n-Cluster Tilting Object

• n-cluster tilting object is defined by its maximal n-rigid property via vanishing Ext^i for 1 ≤ i < n in abelian or triangulated categories.
• It generalizes classical tilting theory, leading to the formulation of higher Auslander–Reiten sequences and establishing profound links with quiver combinatorics.
• Its investigation drives advancements in representation theory, categorification, and mutation theory, with applications in both algebra and geometry.

An $n$-cluster tilting object is a fundamental notion in higher Auslander–Reiten theory and higher representation theory, representing a categorical generalization of classical tilting theory, with deep connections to homological algebra, the combinatorics of quivers, and the structure of triangulated and abelian categories.

1. Definition and Fundamental Properties

An $n$-cluster tilting object $T$ in an abelian or triangulated category $\mathcal{A}$ (for example, $\mathcal{A} = \mathrm{mod}\text{-}A$ for a finite-dimensional algebra $A$) is characterized by a strong homological orthogonality property up to degree $n-1$. Explicitly, $T$ is $n$-cluster tilting if:

$$\operatorname{add}(T) = \{ X \in \mathcal{A}\ |\ \operatorname{Ext}^i_A(T,X)=0\ \forall\,1 \leq i < n \} = \{ Y \in \mathcal{A}\ |\ \operatorname{Ext}^i_A(Y,T)=0\ \forall\,1 \leq i < n \}$$

Here, $\operatorname{add}(T)$ denotes the full subcategory consisting of all direct summands of finite direct sums of $T$. This condition places $T$ as a "maximal $n$-rigid" object and directly generalizes the classical notion of tilting objects (the $n=1$ case), which generate abelian or triangulated categories by vanishing $\operatorname{Ext}^1$. The $n$-cluster tilting subcategory $\mathcal{M} = \operatorname{add}(T)$ is then both functorially finite, generating, and cogenerating.

The $n$-cluster tilting property is pivotal in defining $n$-abelian categories and underpins the emergence of higher analogues of Auslander–Reiten sequences (the $n$-almost split sequences).

2. Relation to Auslander–Reiten Theory and Higher Homological Algebra

$n$-cluster tilting objects play a crucial role in the context of higher Auslander–Reiten theory, providing a categorical arena where the analogues of almost split sequences, Auslander–Reiten translations, and homological finiteness admit higher analogues.

For an $n$-cluster tilting subcategory $\mathcal{M} = \operatorname{add}(T)$, one has:

• Existence of $n$-almost split sequences: $n$-exact sequences of the form

$$0 \to X_{n+1} \to X_{n} \to \dots \to X_1 \to X_0 \to 0$$

that generalize classical Auslander–Reiten sequences, with $X_i \in \mathcal{M}$.

• The existence of the so-called higher Auslander–Reiten (or higher almost split) translations: For example, for indecomposable non-projective $X \in \mathcal{M}$,

$\tau_n(X) = D \mathrm{Ext}^n_A(X, A)$

provides an auto-equivalence (or a correspondence) between non-projective and non-injective objects in $\mathcal{M}$, generalizing the classical $\tau$-translate. This appears in precise form for representation-directed algebras and their $n$-cluster tilting subcategories (Vaso, 2017).

Moreover, the $n$-abelian or $(n+2)$-angulated category structures, elaborated by Jasso and others, emerge naturally as the categorical environment generated by $n$-cluster tilting objects—see, for example, the stable categories of Gorenstein projectives or singularity categories in the setting of $n\mathbb{Z}$-cluster tilting subcategories (Asadollahi et al., 2019, Herschend et al., 2022).

3. Characterization and Existence: Combinatorics and Finiteness

Existence and explicit construction of $n$-cluster tilting objects are subject to strong combinatorial and arithmetic constraints determined by the algebra $A$ and its quiver:

• In the setting of representation-directed algebras and Nakayama algebras, an explicit characterization is given via the bijection induced by the $n$-Auslander–Reiten translation $\tau_n$ and the conditions on syzygies/cosyzygies of indecomposable modules (Vaso, 2017, Vaso, 2018).
• For radical square zero algebras $A = kQ/J^2$, the existence of an $n$-cluster tilting subcategory imposes representation-finiteness, string algebra structure, and strong restrictions on the quiver $Q$, such as combinatorial "admissibility" and divisibility conditions on path lengths (Vaso, 2021).
• In the finite-dimensional setting, the question of whether every $n$-cluster tilting subcategory is of finite type (i.e., generated additively by a finite set of indecomposable objects) is generally open for $n \ge 2$, but all known examples are finite (Fazelpour et al., 19 Jun 2024, Ebrahimi et al., 2019).

A crucial result states that

$\operatorname{Add}(\mathcal{M})$

is an $n$-cluster tilting subcategory of $\mathrm{Mod}\text{-}A$ if and only if $\mathcal{M}$ is of finite type (has an additive generator) (Fazelpour et al., 19 Jun 2024, Ebrahimi et al., 2019).

4. Homological and Representation-Theoretic Applications

$n$-cluster tilting objects provide powerful tools for the paper and classification of representation-finite and higher Auslander algebras, with implications such as:

• Generalization of Gabriel’s theorem: For $n$-representation-finite algebras, the indecomposable summands of $n$-cluster tilting modules correspond injectively to the set of cluster-roots (positive roots) of the Euler quadratic form associated to the algebra (Mizuno, 2012). The dimension vector $\underline{\dim} X$ of a cluster-indecomposable module $X$ satisfies $q_A(\underline{\dim} X) = 1$, and every cluster-indecomposable module is uniquely determined by its dimension vector.
• Explicit description of quivers and relations after $n$-APR tilting: $n$-cluster tilting theory yields induction of new $n$-representation-finite algebras and the associated combinatorial data for their quivers and relations (see $n$-APR tilting modules and generalizations of BGP reflection functors) (Mizuno, 2012).
• In the context of noncommutative projective schemes and AS-Gorenstein algebras, $(d-1)$-cluster tilting modules mediate derived equivalence and allow the improvement of homological invariants, leading to Morita-type equivalences of noncommutative projective spaces (Ueyama, 2016).
• In geometric and categorified settings—such as cluster categories, weighted projective lines, and m-cluster categories—$n$-cluster tilting objects (and their higher analogues) underlie the construction of derived equivalences, explicit combinatorial models (such as in type D, Euclidean type, and for quivers without loops), and the classification of stable endomorphism algebras (Chen et al., 2013, Jacquet-Malo, 2017, Xu et al., 2014).

5. Generalizations: $n$-Precluster and $n\mathbb{Z}$-Cluster Tilting, Gorenstein Conditions

Important generalizations adapt the notion of $n$-cluster tilting to broader and more flexible contexts:

• $n$-precluster tilting subcategories (and $\tau_n$-selfinjective algebras) weaken the maximality of orthogonality; these admit higher Auslander–Gorenstein correspondence and arise in the theory of $n$-minimal Auslander–Gorenstein algebras (Iyama et al., 2016).
• $n\mathbb{Z}$-cluster tilting subcategories require closure under $n$-syzygies and $n$-cosyzygies (i.e., being stable under "higher translation vectors"), and lead to special types of $n$-Gorenstein cluster tilting subcategories when the algebra is Iwanaga–Gorenstein (Asadollahi et al., 2019). For Nakayama algebras, a full classification is now available, showing that selfinjective Nakayama algebras may have multiple such subcategories, and every $n\mathbb{Z}$-cluster tilting subcategory in the module category induces one in the singularity category (Herschend et al., 2022).
• Relative cluster tilting objects ($T[1]$-cluster tilting) and their mutation theory generalize the notion to settings lacking Calabi–Yau properties, and allow bijections with support $\tau$-tilting modules (Yang et al., 2015).

These generalizations are critical for capturing the deeper categorical symmetries and for connecting representation theory to singularity categories, Gorenstein homological algebra, and categorical lifts of derived equivalence.

6. Lattice and K-theoretic Structure, Grothendieck Groups, and Mutation

The paper of the lattice of $n$-cluster tilting subcategories, and their relation to K-theory, has led to new structural insights:

• For radical square zero algebras (outside cyclically oriented extended Dynkin type), the set of $n$-cluster tilting subcategories forms a lattice isomorphic to the opposite of the lattice of divisors of an admissible integer determined by the quiver (Vaso, 2021).
• In finite type, the alternating sums of classes in $n$-almost split sequences form a basis for the relations in the Grothendieck group of $\mathcal{M}$ (and hence in $K_0(\mathcal{M})$), tightly connecting homological algebra to K-theoretic invariants (Diyanatnezhad et al., 2021).
• The mutation theory for $n$-cluster tilting (generalizations of Fomin–Zelevinsky mutation, cluster-tilting mutation, and exchange triangles) and combinatorial models (e.g., flips in polygons, colored quivers) are key both for applications to categorification and for classifying exchange graphs and endomorphism algebras (Chen et al., 2013, Jacquet-Malo, 2017, Grimeland, 2016).

7. Open Problems and Further Directions

Several major questions and directions remain central in the field:

• Finiteness: Iyama's question asks whether every $n$-cluster tilting subcategory in the module category over an artin algebra is of finite type for $n \ge 2$. Results show that this is intimately related to rigidity of filtered colimits, vanishing of $\operatorname{Ext}$ for limits, and purity properties (Fazelpour et al., 19 Jun 2024, Ebrahimi et al., 2019).
• Construction and classification: Determining precisely which algebras admit nontrivial $n$-cluster tilting objects remains a nuanced problem, contingent on combinatorics of the quiver, homological dimensions, and exact sequences—see explicit classifications in Nakayama, radical square zero, and representation-directed cases (Vaso, 2017, Herschend et al., 2022, Vaso, 2021).
• Interactions with geometry and categorification: The passage from module categories to singularity categories, the interaction with noncommutative projective geometry, and the higher analogues of cluster categories indicate rich connections to algebraic geometry, mathematical physics, and higher categorical invariants.

The robust categorical, homological, and combinatorial structure afforded by $n$-cluster tilting objects reveals a landscape of deep relationships between module theory, higher-dimensional Auslander–Reiten theory, and the algebraic geometry of both commutative and noncommutative spaces. The development of their theory continues to produce substantial new insights, methods, and classifications across modern representation theory.