Relative Cluster-Tilting Objects

• Relative cluster-tilting objects are generalizations of classical cluster-tilting that incorporate a fixed reference object or subcategory to control orthogonality conditions.
• They are defined in various settings—including triangulated, extriangulated, and dg-enhanced categories—by modifying standard Ext^1 criteria with relative notions.
• These constructions establish support τ-tilting correspondences, underpin mutation theories, and facilitate quotient and module-category realizations.

Relative cluster-tilting objects are not governed by a single universal definition. The term is used for several closely related constructions in triangulated, extriangulated, Frobenius, and dg-enhanced settings, all of which modify ordinary cluster-tilting by introducing a reference datum: a fixed cluster-tilting object $T$, a rigid subcategory $\mathcal R$, a fully rigid subcategory $\mathcal C$, a quotient by projective-injective or frozen summands, or a dg morphism $B\to A$. In the foundational triangulated formulation, an object $X$ is relative cluster-tilting when $[T[1]](X,X[1])=0$ and $|X|=|T|$, and later work identifies the same class with ghost or $T[1]$-cluster-tilting objects; broader theories replace $T$ by rigid or fully rigid subcategories and recover support $\tau$-tilting or support tilting in associated abelian quotients (Yang et al., 2015, Yang et al., 2017, Wu, 2021).

1. Terminological scope and common structural pattern

Across the literature, “relative cluster-tilting” denotes a family of analogues of ordinary cluster-tilting rather than a single fixed axiom system. The reference object or subcategory determines which extensions are ignored, which region of the ambient category is relevant, and which quotient or module category receives the corresponding support-tilting data. A common pattern is the replacement of absolute $\mathcal R$0-orthogonality by orthogonality modulo morphisms factoring through a designated summand or subcategory, or by maximal rigidity inside a controlled extension-closed subcategory.

Framework	Relative notion	Reference datum
Triangulated category with cluster-tilting object	relative or $\mathcal R$1-cluster-tilting object	fixed $\mathcal R$2
Triangulated category with rigid subcategory	two-term weak $\mathcal R$3-cluster-tilting	rigid $\mathcal R$4
Extriangulated category with fully rigid subcategory	maximal $\mathcal R$5-rigid	fully rigid $\mathcal R$6
Extriangulated category with rigid object	maximal $\mathcal R$7-rigid	rigid $\mathcal R$8
Relative cluster category / Higgs category	$\mathcal R$9-cluster-tilting object in a Frobenius extriangulated category	dg morphism $\mathcal C$0, idempotent $\mathcal C$1

The terminological variation is explicit in the sources. Some papers introduce “relative cluster-tilting” directly, some replace it by “ghost cluster tilting,” some work with “two-term weak $\mathcal C$2-cluster tilting,” and some state that the closest formal notion is maximal relative rigid or maximal $\mathcal C$3-rigid rather than a separately named relative cluster-tilting object (Yang et al., 2015, Zhou et al., 2018, Liu et al., 2020, Wu, 2021).

2. Relative cluster-tilting in triangulated categories with a fixed cluster-tilting object

In the object-level formulation of Yang–Zhu, the ambient category is a $\mathcal C$4-linear, Hom-finite, Krull–Schmidt triangulated category $\mathcal C$5 with Serre functor and a fixed cluster-tilting object $\mathcal C$6. For any objects $\mathcal C$7, the notation $\mathcal C$8 denotes morphisms $\mathcal C$9 factoring through $B\to A$0. An object $B\to A$1 is $B\to A$2-rigid, or relative rigid, when

$B\to A$3

It is relative cluster-tilting, or $B\to A$4-cluster tilting, when it is $B\to A$5-rigid and

$B\to A$6

An almost relative cluster-tilting object satisfies the same vanishing with $B\to A$7 (Yang et al., 2015).

The later “ghost” formulation replaces the size condition by a maximal orthogonality condition. An object $B\to A$8 is ghost cluster tilting if

$B\to A$9

When the ambient triangulated category has a Serre functor and a cluster-tilting object, this definition is equivalent to the earlier relative cluster-tilting definition. In the same setting, relative cluster-tilting objects are precisely maximal ghost rigid objects (Yang et al., 2017).

This theory is designed to match $X$0-tilting over the endomorphism algebra

$X$1

The quotient equivalence

$X$2

transports relative rigidity to $X$3-rigidity. If $X$4 with $X$5 the maximal direct summand in $X$6, the correspondence sends $X$7 to

$X$8

and induces bijections between $X$9-rigid objects and $[T[1]](X,X[1])=0$0-rigid pairs, and between $[T[1]](X,X[1])=0$1-cluster-tilting objects and support $[T[1]](X,X[1])=0$2-tilting modules. The same framework provides a partial order and mutation theory, and every basic almost $[T[1]](X,X[1])=0$3-cluster-tilting object has exactly two complements (Yang et al., 2015).

A major simplification occurs in the $[T[1]](X,X[1])=0$4-Calabi–Yau case. There, $[T[1]](X,X[1])=0$5-rigidity coincides with ordinary rigidity, and $[T[1]](X,X[1])=0$6-cluster-tilting objects coincide with ordinary cluster-tilting objects. Outside the $[T[1]](X,X[1])=0$7-CY case, relative cluster-tilting is strictly broader than classical cluster-tilting (Yang et al., 2015, Yang et al., 2017).

3. Two-term relative cluster-tilting with respect to a rigid subcategory

A more general formulation fixes a rigid subcategory $[T[1]](X,X[1])=0$8 of a Hom-finite Krull–Schmidt triangulated category $[T[1]](X,X[1])=0$9. The relevant “two-term” region is

$|X|=|T|$0

A subcategory $|X|=|T|$1 is $|X|=|T|$2-rigid if

$|X|=|T|$3

and it is two-term $|X|=|T|$4-rigid if moreover $|X|=|T|$5. It is two-term weak $|X|=|T|$6-cluster tilting if

$|X|=|T|$7

and

$|X|=|T|$8

Adding contravariant finiteness yields the notion of two-term $|X|=|T|$9-cluster-tilting subcategory (Zhou et al., 2018).

The structural theorem states that, inside the two-term region,

$T[1]$0

The restricted Yoneda functor

$T[1]$1

induces an equivalence

$T[1]$2

and therefore a bijection between two-term $T[1]$3-rigid subcategories and $T[1]$4-rigid pairs, as well as a bijection between two-term weak $T[1]$5-cluster-tilting subcategories and support $T[1]$6-tilting pairs (Zhou et al., 2018).

This two-term theory interpolates between earlier frameworks. If $T[1]$7 is cluster-tilting, then $T[1]$8, so the restriction to two-term objects disappears and one recovers $T[1]$9-cluster-tilting theory. If $T$0 is silting, then two-term weak $T$1-cluster-tilting subcategories are exactly two-term silting subcategories (Zhou et al., 2018).

The mutation theory was sharpened in 2024. For an $T$2-functorially finite two-term $T$3-rigid subcategory $T$4, there are canonical left and right completions $T$5 and $T$6, constructed from left approximations of objects in $T$7 and right approximations of objects in $T$8. Any almost complete two-term weak $T$9-cluster-tilting subcategory has exactly these two completions. The exchange triangles carry connecting morphisms factoring through $\tau$0, and the same formalism transfers to support $\tau$1-tilting in functor categories and abelian categories (Liu et al., 2024).

The relative nature of the theory is substantive rather than cosmetic. The 2024 paper gives examples where a completion is two-term weak $\tau$2-cluster tilting but is neither cluster-tilting nor rigid in the ordinary sense, so the relative notion is genuinely broader than its classical $\tau$3-CY specialization (Liu et al., 2024).

4. Extriangulated analogues: maximal relative rigid and support tilting

In extriangulated categories, the role of relative cluster-tilting is often played by maximal relative rigid subcategories rather than by an object-level orthogonality axiom identical to the triangulated one. One approach fixes a fully rigid subcategory $\tau$4 of an extriangulated category $\tau$5 with enough projectives and injectives, together with a twin cotorsion pair

$\tau$6

The quotient $\tau$7 is abelian, and under the hereditary finite-length hypothesis the image of a cluster-tilting subcategory of $\tau$8 is support tilting in $\tau$9. Conversely, every support tilting subcategory $\mathcal R$00 of the quotient lifts to a unique maximal $\mathcal R$01-rigid subcategory $\mathcal R$02, where $\mathcal R$03. If $\mathcal R$04, this yields a bijection between basic maximal $\mathcal R$05-rigid objects and basic support tilting objects (Liu et al., 2020).

A second extriangulated formulation begins from a rigid object $\mathcal R$06 with no projective direct summand. Writing

$\mathcal R$07

one defines $\mathcal R$08 to be $\mathcal R$09-rigid when

$\mathcal R$10

A basic object is maximal $\mathcal R$11-rigid if no non-projective indecomposable $\mathcal R$12 can be added while preserving $\mathcal R$13-rigidity. The algebra

$\mathcal R$14

controls the associated module category, and the functor $\mathcal R$15 yields a bijection between basic $\mathcal R$16-rigid objects and basic $\mathcal R$17-rigid pairs, and between basic maximal $\mathcal R$18-rigid objects and basic support $\mathcal R$19-tilting pairs (Liu et al., 2019).

These extriangulated theories clarify an important point of terminology. Several papers state explicitly that they do not introduce a new formal notion called “relative cluster-tilting” in their own text; instead, maximal $\mathcal R$20-rigid or maximal $\mathcal R$21-rigid is the operative analogue. A plausible implication is that, outside the $\mathcal R$22-CY triangulated setting, the correct general replacement for cluster-tilting is often maximal relative rigidity together with a support-tilting correspondence rather than a literal transplant of the classical two-sided orthogonality formula (Liu et al., 2020, Liu et al., 2019).

Mutation survives in this broader setting. In the $\mathcal R$23-rigid extriangulated theory, every basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable complements, with exchange $\mathcal R$24-triangles controlled by minimal $\mathcal R$25-approximations, directly paralleling support $\mathcal R$26-tilting mutation (Liu et al., 2019).

5. Quotient and module-category realizations

A precursor to later relative theories appears in the module-category realization of cluster categories. For a finite-dimensional hereditary algebra $\mathcal R$27, the cluster category

$\mathcal R$28

has the standard fundamental domain $\mathcal R$29 whose indecomposables are the indecomposable $\mathcal R$30-modules together with $\mathcal R$31. The paper realizes this domain inside $\mathcal R$32, where

$\mathcal R$33

and proves that the indecomposable $\mathcal R$34-modules of projective dimension $\mathcal R$35 are exactly the indecomposable objects in the realized domain $\mathcal R$36 together with the indecomposable projective-injective $\mathcal R$37-modules. Equivalently,

$\mathcal R$38

The main translation theorem states that if $\mathcal R$39 represents an object of $\mathcal R$40, then $\mathcal R$41 is cluster-tilting in $\mathcal R$42 if and only if

$\mathcal R$43

is a basic tilting $\mathcal R$44-module (Cappa et al., 2011).

This is not yet called relative cluster-tilting, but it is a module-theoretic realization of cluster-tilting relative to a chosen fundamental domain and to a distinguished projective-injective summand. The same relative viewpoint governs the quiver formula for the cluster-tilted algebra: extra arrows in $\mathcal R$45 are read off from minimal relations in

$\mathcal R$46

A plausible interpretation is that this paper supplies an early template for later relative theories: cluster-tilting becomes ordinary tilting only after one enlarges by a fixed projective-injective part (Cappa et al., 2011).

A different quotient-relative perspective is orbit-theoretic. If $\mathcal R$47 is a locally finite triangulated category and $\mathcal R$48 an autoequivalence such that the orbit category $\mathcal R$49 is triangulated, then there is a bijection between $\mathcal R$50-periodic $\mathcal R$51-cluster-tilting subcategories of $\mathcal R$52 and $\mathcal R$53-cluster-tilting subcategories of $\mathcal R$54. Thus periodicity under the orbit functor is exactly the descent condition for cluster-tilting through the quotient. In Dynkin-derived settings, this criterion becomes a classification of which periodic cluster-tilting subcategories survive in orbit categories, and the periodicity can be read from symmetries of the quivers of the corresponding cluster-tilted algebras (Grimeland, 2016).

6. Higher and dg-relative theories

The most direct dg-relative theory starts from a morphism of dg algebras

$\mathcal R$55

under assumptions that $\mathcal R$56 and $\mathcal R$57 are homologically smooth, $\mathcal R$58 is connective, $\mathcal R$59 is finite-dimensional, and $\mathcal R$60 carries a left $\mathcal R$61-Calabi–Yau structure. The associated relative cluster category is

$\mathcal R$62

Inside it, a relative fundamental domain $\mathcal R$63 is defined, and its image is the Higgs category $\mathcal R$64. The decisive theorem is that $\mathcal R$65 is a Frobenius extriangulated category with projective-injective objects

$\mathcal R$66

that $\mathcal R$67 is an $\mathcal R$68-cluster-tilting subcategory of $\mathcal R$69, and that

$\mathcal R$70

Here $\mathcal R$71 is the homotopy cofiber of $\mathcal R$72. In this framework, $\mathcal R$73 is a genuinely relative cluster-tilting object: the frozen or coefficient part survives as the projective-injective subcategory $\mathcal R$74, and ordinary generalized cluster-tilting is recovered only after stabilization (Wu, 2021).

This relative dg construction applies to relative Ginzburg dg algebras from ice quivers with potential and to higher Auslander algebras. For a Jacobi-finite ice quiver with potential $\mathcal R$75, the free module $\mathcal R$76 is a cluster-tilting object in the corresponding Higgs category, with endomorphism algebra the relative Jacobian algebra

$\mathcal R$77

The stable quotient of the Higgs category recovers the usual cluster category of the unfrozen quiver (Wu, 2021).

Higher complement phenomena also admit a canonical relative form. In generalized $\mathcal R$78-cluster categories attached to strongly $\mathcal R$79-Calabi–Yau dg algebras, an almost complete $\mathcal R$80-cluster-tilting $\mathcal R$81-object arises by deleting a distinguished indecomposable summand $\mathcal R$82 from the canonical $\mathcal R$83-cluster-tilting object. Such objects have at least $\mathcal R$84 complements in general, and under stronger hypotheses—most notably for good completed deformed preprojective dg algebras with finite-dimensional $\mathcal R$85 and a no-loop condition at the relevant vertex—they have exactly $\mathcal R$86 complements, with an $\mathcal R$87-periodicity pattern among the mutation-generated complements (Guo, 2012).

Taken together, these higher results show that “relative cluster-tilting” extends well beyond the original $\mathcal R$88-formalism. It can mean cluster-tilting in a Frobenius extriangulated category carrying frozen projective-injective summands, maximal relative rigidity in an extriangulated quotient, or higher completion theory for almost complete objects obtained by deleting a distinguished summand. The common invariant is the same: cluster-tilting data are measured relative to an ambient control object or subcategory, and the resulting structures are organized by support-tilting correspondences, quotient functors, and mutation.