Tensorly faithful Wakamatsu tilting modules: classifying rings that force tensorly faithfulness and conditions weaker than the generalized Auslander-Reiten conjecture implying the Wakamatsu tilting conjecture

The Wakamatsu tilting conjecture (WTC) predicts that every Wakamatsu tilting module of finite projective dimension over an associative ring is a classical tilting module. Divaani-Aazar, Fallah and Tousi (2024/2025) proved that the generalized Auslander-Reiten conjecture (GARC) implies WTC, and that any Wakamatsu tilting module of finite projective dimension that is tensorly faithful is in fact projective; since Diveris's counterexample refutes GARC over general rings, the conjecture's validity is currently known only for restricted classes such as left max and left Kasch rings, left Artinian local rings, group rings of a finite group over a commutative Artinian ring, and the commutative noetherian case. This record states the derived open problem left by that work: characterize the rings (or bimodules) for which every Wakamatsu tilting module is tensorly faithful beyond the known classes, and find hypotheses weaker than or independent of GARC that still imply WTC in its natural generality.

Background

Tilting theory studies modules that induce equivalences between derived categories. A right module $T_R$ over an associative ring $R$ with identity is $n$-tilting (Miyashita) if it lies in the class $\mathrm{gen}^*(R)$ (it admits a long $\mathrm{Ext}$-exact coresolution by direct summands of finite sums of $T$), has $\mathrm{pd}(T_R)\le n$, satisfies $\mathrm{Ext}^i_R(T,T)=0$ for $i>0$, and admits an exact coresolution $0\to R\to T_0\to\cdots\to T_n\to 0$ with each $T_i$ in $\mathrm{add}(T)$. Wakamatsu relaxed this by keeping $T_R\in\mathrm{gen}^*(R)$ and $\mathrm{Ext}^i_R(T,T)=0$ for $i>0$ while allowing infinite projective dimension and an infinite coresolution $0\to R\to T_0\to T_1\to\cdots$ with $T_i\in\mathrm{add}(T)$ that stays exact after applying $\mathrm{Hom}_R(-,T)$; such modules are called Wakamatsu tilting. Equivalently (Wakamatsu 2004), a bimodule $_S T_R$ is Wakamatsu tilting when $_S T$ and $T_R$ both lie in the corresponding $\mathrm{gen}^*$ classes, $R\cong\mathrm{End}(_S T)$, $S\cong\mathrm{End}(T_R)$, and $\mathrm{Ext}$ vanishes on both sides; these are exactly the semidualizing $(S,R)$-bimodules of Holm and White. The Wakamatsu tilting conjecture (WTC), posed by Beligiannis and Reiten, asserts that every Wakamatsu tilting $R$-module of finite projective dimension is already a tilting module; it is one of the classical homological conjectures of the representation theory of Artin algebras (rings that are finitely generated as modules over an Artinian commutative center, e.g. finite-dimensional algebras over a field), connected to the finitistic dimension, Nakayama, and Gorenstein symmetry conjectures (Mantese and Reiten proved that the finitistic dimension conjecture implies WTC and that WTC holds for Gorenstein Artin algebras). The generalized Auslander-Reiten conjecture (GARC) for a ring $S$ states that if $\mathrm{Ext}^i_S(M,M\oplus S)=0$ for all $i>n$ then $\mathrm{pd}(_S M)\le n$; it is open for Artin algebras but fails for general associative rings: Diveris presented a counterexample in Example 3.7 of 'Finitistic extension degree'. A right $R$-module $T_R$ is tensorly faithful (originally 'faithfully module' in Holm-White) if the functor $T\otimes_R-$ is faithful, i.e. $T\otimes_R N\neq 0$ for every nonzero left $R$-module $N$. Divaani-Aazar, Fallah and Tousi established two routes from these conditions to WTC over arbitrary rings: (i) if GARC holds for the endomorphism ring $S$, then a Wakamatsu tilting $_S T_R$ with $\mathrm{pd}(T_R)<\infty$ is tilting; (ii) any Wakamatsu tilting $R$-module of finite projective dimension that is tensorly faithful is even projective. For the second route they proved that $T_R$ is tensorly faithful whenever $R=\mathrm{End}(_S T)$ is a left max ring (every nonzero left module has a maximal submodule) and left Kasch (every simple left module embeds in $R$), recovering WTC for left Artinian local rings and for group rings $A[G]$ of a finite group $G$ over a commutative Artinian ring $A$; Holm-White's results give tensorly faithfulness over commutative noetherian rings when $S=R$. Cruz (2024) characterized WTC over finite-dimensional algebras via self-orthogonal $n$-quasi-generators. No conditions beyond these classes are known, and GARC being false in general blocks the first route as a uniform argument.

Problem Statement

For an associative ring $R$ with identity, determine when the functor $T\otimes_R-$ is faithful for every Wakamatsu tilting right $R$-module $T$ (equivalently, every semidualizing $(S,R)$-bimodule $_S T_R$ with $T_R$ Wakamatsu tilting), and use this — or any other hypothesis strictly weaker than or independent of the generalized Auslander-Reiten conjecture — to extend the Wakamatsu tilting conjecture beyond the currently known classes. Concretely: (i) characterize, or give verifiable ring-theoretic conditions on $R$ or on the pair $(S,R)$, under which every Wakamatsu tilting $R$-module is tensorly faithful, going beyond the two known families: $R=\mathrm{End}(_S T)$ left max and left Kasch (which covers left Artinian local rings and group rings $A[G]$ with $A$ commutative Artinian), and $R$ commutative noetherian with $S=R$; and (ii) since the generalized Auslander-Reiten conjecture fails for general associative rings (Diveris's counterexample) while Theorem 1.1 of Divaani-Azar-Fallah-Tousi derives WTC from it, find hypotheses on $R$ or on the endomorphism ring $S$ that are weaker than or independent of GARC — and not merely a restatement of the Wakamatsu tilting conjecture itself nor a hypothesis already known to imply it (such as the finitistic dimension conjecture) — and still imply that every Wakamatsu tilting $R$-module of finite projective dimension is a tilting module. Both parts serve one objective: establishing the Wakamatsu tilting conjecture in its natural generality by replacing the two routes (tensorly faithfulness via left max and Kasch, and GARC) that the 2024/2025 source leaves open. The natural generality of the conjecture is over arbitrary associative rings (or at least all Artin algebras). For (i), going beyond the two known families means a criterion deciding tensorly faithfulness for every Wakamatsu tilting module over an infinite family of rings strictly containing them, or an explicit ring together with a Wakamatsu tilting module of infinite projective dimension that is provably not tensorly faithful.

References

1. Kamran Divaani-Azar, Ali Mahin Fallah, Massoud Tousi, 'On the Wakamatsu tilting conjecture', Archiv der Mathematik (2025), DOI 10.1007/s00013-025-02147-5; arXiv:2407.06353, https://arxiv.org/abs/2407.06353
2. K. Diveris, 'Finitistic extension degree', Algebras and Representation Theory 17(2) (2014) 495-506
3. Henrik Holm, Diana White, 'Foxby equivalence over associative rings', Journal of Mathematics of Kyoto University 47(4) (2007) 781-808, DOI 10.1215/kjm/1250692289, https://doi.org/10.1215/kjm/1250692289
4. Takayoshi Wakamatsu, 'Tilting modules and Auslander's Gorenstein property', Journal of Algebra 275(1) (2004) 3-39, DOI 10.1016/j.algebra.2003.12.008, https://doi.org/10.1016/j.algebra.2003.12.008
5. Francesca Mantese, Idun Reiten, 'Wakamatsu tilting modules', Journal of Algebra 278(2) (2004) 532-552, DOI 10.1016/j.jalgebra.2004.03.023, https://doi.org/10.1016/j.jalgebra.2004.03.023
6. Tiago Cruz, 'Higher Morita-Tachikawa correspondence', Bulletin of the London Mathematical Society 56(8) (2024) 2647-2660, DOI 10.1112/blms.13090, https://doi.org/10.1112/blms.13090
7. Kamran Divaani-Azar, 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
8. Zhenxian Chen, Changchang Xi, 'Singular equivalences and homological conjectures', arXiv:2603.20643 (2026), https://arxiv.org/abs/2603.20643
9. Dajun Liu, Jiaxuan Feng, Hanpeng Gao, 'Projectively Wakamatsu Tilting Modules over One-Point Extensions', arXiv:2604.10118 (2026), https://arxiv.org/abs/2604.10118
10. Guoqiang Zhao, Juxiang Sun, 'Tilting pairs and Wakamatsu tilting pairs of subcategories over cleft extensions', arXiv:2605.20720 (2026), https://arxiv.org/abs/2605.20720
11. Zhiwei Zhu, Jiaqun Wei, 'Wakamatsu-tilting subcategories in extriangulated categories', Communications in Algebra 54(5) (2025) 2159-2174, DOI 10.1080/00927872.2025.2578209; arXiv:2503.00010, https://arxiv.org/abs/2503.00010