arrow in $\Gamma(\mathop{\text{mod}}A)$, then either $Y$ or $\tau_{A}Y\in\Sigma$. 2. (2)

If $Y\in\Sigma$ and $X\rightarrow Y$ is an arrow in $\Gamma(\mathop{\text{mod}}A)$, then either $X$ or $\tau_{A}^{-1}X\in\Sigma$. 3. (3)

For every sectional path $X=X_{0}\rightarrow X_{1}\rightarrow X_{2}\rightarrow\dots\rightarrow X_{s}=Y$ in $\Gamma(\mathop{\text{mod}}A)$ with $X,Y\in\Sigma$, we have $X_{i}\in\Sigma$, for $i=0,1,\dots,s.$ 4. (4)

The number of indecomposable $A$-modules in $\Sigma$ equals the number of non-isomorphic summands of $T$, where $T$ is a tilting $A$-module.

There is a relationship between tilted and cluster-tilted algebras given in terms of slices and local slices.

Theorem 2.8**.**

[3, Corollary 20]*

Let $C$ be a tilted algebra and $B$ the corresponding cluster-tilted algebra. Then any slice in $\mathop{\emph{mod}}C$ embeds as a local slice in $\mathop{\emph{mod}}B$ and any local slice $\Sigma$ in $\mathop{\emph{mod}}B$ arises in this way.*

The existence of local slices in a cluster-tilted algebra gives rise to the following definition. The unique connected component of $\Gamma(\mathop{\text{mod}}B)$ that contains local slices is called the $\it{transjective~{}component}$.

The next result says a slice in a tilted algebra together with its $\tau$ and $\tau^{-1}$ translates full embeds in the cluster-tilted algebra.

Proposition 2.9**.**

[4, Proposition 3]*

Let $C$ be a tilted algebra, $\Sigma$ a slice, $M\in\Sigma$, and $B$ the corresponding cluster-tilted algebra.*

1. (1)

$\tau_{C}M\cong\tau_{B}M$. 2. (2)

$\tau_{C}^{-1}M\cong\tau_{B}^{-1}M$.

In [3], the authors gave an example of an indecomposable transjective module over a cluster-tilted algebra that does not lie on a local slice. It was proved in [5] the number of such modules is finite.

Proposition 2.10**.**

[5, Corollary 3.8]*

Let $B$ be a cluster-tilted algebra. Then the number of isomorphism classes of indecomposable transjective $B$-modules that do not lie on a local slice is finite.*

2.5 $\tau$-tilting finite algebras

Following [1] we state the following definition.

Definition 2.11**.**

A $C$-module $M$ is $\tau_{C}$-rigid if $\text{Hom}_{C}(M,\tau_{C}M)=0$. A $\tau_{C}$-rigid module $M$ is $\tau_{C}$-tilting if the number of pairwise, non-isomorphic, indecomposable summands of $M$ equals the number of isomorphism classes of simple $C$-modules.

It follows from the Auslander-Reiten formulas that any $\tau_{C}$-rigid module is rigid and the converse holds if the projective dimension is at most 1. In particular, any partial tilting module is a $\tau_{C}$-rigid module, and any tilting module is a $\tau_{C}$-tilting module. Thus, we can regard $\tau_{C}$-tilting theory as a generalization of classic tilting theory. Following [10], we have the following definition.

Definition 2.12**.**

Let $A$ be a finite dimensional algebra. We say that $A$ is $\tau_{A}$-$\it{tilting~{}finite}$ if there are only finitely many isomorphism classes of basic $\tau_{A}$-tilting $A$-modules.

The authors provide several equivalent conditions for an algebra $A$ to be $\tau_{A}$-tilting finite. In particular, we need the following.

Lemma 2.13**.**

[10, Corollary 2.9.]*

$A$ is $\tau_{A}$-tilting finite if and only if there are only finitely many isomorphism classes of indecomposable $\tau_{A}$-rigid $A$-modules.*

2.6 A criterion for representation-finiteness

We will need the following criterion for an algebra to be representation-finite.

Theorem 2.14**.**

[6, IV Theorem 5.4.]*

Assume $A$ is a basic and connected finite dimensional algebra. If $\Gamma(\mathop{\emph{mod}}A)$ admits a finite connected component $\mathcal{C}$, then $\mathcal{C}=\Gamma(\mathop{\emph{mod}}A)$. In particular, $A$ is representation-finite.*

3 Main Result

We are now ready to prove our main theorem.

Theorem 3.1**.**

Let $B$ be a cluster-tilted algebra. Then $B$ is $\tau_{B}$-tilting finite if and only if $B$ is representation-finite.

Proof.

The sufficiency is obvious so we prove the necessity. Assume $B$ is $\tau_{B}$-tilting finite but representation-infinite. By Theorems 2.6 and 2.8, we know the transjective component of $\Gamma(\mathop{\text{mod}}B)$ exists. Since $B$ is representation-infinite, Theorem 2.14 guarantees the transjective component must be infinite. By Proposition 2.10 and the fact that the transjective component is infinite, we must have an infinite number of indecomposable transjective $B$-modules which lie on a local slice. Let $M$ be such a $B$-module. Theorem 2.8 guarantees there exists a tilted algebra $C$ and a slice $\Sigma$ such that $M$ is a $C$-module and $M\in\Sigma$. It follows from parts $(2)$ and $(3)$ of the definition of a slice that $M$ is $\tau_{C}$-rigid. By Proposition 2.9, we know $\tau_{C}M\cong\tau_{B}M$. This implies $M$ is $\tau_{B}$-rigid. Since $M$ was arbitrary, we have shown there exists an infinite number of indecomposable transjective $B$-modules which are $\tau_{B}$-rigid. This is a contradiction to our assumption that $B$ was $\tau_{B}$-tilting finite and Lemma 2.13. We conclude $B$ must be representation-finite.

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