The Wakamatsu Tilting Conjecture: is every Wakamatsu tilting module of finite projective dimension over an Artin algebra tilting?

The Wakamatsu tilting conjecture (WTC), formulated by Beligiannis and Reiten, asks whether every Wakamatsu tilting module of finite projective dimension over an Artin algebra is a tilting module. It is one of the classical homological conjectures of the representation theory of algebras: the finitistic dimension conjecture implies it, it implies the Gorenstein symmetry and generalized Nakayama conjectures, and Divaani-Aazar, Fallah and Tousi (2025) proved that the generalized Auslander-Reiten conjecture implies it. WTC is known only for special classes (Gorenstein algebras, representation-finite and torsionless-finite algebras, left max and Kasch rings, left Artinian local rings, group rings of finite groups over commutative Artinian rings).

Background

An Artin algebra is a ring that is finitely generated as a module over a commutative artinian ring; a finite-dimensional algebra over a field is the basic example, and all modules considered are finitely generated. Tilting theory studies modules that encode derived equivalences between algebras. Following Miyashita, a right module $T_R$ over a ring $R$ is $n$-tilting if it has projective dimension at most $n$, is self-orthogonal (i.e. $\operatorname{Ext}^i_R(T,T)=0$ for all $i>0$), lies in the class $\operatorname{gen}^{*}(R)$ of modules admitting an exact sequence $\cdots\to R_1\to R_0\to T\to 0$ with each $R_i$ a direct summand of a finite direct sum of copies of $R$ and $\operatorname{Ext}^1_R(R,-)$ vanishing on the kernels of its maps, and there is an exact coresolution $0\to R\to T_0\to T_1\to\cdots\to T_n\to 0$ with each $T_i$ in $\operatorname{add}(T)$, the class of direct summands of finite direct sums of copies of $T$. Wakamatsu (1988) generalized tilting modules by dropping the finite projective dimension requirement: a right module $T_R$ is Wakamatsu tilting if $T_R\in\operatorname{gen}^{*}(R)$, $\operatorname{Ext}^i_R(T,T)=0$ for all $i>0$, and there is an infinite coresolution $0\to R\to T_0\to T_1\to T_2\to\cdots$ with each $T_i\in\operatorname{add}(T)$ that stays exact after applying the functor $\operatorname{Hom}_R(-,T)$. Every $n$-tilting module is Wakamatsu tilting, and the converse is the content of the conjecture. A bimodule $_S T_R$ is a Wakamatsu tilting module precisely when it is a semidualizing $(S,R)$-bimodule in the sense of Holm and White, i.e. $T\in\operatorname{gen}^{*}$ on both sides, $R\cong\operatorname{End}(_S T)^{\operatorname{op}}$ and $S\cong\operatorname{End}(T_R)^{\operatorname{op}}$, and $T$ is self-orthogonal over both rings; this links the conjecture to the study of semidualizing modules over commutative rings, where the analogous objects are classified only in special cases. The Wakamatsu tilting conjecture (WTC), stated by Beligiannis and Reiten in their 2007 memoir, asserts that over an Artin algebra every Wakamatsu tilting module of finite projective dimension is a tilting module. It sits among the interrelated homological conjectures for Artin algebras: the finitistic dimension conjecture (FDC), that the supremum of projective dimensions of all finitely generated modules of finite projective dimension is finite; the Gorenstein symmetry conjecture (GSC), that the left and right self-injective dimensions of the algebra are simultaneously finite (injective dimension being the length of a minimal injective resolution); the generalized Nakayama conjecture (GNC), that every indecomposable injective appears in the minimal injective resolution of the regular module; the generalized Auslander-Reiten conjecture (GARC), that a finitely generated module $M$ satisfying $\operatorname{Ext}^i_R(M,M\oplus R)=0$ for all $i>n$ must have projective dimension at most $n$; and the Auslander-Reiten conjecture (ARC), its case $n=0$. The known implications are FDC $\Rightarrow$ WTC (Mantese and Reiten, who also proved WTC for Gorenstein Artin algebras, i.e. algebras of finite injective dimension on both sides), WTC $\Rightarrow$ GSC and WTC $\Rightarrow$ GNC (Beligiannis and Reiten), and GARC $\Rightarrow$ WTC (Divaani-Aazar, Fallah and Tousi). GARC holds for algebras in which every module has an ultimately closed projective resolution, in particular representation-finite, torsionless-finite and radical-square-zero algebras (Wei). Since GARC implies the still-open ARC, GARC is open for Artin algebras, and Schulz's 1994 example of a non-projective module without self-extensions shows that GARC fails over general associative rings; this blocks the GARC route to WTC for arbitrary rings. A second route introduced in the 2025 paper of Divaani-Aazar, Fallah and Tousi uses tensorly faithfulness: a right module $T_R$ is tensorly faithful if the functor $T\otimes_R-$ is faithful, equivalently $T\otimes_R M\neq 0$ for every nonzero left $R$-module $M$. They prove that a Wakamatsu tilting module of finite projective dimension that is tensorly faithful is projective, and that every Wakamatsu tilting module over a left max ring (every nonzero module has a maximal submodule) that is left Kasch (every simple left module embeds in the ring) is tensorly faithful; this yields WTC for left Artinian local rings and for group rings $A[G]$ of a finite group $G$ over a commutative artinian ring $A$. Over a commutative noetherian ring with $S=R$, Holm and White's results imply every Wakamatsu tilting module is tensorly faithful, but over noncommutative rings tensorly faithfulness can fail (the source gives an explicit 1-tilting module over a quiver algebra that is not tensorly faithful). Cruz reformulated the conjecture through the higher Morita-Tachikawa correspondence: WTC holds for all finite-dimensional algebras if and only if over every such algebra each self-orthogonal $n$-quasi-generator, meaning a module $M$ with $\operatorname{Ext}^i(M,M\oplus R)=0$ for $i>n$ that generates the modules annihilated by it, has projective dimension at most $n$. Despite these advances, no proof or counterexample is known for the general Artin algebra case.

Problem Statement

Let $\Lambda$ be an Artin algebra. The Wakamatsu tilting conjecture asks: must every Wakamatsu tilting $\Lambda$-module $T$ of finite projective dimension be a tilting module? The source paper studies the same question in Wakamatsu's stronger setting of arbitrary associative rings with identity, in the bimodule form: if $_S T_R$ is a Wakamatsu tilting bimodule (equivalently, a semidualizing $(S,R)$-bimodule) with $\operatorname{pd}(T_R)<\infty$, must $T_R$ admit a finite exact coresolution $0\to R\to T_0\to\cdots\to T_n\to 0$ with each $T_i\in\operatorname{add}(T)$? A positive answer for all rings implies the Artin algebra case, but not conversely. The Artin algebra case would be settled either by a proof that WTC holds for every Artin algebra, or by exhibiting a single Artin algebra with a Wakamatsu tilting module of finite projective dimension that is not tilting.

References

1. Kamran Divaani-Aazar, Ali Mahin Fallah, Massoud Tousi, 'On the Wakamatsu tilting conjecture', Archiv der Mathematik 125(3) (2025) 291-302, DOI 10.1007/s00013-025-02147-5 (preprint arXiv:2407.06353), https://doi.org/10.1007/s00013-025-02147-5
2. Takayoshi Wakamatsu, 'On modules with trivial self-extensions', Journal of Algebra 114 (1988) 106-114, DOI 10.1016/0021-8693(88)90215-3, https://doi.org/10.1016/0021-8693(88)90215-3
3. Takayoshi Wakamatsu, 'Tilting modules and Auslander's Gorenstein property', Journal of Algebra 275 (2004) 3-39, DOI 10.1016/j.jalgebra.2003.12.008, https://doi.org/10.1016/j.jalgebra.2003.12.008
4. Apostolos Beligiannis, Idun Reiten, 'Homological and homotopical aspects of torsion theories', Memoirs of the American Mathematical Society 188 (883) (2007), DOI 10.1090/memo/0883, https://doi.org/10.1090/memo/0883
5. Francesca Mantese, Idun Reiten, 'Wakamatsu tilting modules', Journal of Algebra 278 (2004) 532-552, DOI 10.1016/j.jalgebra.2004.03.023, https://doi.org/10.1016/j.jalgebra.2004.03.023
6. Rainer Schulz, 'A non-projective module without self-extensions', Archiv der Mathematik 62 (1994) 497-500, DOI 10.1007/BF01193735, https://doi.org/10.1007/BF01193735
7. Jiaqun Wei, 'Auslander bounds and homological conjectures', Revista Matemática Iberoamericana 27 (2011) 871-884, DOI 10.4171/rmi/655, https://doi.org/10.4171/rmi/655
8. Tiago Cruz, 'Higher Morita-Tachikawa correspondence', Bulletin of the London Mathematical Society 56 (2024) 2647-2660, DOI 10.1112/blms.13090, https://doi.org/10.1112/blms.13090
9. Henrik Holm, Diana White, 'Foxby equivalence over associative rings', Journal of Mathematics of Kyoto University 47 (2007) 781-808, DOI 10.1215/kjm/1250692289, https://doi.org/10.1215/kjm/1250692289
10. Haruhisa Enomoto, 'Maximal self-orthogonal modules and a new generalization of tilting modules', arXiv:2301.13498 (2023), https://arxiv.org/abs/2301.13498
11. Kamran Divaani-Aazar, Ali Mahin Fallah, Massoud Tousi, 'Equivalence classes of Wakamatsu tilting modules and preenveloping and precovering subcategories', arXiv:2512.11600 (2025), https://arxiv.org/abs/2512.11600
12. Dajun Liu, Jiaxuan Feng, Hanpeng Gao, 'Projectively Wakamatsu Tilting Modules over One-Point Extensions', arXiv:2604.10118 (2026), https://arxiv.org/abs/2604.10118