Cluster-tilting Objects Research Articles

Maximal rigid objects without loops in connected 2-CY categories are cluster-tilting objects

In this paper, we study Conjecture II.1.9 of [A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math. 145(4) (2009) 1035–1079], which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster-tilting object in a connected Hom-finite triangulated 2-CY category [Formula: see text]. We obtain some conditions equivalent to the conjecture, and by using them we prove the conjecture.

A relation between tilting graphs and cluster-tilting graphs of hereditary algebras

We give the condition of isomorphisms between tilting graphs and cluster-tilting graphs of hereditary algebras. As a conclusion, it is proved that a graph is a skeleton graph of Stasheff polytope if and only if it is both the tilting graph of a hereditary algebra and also the cluster-tilting graph of another hereditary algebra. At last, when comparing such uniformity, the geometric realizations of simplicial complexes associated with tilting modules and cluster-tilting objects are discussed respectively.

τ-Tilting theory and ⁎-modules

Recently, Adachi, Iyama and Reiten [1] introduced support τ-tilting modules which is an interesting generalization of classical tilting modules and showed that they have close relations with silting complexes and cluster-tilting objects. In this short paper, we prove that support τ-tilting modules over artin algebras are just quasi-tilting modules in sense of Colpi, D'Este and Tonolo [6].

Projective Dimension of Modules over Cluster-Tilted Algebras

We study the projective dimension of finitely generated modules over cluster-tilted algebras End𝒞(T) where T is a cluster-tilting object in a cluster category 𝒞. It is well-known that all End𝒞(T)-modules are of the form Hom𝒞(T, M) for some object M in 𝒞, and since End𝒞(T) is Gorenstein of dimension 1, the projective dimension of Hom𝒞(T, M) is either zero, one or infinity. We define in this article the ideal IM of End𝒞(T[1]) given by all endomorphisms that factor through M, and show that the End𝒞(T)-module Hom𝒞(T, M) has infinite projective dimension precisely when IM is non-zero. Moreover, we apply the results above to characterize the location of modules of infinite projective dimension in the Auslander-Reiten quiver of cluster-tilted algebras of type A and D.

Cluster equivalence and graded derived equivalence.

In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster-tilting objects in the generalized cluster categories: We first observe that cluster-tilting objects in generalized cluster categories are in natural bijection with cluster-tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that in general, if two algebras are cluster equivalent, then (under certain conditions) the algebras can be graded in such a way that the categories of graded modules are derived equivalent. To this end we introduce mutation of graded quivers with potential, and show that this notion reflects mutation in derived categories.

-tilting theory

AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

Lifting to maximal rigid objects in 2-Calabi-Yau triangulated categories

We show that a tilting module over the endomorphism algebra of a maximal rigid object in a 2-Calabi-Yau triangulated category lifts to a maximal rigid object in this 2-Calabi-Yau triangulated category. This strengthens recent work of Fu and Liu for cluster-tilting objects.

Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings

Let $R$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and van den Bergh we study the question of when the endomorphism ring of $M$ has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced $1$-dimensional hypersurfaces.

Lifting to Cluster-tilting Objects in Higher Cluster Categories

Let d > 1 be a positive integer. In this note, we consider the d-cluster-tilted algebras, i.e., algebras which appear as endomorphism rings of d-cluster-tilting objects in higher cluster categories (d-cluster categories). We show that tilting modules over such algebras lift to d-cluster-tilting objects in the corresponding higher cluster category.

On Tropical Friezes Associated with Dynkin Diagrams

Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a certain addition formula. We show that when $\mathcal{C}$ is the cluster category of a Dynkin quiver, the tropical friezes on ${\mathcal{C}}$ are in bijection with the $n$-tuples in ${\mathbb{Z}}^n$, any tropical frieze $f$ on $\mathcal{C}$ is of a special form, and there exists a cluster-tilting object such that $f$ simultaneously takes non-negative values or non-positive values on all its indecomposable direct summands. Using similar techniques, we give a proof of a conjecture of Ringel for cluster-additive functions on stable translation quivers.

Graded mutation in cluster categories coming from hereditary categories with a tilting object

We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique for recovering a cluster-tilting object from its graded quiver in the cluster category of cohX.

Fundamental domains of cluster categories inside module categories

Let H be a finite dimensional hereditary algebra over an algebraically closed field, and let CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H), and of the cluster-tilting objects, in terms of the category modΓ of finitely generated modules over a suitable tilted algebra Γ. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.

Algebras from surfaces without punctures

We introduce a new class of finite-dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial cluster-tilting objects in generalized cluster categories, we compute the invariant of Avella-Alaminos and Geiss for surface algebras and we provide a geometric model for the module category of surface algebras.

Cluster characters II: a multiplication formula

Let 𝒞 be a Hom-finite triangulated 2-Calabi–Yau category with a cluster-tilting object. Under some constructibility assumptions on 𝒞 which are satisfied, for instance, by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra of Dynkin type, we prove a multiplication formula for the cluster character associated with any cluster-tilting object. This formula generalizes those obtained by Caldero–Keller for representation finite path algebras and by Xiao–Xu for finite-dimensional path algebras. We prove an analogous formula for the cluster character defined by Fu–Keller in the set-up of Frobenius categories. It is similar to a formula obtained by Geiss–Leclerc–Schröer in the context of preprojective algebras.

Mutation of cluster-tilting objects and potentials

We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.

Generic Cluster Characters

Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to $T$. If $\CC$ has a cluster structure in the sense of Buan-Iyama-Reiten-Scott, $\mathcal G^T(\CC)$ contains the set of cluster monomials of the corresponding cluster algebra. Moreover, these sets coincide if $\mathcal C$ has finitely many indecomposable objects. When $\CC$ is the cluster category of an acyclic quiver and $T$ is the canonical cluster-tilting object, this set coincides with the set of generic variables previously introduced by the author in the context of acyclic cluster algebras. In particular, it allows to construct $\Z$-linear bases in acyclic cluster algebras.

Cluster-concealed algebras

The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects T in cluster categories; we call such an algebra cluster-concealed in case T is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If C is a representation-finite cluster-tilted algebra, then the indecomposable C-modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra C, we are going to describe the dimension vectors of the indecomposable C-modules in terms of the root system of a quadratic form. The roots may have both positive and negative coordinates and we have to take absolute values.

The Cluster Complex of an Hereditary Artin Algebra

We show that the cluster complex of an hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the cluster-tilting objects form one equivalence class under mutation, which is a necessary first step towards proving a bijection with the set of clusters in the corresponding cluster algebra.

Caldero–Keller Approach to the Denominators of Cluster Variables

Buan, Marsh, and Reiten proved that if a cluster-tilting object T in a cluster category 𝒞 associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in 𝒞, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T. In this article, we give an alternative proof of this result using the Caldero–Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.

Constructing tilted algebras from cluster-tilted algebras

Any cluster-tilted algebra is the relation extension of a tilted algebra. Given the distribution of a cluster-tilting object in the Auslander–Reiten quiver of the cluster category, we present a method to construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.