Wakamatsu Tilting Conjecture Overview

• Wakamatsu Tilting Conjecture is a major open problem in generalized tilting theory, proposing that a finite projective dimension forces a selforthogonal Wakamatsu tilting module to be classical.
• It integrates module-theoretic, cotorsion pair, and categorical frameworks, connecting tilting, cotilting, and repetitive equivalences in representation theory.
• Positive cases under conditions like finite representation type and tensor faithfulness provide practical insights and guide ongoing research in homological algebra.

The Wakamatsu Tilting Conjecture (WTC) is a central open problem in generalized tilting theory. In a standard form for Artin algebras, it asserts that if $T_R$ is a Wakamatsu tilting $R$-module with finite projective dimension, then $T$ is a tilting $R$-module; in a common broader formulation, a selforthogonal Wakamatsu-tilting module or bimodule with finite projective or injective dimension should collapse to a classical tilting or cotilting object rather than produce genuinely new examples (Divaani-Aazar et al., 2024, Chen et al., 2016). The conjecture lies at the intersection of tilting theory, cotorsion pairs, semidualizing bimodules, repetitive and stable equivalence, and a web of homological conjectures. Several important special cases are known, and a substantial categorical framework has been built around the conjecture, but the general case remains open (Enomoto, 2023).

1. Standard formulation and basic objects

For a right $R$-module $T_R$, an $n$-tilting module is defined by three conditions: $T_R\in \operatorname{gen}(R_R)$ and $\operatorname{pd}_R(T)\le n$; $\operatorname{Ext}^i_R(T,T)=0$ for all $R$0; and there is an exact sequence

$R$1

with all $R$2 (Divaani-Aazar et al., 2024). A Wakamatsu tilting module keeps the generation and self-orthogonality conditions but replaces the finite tilting resolution by an infinite exact sequence

$R$3

with each $R$4, required to remain exact after applying $R$5 (Divaani-Aazar et al., 2024). Every $R$6-tilting module is therefore Wakamatsu tilting, but the converse is precisely what WTC asks under finite projective dimension.

A bimodule formulation is often more symmetric. If $R$7 is a bimodule with $R$8, then Wakamatsu tilting can be expressed through semidualizing conditions: $R$9

$T$0

and

$T$1

This equivalence between Wakamatsu tilting bimodules and semidualizing bimodules is used repeatedly in modern treatments (Divaani-Aazar et al., 2024).

A common source of ambiguity is the relation between WTC and broader “selforthogonal implies tilting” statements. The conjecture is not merely about self-orthogonality; it is specifically about Wakamatsu tilting modules, which already carry a strong infinite coresolution condition. This distinction is essential in later variants, such as projectively Wakamatsu tilting modules, Wakamatsu-tilting subcategories, and Wakamatsu-silting complexes (Enomoto, 2023, Wei, 2013).

2. Auslander classes, cotorsion pairs, and reformulations

The homological infrastructure around WTC is built from Auslander-type classes and cotorsion pairs. For a selforthogonal module $T$2, one considers subcategories such as

$T$3

and the class of modules admitting infinite $T$4-resolutions or $T$5-coresolutions with successive syzygies remaining in suitable Ext-orthogonals. In the Artin algebra setting, these appear as the Auslander class $T$6 and the co-Auslander class $T$7, which organize the modules most tightly controlled by $T$8 (Wei, 2016, Wei, 2016).

Cotorsion pairs supply a second, and often more flexible, reformulation. For a complete hereditary cotorsion pair $T$9, its kernel is

$R$0

A major result shows that $R$1 for some tilting module $R$2 if and only if several homological conditions hold, including bounded projective dimension on the kernel and bounded Gorenstein projective dimension on the left side of the cotorsion pair (Wang et al., 2018). This kernel criterion becomes especially relevant for WTC because a Wakamatsu tilting module $R$3 canonically produces cotorsion pairs from the exact sequence

$R$4

where each $R$5 and the kernels $R$6 lie in $R$7. Writing

$R$8

one obtains complete hereditary cotorsion pairs from $R$9 and from $R$0, and $R$1 is $R$2-tilting if and only if the kernels of those cotorsion pairs are additive closures of $R$3-tilting modules (Wang et al., 2018). In that sense, WTC can be read as a statement that for Wakamatsu tilting modules of finite projective dimension, these associated cotorsion kernels should automatically be tilting kernels.

A third reformulation uses equivalence classes of Wakamatsu tilting modules over arbitrary rings. An equivalence relation defined through invertible bimodules identifies Wakamatsu tilting modules that differ only by Morita-twisting of the left endomorphism ring. Under this relation, equivalence classes of Wakamatsu tilting modules correspond to preenveloping coresolving subcategories with Ext-projective generators, and also to resolving subcategories with Ext-injective cogenerators (Divaani-Aazar et al., 12 Dec 2025). Over complete algebras, including Artin algebras, this equivalence collapses to ordinary isomorphism of basic modules, recovering Mantese–Reiten-type correspondences (Divaani-Aazar et al., 12 Dec 2025). This suggests that WTC can be approached not only module by module, but also via the geometry of the associated resolving and coresolving subcategories.

3. Categorical avatars: stable, repetitive, exact, and extriangulated settings

A distinctive feature of Wakamatsu tilting theory is that it naturally produces equivalences weaker than derived equivalences but still strongly triangulated in flavor. For a good Wakamatsu-tilting bimodule $R$4, Wakamatsu constructed an explicit stable equivalence between the stable module categories of the trivial extension algebras $R$5 and $R$6. Chen and Wei showed that this functor is in fact a triangle functor, hence a triangle equivalence between the stable module categories (Chen et al., 2016). The construction relies on complete hereditary cotorsion pairs $R$7 and $R$8 with

$R$9

and equivalences induced by

$T_R$0

The resulting stable equivalence is therefore not merely formal; it is controlled by the same relative homological algebra that underlies WTC (Chen et al., 2016).

An allied result shows that a good Wakamatsu-tilting module induces an equivalence between the stable categories of the repetitive algebras of $T_R$1 and $T_R$2 (Wei, 2016). Since Happel’s embedding identifies bounded derived categories with stable categories of repetitive algebras when global dimension is finite, this repetitive equivalence is a derived-like shadow of classical tilting theory. The paper makes the analogy explicit: a good Wakamatsu-tilting module plays a role for repetitive equivalence similar to the role of a tilting module for derived equivalence (Wei, 2016).

Exact and extriangulated category frameworks push this further. Any exact category with enough projectives and injectives can be embedded into a category of the form $T_R$3 for a Wakamatsu tilting subcategory $T_R$4, and when higher kernels exist, the same setup is forced to come from a cotilting subcategory (Enomoto, 2016). Zhu and Wei then showed in the extriangulated setting that Wakamatsu-tilting subcategories, Wakamatsu-cotilting subcategories, $T_R$5-tilting subcategories, and $T_R$6-cotilting subcategories all coincide (Zhu et al., 17 Feb 2025). This unification is significant because it removes part of the formal asymmetry between tilting and cotilting sides and suggests that WTC may be approached through intrinsically categorical, rather than purely module-theoretic, structures.

Recollement and cleft-extension techniques furnish yet another layer. Wakamatsu tilting subcategories can be glued along recollements of abelian categories, and under natural exactness and closure hypotheses the converse descent also holds (Wang et al., 2024). More recently, cleft extensions of abelian categories were shown to preserve and reflect both tilting pairs and Wakamatsu tilting pairs, with explicit Tor-type conditions when specialized to $T_R$7-extensions and tensor rings (Zhao et al., 20 May 2026). This suggests a reduction strategy: in favorable ring extensions, Wakamatsu tilting data can be transferred between a base ring and an extension, potentially allowing local-to-global attacks on WTC.

4. Established implications, positive cases, and invariance

Several important positive results are known, though they cover different regimes and use different methods.

Setting	Conclusion	Source
Left max and Kasch rings	Any Wakamatsu tilting module of finite projective dimension is projective; hence WTC holds	(Divaani-Aazar et al., 2024)
Left Artinian local rings	WTC holds	(Divaani-Aazar et al., 2024)
Group rings $T_R$8 with $T_R$9 commutative Artinian and $n$0 finite	WTC holds	(Divaani-Aazar et al., 2024)
Representation-finite Artin algebras	WTC holds	(Enomoto, 2023)
Representation-finite Iwanaga–Gorenstein Artin algebras	Every Wakamatsu tilting module is tilting	(Enomoto, 2023)
Stable equivalence classes of Artin algebras with neither nodes nor semisimple direct summands	WTC is preserved under stable equivalence	(Sun et al., 12 Jul 2025)

One major route runs through the generalized Auslander–Reiten conjecture (GARC). It was proved that if GARC holds for all associative rings with identity, then WTC also holds for all such rings (Divaani-Aazar et al., 2024). More concretely, if $n$1 is a Wakamatsu tilting bimodule with $n$2 and GARC holds for $n$3, then $n$4 is tilting (Divaani-Aazar et al., 2024). This places WTC in the standard implication diagram together with the Finitistic Dimension Conjecture, the Generalized Nakayama Conjecture, the Gorenstein Symmetry Conjecture, and ARC/GARC (Divaani-Aazar et al., 2024).

A second route uses tensor faithfulness. If a Wakamatsu tilting module $n$5 has finite projective dimension and the functor

$n$6

is faithful on nonzero modules, then $n$7 is projective (Divaani-Aazar et al., 2024). This criterion yields WTC for left max and Kasch rings, including left Artinian local rings and group rings of finite groups over commutative Artinian rings (Divaani-Aazar et al., 2024).

A third route is specific to finite representation type. The introduction of projectively Wakamatsu tilting modules sharpened the structure of Wakamatsu tilting in that setting. Under a finiteness condition on $n$8, and in particular for representation-finite algebras, the following notions coincide: projectively Wakamatsu tilting, Wakamatsu tilting, maximal self-orthogonal, and self-orthogonal modules with $n$9 (Enomoto, 2023). Since projectively Wakamatsu tilting modules of finite projective dimension are tilting, this yields WTC for representation-finite algebras (Enomoto, 2023).

A more recent invariance result interprets WTC through $T_R\in \operatorname{gen}(R_R)$0-left approximation dimensions. For stably equivalent Artin algebras with neither nodes nor semisimple direct summands, these approximation dimensions are preserved, giving bijections between basic Wakamatsu tilting modules and between basic tilting modules, and implying that WTC is preserved under such stable equivalences (Sun et al., 12 Jul 2025). This does not prove WTC in a new absolute class, but it shows that WTC is an invariant of a large stable-equivalence class once the usual Auslander–Reiten obstructions are removed (Sun et al., 12 Jul 2025).

5. Derived, $T_R\in \operatorname{gen}(R_R)$1-tilting, and projective-Wakamatsu variants

The conjectural landscape extends beyond the original module-theoretic formulation. One prominent generalization replaces modules by complexes. A Wakamatsu-silting complex is a semi-selforthogonal complex $T_R\in \operatorname{gen}(R_R)$2 in $T_R\in \operatorname{gen}(R_R)$3 such that the regular module $T_R\in \operatorname{gen}(R_R)$4 lies in the positive-shift closure of the Auslander class $T_R\in \operatorname{gen}(R_R)$5; Wakamatsu-tilting modules occur precisely as degree-zero cases (Wei, 2013). The corresponding conjecture states that every compact Wakamatsu-silting complex is silting. This generalizes the classical WTC, and it was shown to lie under the Finitistic Dimension Conjecture: if the finitistic dimension conjecture holds for $T_R\in \operatorname{gen}(R_R)$6, then every compact Wakamatsu-silting complex over $T_R\in \operatorname{gen}(R_R)$7 is silting (Wei, 2013). This derived formulation makes explicit that WTC is part of a larger boundedness-versus-generation problem.

Another nearby family of conjectures arises in $T_R\in \operatorname{gen}(R_R)$8-tilting theory. A self-orthogonal $T_R\in \operatorname{gen}(R_R)$9-tilting module is conjectured to be $\operatorname{pd}_R(T)\le n$0-tilting, and this “Self-orthogonal $\operatorname{pd}_R(T)\le n$1-tilting Conjecture” is placed in a chain

$\operatorname{pd}_R(T)\le n$2

where SWC is the Self-orthogonal Wakamatsu-tilting Conjecture and SFC is the Self-orthogonal Faithful Conjecture (Chen et al., 5 Jan 2025). Finite global delooping level of the endomorphism algebra $\operatorname{pd}_R(T)\le n$3 forces a self-orthogonal $\operatorname{pd}_R(T)\le n$4-tilting module to be $\operatorname{pd}_R(T)\le n$5-tilting (Chen et al., 5 Jan 2025). This is not WTC itself, but it closely parallels the Wakamatsu philosophy: generalized tilting plus strong homological finiteness should collapse to classical tilting.

Projectively Wakamatsu tilting modules form another intermediate class. By definition, a module $\operatorname{pd}_R(T)\le n$6 is projectively Wakamatsu tilting if it is self-orthogonal and an Ext-progenerator of $\operatorname{pd}_R(T)\le n$7; equivalently, $\operatorname{pd}_R(T)\le n$8 (Enomoto, 2023). This class sits between classical tilting and general Wakamatsu tilting,

$\operatorname{pd}_R(T)\le n$9

and in the representation-finite case all three notions collapse together with maximal self-orthogonality (Enomoto, 2023). One-point extension results show that projectively Wakamatsu tilting modules can be lifted along $\operatorname{Ext}^i_R(T,T)=0$0 by

$\operatorname{Ext}^i_R(T,T)=0$1

under the condition $\operatorname{Ext}^i_R(T,T)=0$2, and that source point extensions of representation-finite algebras admit a complete classification

$\operatorname{Ext}^i_R(T,T)=0$3

together with mutation preservation under suitable Ext-vanishing hypotheses (Liu et al., 11 Apr 2026). These results concern the rigid core of Wakamatsu tilting theory rather than WTC directly, but they illustrate how generalized tilting data can still be controlled under standard algebra constructions.

6. Status, interpretation, and outlook

WTC remains open in general. No paper discussed here proves the conjecture in full generality, and several structural papers explicitly do not claim to do so. Chen–Wei’s result on trivial extension algebras proves that Wakamatsu’s stable equivalence is triangulated, not that Wakamatsu tilting modules of finite projective dimension are tilting (Chen et al., 2016). Likewise, repetitive equivalences induced by good Wakamatsu-tilting modules establish a strong homological relation between $\operatorname{Ext}^i_R(T,T)=0$4 and $\operatorname{Ext}^i_R(T,T)=0$5, but not a derived equivalence in general (Wei, 2016). A recurrent misconception is therefore that every triangulated or stable equivalence arising from a Wakamatsu-tilting module is already evidence that the module is classical tilting. The available results show a much subtler picture: such equivalences are abundant and structurally close to derived equivalence, yet they do not by themselves collapse Wakamatsu tilting to tilting.

What the current literature does show is that WTC is deeply embedded in a coherent homological program. Its known positive cases arise from at least four mechanisms: finiteness of representation type, tensor faithfulness, implication from broader conjectures such as GARC or the finitistic dimension conjecture, and stability under certain equivalences (Divaani-Aazar et al., 2024, Enomoto, 2023, Sun et al., 12 Jul 2025). At the same time, exact-category, extriangulated, recollement, cleft-extension, and repetitive-category frameworks have made it possible to transport Wakamatsu tilting structures far beyond the original module-theoretic setting (Enomoto, 2016, Zhu et al., 17 Feb 2025, Zhao et al., 20 May 2026).

A plausible implication is that progress on WTC is likely to come from comparing these frameworks rather than from an isolated module-theoretic argument. Cotorsion-pair kernels, delooping levels, repetitive equivalence, and stable-equivalence invariance all point in the same direction: Wakamatsu tilting behaves like a derived-type notion whose genuinely new phenomena should survive only in the presence of infinite homological complexity. WTC asserts that finite projective dimension is exactly the threshold where that extra generality disappears.