Cluster Tilting Object in Calabi–Yau Theory

• Cluster tilting objects are maximally rigid entities in 2-Calabi–Yau triangulated categories that satisfy vanishing Hom conditions, ensuring their maximality.
• They exhibit mutation dynamics via exchange triangles, where the replacement of an indecomposable summand yields exactly two distinct complements in the 2-CY setting.
• They correspond bijectively with torsion classes and support τ-tilting modules, thereby bridging representation theory with the categorification of cluster algebras.

A cluster tilting object is a fundamental concept in the study of Calabi–Yau triangulated categories, higher representation theory, and categorifications of cluster algebras. It generalizes classical tilting theory to the setting of triangulated and, more broadly, Calabi–Yau categories, encoding rigidity, maximality, and mutation behavior that parallel structural properties of cluster algebras and their combinatorics.

1. Definition and Characterization

Let $\mathcal{C}$ be a Hom-finite, Krull–Schmidt, triangulated category over an algebraically closed field $k$, equipped with a Serre functor $\mathbb{S}$. If $\mathcal{C}$ is 2-Calabi–Yau (i.e., $\mathbb{S} \cong [2]$), a cluster tilting object $T \in \mathcal{C}$ is defined by two principal requirements:

• Rigidity: $\operatorname{Hom}_{\mathcal{C}}(T, T[1]) = 0$.
• Maximality: For any $X \in \mathcal{C}$, if $\operatorname{Hom}_{\mathcal{C}}(T, X[1]) = 0$, then $X$ is isomorphic to a direct summand of a finite direct sum of $T$ (denoted $X \in \operatorname{add} T$).

This is equivalently stated as

$$\operatorname{add} T = \{\, X \in \mathcal{C} \mid \operatorname{Hom}_{\mathcal{C}}(T, X[1]) = 0\, \} = \{\, X \in \mathcal{C} \mid \operatorname{Hom}_{\mathcal{C}}(X, T[1]) = 0\, \}$$

by the 2-Calabi–Yau property, reflecting the symmetry of the Serre duality in $\mathcal{C}$ (Adachi et al., 2012).

The notion generalizes to higher Calabi–Yau categories and $d$-cluster-tilting, where $d \geq 2$, replacing $[1]$ with higher shifts and corresponding Ext-vanishing conditions (Guo, 2010, Holm et al., 2012).

2. Mutation and Complements: Exchange Graph Structure

A distinctive feature of cluster-tilting objects is their mutation theory, governed by the two-complement property in 2-Calabi–Yau settings:

• An almost complete cluster-tilting object $U$ in $\mathcal{C}$ (that is, $U$ with $|U| = |T| - 1$, rigid and missing one indecomposable summand compared to some cluster-tilting object $T$) is a direct summand of precisely two non-isomorphic cluster-tilting objects, i.e., there exist exactly two ways to add an indecomposable $X$ (called a complement) such that $U \oplus X$ is cluster-tilting (Adachi et al., 2012).

The transition between cluster-tilting objects via exchange of indecomposable summands is encoded by exchange triangles:

$$X \to U' \to Y \to X[1] \quad \text{and} \quad Y' \to U'' \to X \to Y'[1]$$

These triangles describe the mutation process at $X$ and produce the two possible complements, yielding a well-structured cluster-tilting graph. Each edge corresponds to exchanging one indecomposable summand, and connectedness is established in broad settings, including categories of coherent sheaves on weighted projective lines of wild type (Fu et al., 2018).

In $d$-Calabi–Yau triangulated categories, the analog holds: an indecomposable summand of a $d$-cluster-tilting object admits exactly $d$ non-isomorphic (complementing) mutations, with structure determined by Iyama–Yoshino exchange triangles (Holm et al., 2012).

3. Bijections with Torsion Classes, Support τ-Tilting, and Silting Theory

A striking connection links cluster-tilting objects to representation theory of algebras via several bijections, particularly when $\mathcal{C}$ is associated to a 2-CY tilted algebra $$A = \operatorname{End}_{\mathcal{C}}(T)^{\mathrm{op}}$$ (Adachi et al., 2012):

Mathematical Structure	Correspondence
Cluster-tilting objects in $\mathcal{C}$	Functorially finite torsion classes in $\mathrm{mod}\,A$
Support $\tau$-tilting $A$-modules	Two-term silting complexes in $K^b(\mathrm{proj}\,A)$

Explicitly, for $X \in \mathcal{C}$ decomposed as $X = X' \oplus X''$ with $X'' \in \operatorname{add} T[1]$ maximal, the image under the canonical functor $$\bar{\cdot}: \mathcal{C}/[T[1]] \rightarrow \mathrm{mod}\,A$$ provides a support τ-tilting pair $(\bar{X}', \bar{X}''[-1])$ in $\mathrm{mod}\,A$. All these maps are bijections respecting the number of indecomposable summands and are compatible with the mutation structures (Adachi et al., 2012, Yang et al., 2017, Yang et al., 2015).

This correspondence extends τ-tilting theory to a completion of classical tilting theory, recovering the two-complement property in situations where classical tilting theory may fail (Adachi et al., 2012).

4. Combinatorial and Geometric Models

Cluster-tilting objects have highly structured combinatorial interpretations. In the case of Dynkin type $A_n$ and related hereditary or tubular settings, cluster-tilting objects correspond to non-crossing partitions, triangulations of polygons, and higher (e.g., $(d+2)$)-angulations for $d$-cluster settings (Chen et al., 2013, Holm et al., 2012). The cluster-tilting exchange graph reflects the combinatorics of quiver mutation (Fomin–Zelevinsky mutation rule) and is tightly linked to the mutation class of endomorphism algebras.

For weighted projective lines, the cluster-tilting approach provides mutation-based methods for classifying tilting objects (e.g., via explicit sequences of mutations leading to all basic tilting objects of desired characteristics) and for determining possible endomorphism algebras as members of a finite mutation class (Chen et al., 2013).

These bijections also correspond to cluster variables and clusters in associated cluster algebras via the Caldero–Chapoton map, further demonstrating the broad combinatorial reach of cluster-tilting theory (Fu et al., 2018, Stovicek et al., 2016).

5. Higher and Generalized Cluster-Tilting Notions

The cluster-tilting paradigm extends to higher dimensions and generalizations:

• $d$-cluster-tilting: In an $(d+1)$-Calabi–Yau triangulated category, an object $T$ is $d$-cluster tilting if $\operatorname{Hom}_{\mathcal{C}}(T, T[i]) = 0$ for $1 \leq i \leq d$ and the vanishing maximally characterizes $T$ as before. Mutations involve $d$ possible complements for each indecomposable summand (Guo, 2010, Holm et al., 2012).
• Relative and ghost cluster-tilting: These generalizations extend the notion of cluster-tilting to arbitrary triangulated settings with a fixed cluster-tilting object $T$, using vanishing of morphisms factoring through $T[1]$ as the rigidity criterion. There exist bijections to support $\tau$-tilting and ghost cluster-tilting, with the classical notion recovered in the 2-CY case (Yang et al., 2017, Yang et al., 2015).
• Weak cluster-tilting: Weakly $d$-cluster-tilting subcategories, which lack functorial finiteness, may exhibit highly variable mutation behavior, with the possibility that an indecomposable summand has fewer than $d$ possible mutations—this presents a contrast to the uniform behavior in the functorially finite (genuine) cluster-tilting case (Holm et al., 2012).

6. Homological, Module-Theoretic, and Geometric Impact

Cluster-tilting objects mediate deep homological and categorical phenomena:

• Endomorphism algebras and Calabi–Yau completions: The endomorphism algebra of a cluster-tilting object often exhibits pronounced homological symmetries (e.g., being 2-CY tilted, Gorenstein, or possessing a Calabi–Yau property relative to an idempotent) and serves as a noncommutative model for important geometric settings (Pressland, 2015).
• Stable categories and singularity theory: In the context of Cohen–Macaulay modules over Gorenstein or quotient singularities, cluster-tilting objects yield derived equivalences and noncommutative resolutions, aligning with the structures appearing in higher-dimensional McKay correspondences (Iyama et al., 2010).
• Higher Auslander–Reiten theory: $d$-cluster-tilting modules and subcategories organize the homological landscape in higher representation theory and encode the maximal $d$-orthogonal subcategories central to these developments (Mizuno, 2012).
• Categorification of cluster algebras: The cluster structure in 2-CY categories with cluster-tilting objects provides a categorical framework for the theory of cluster algebras, with categorical mutations paralleling combinatorial exchange relations (Fu et al., 2018, Stovicek et al., 2016).

These multiple roles underscore the centrality of cluster-tilting objects in modern algebra, representation theory, and categorification frameworks. The explicit construction, mutation theory, and categorical correspondence with torsion classes and silting theory fuel broad applications across algebraic and geometric disciplines.

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Key references:

• (Adachi et al., 2012) (τ-tilting theory; main definitions and bijections for cluster-tilting objects)
• (Chen et al., 2013) (explicit construction and mutation in weighted projective lines)
• (Fu et al., 2018) (connectedness and combinatorics of cluster-tilting graphs)
• (Guo, 2010, Holm et al., 2012) (generalized and higher cluster-tilting, $d$-Calabi–Yau categories)
• (Pressland, 2015) (internal Calabi–Yau properties and endomorphism algebras of cluster-tilting objects)
• (Yang et al., 2017, Yang et al., 2015) (relative and ghost cluster-tilting; functorial finiteness and links to $\tau$-tilting)
• (Stovicek et al., 2016) (cluster structure and infinite context)
• (Iyama et al., 2010) (cluster-tilting in CM categories and connections to singularity theory)