For a faithfully balanced selforthogonal bimodule RωS_R\omega_S over a left noetherian ring RR and a right noetherian ring SS — equivalently, a Wakamatsu tilting module Rω_R\omega with S=End⁡(Rω)S=\operatorname{End}(_R\omega) — Zhao and Yin (2013) posed the structural question of when the class Tω(R)T_\omega(R) of ω\omega-∞\infty-torsionfree finitely generated left RR-modules coincides with the class Gω(R)G_\omega(R) of modules of generalized Gorenstein dimension zero with respect to ω\omega. Since Gω(R)=⊥Rω∩Tω(R)G_\omega(R)={}^{\perp_R}\omega\cap T_\omega(R) always holds, the question is exactly when every ω\omega-∞\infty-torsionfree finitely generated module is Ext⁡\operatorname{Ext}-orthogonal to ω\omega. The source paper resolves only the diagonal case ω=RRR\omega={}_RR_R, where the equality is equivalent to ⊥RR{}^{\perp_R}R having the RR-torsionless property. The general bimodule case appears to remain open.

Background

Let RR be an associative ring with identity, mod⁡R\operatorname{mod}R the category of finitely generated left RR-modules, and Ext⁡Ri\operatorname{Ext}^i_R the derived functors of Hom⁡R\operatorname{Hom}_R, which measure the failure of Hom⁡R(−,N)\operatorname{Hom}_R(-,N) to preserve exactness. A ring is (left) noetherian if every left ideal is finitely generated, so that finitely generated modules are well behaved homologically.

A finitely generated left RR-module Rω_R\omega is called selforthogonal if Ext⁡Ri(ω,ω)=0\operatorname{Ext}^i_R(\omega,\omega)=0 for all i≥1i\ge 1, and a Wakamatsu tilting module (Wakamatsu 2004; Mantese–Reiten 2004) if, in addition, Rω_R\omega admits the following replacement for projectivity: there is an exact sequence 0→RR→ω0→ω1→⋯0\to{}_RR\to\omega_0\to\omega_1\to\cdots with each ωi\omega_i in add⁡Rω\operatorname{add}_R\omega (the direct summands of finite direct sums of copies of ω\omega) which stays exact after applying Hom⁡R(−,ω)\operatorname{Hom}_R(-,\omega). Equivalently, writing S=End⁡(Rω)S=\operatorname{End}(_R\omega), the bimodule RωS_R\omega_S is faithfully balanced (the natural maps R→End⁡(ωS)R\to\operatorname{End}(\omega_S) and Sop→End⁡(Rω)S^{\mathrm{op}}\to\operatorname{End}(_R\omega) are isomorphisms) and selforthogonal on both sides. Classical tilting modules, and semidualizing bimodules over noetherian rings (faithfully balanced selforthogonal bimodules that are finitely generated on both sides), are special cases. Wakamatsu tilting modules are the basic objects of the generalized tilting theory studied intensively because of their connection with the Wakamatsu tilting conjecture, which predicts that over an Artin algebra a Wakamatsu tilting module of finite injective dimension is cotilting.

Two module classes relative to such a bimodule RωS_R\omega_S drive the present question (the framework is due to Zhaoyong Huang and collaborators; see Huang 2001, Huang–Tang 2001, Huang 2009). For M∈mod⁡RM\in\operatorname{mod}R with projective presentation P1→P0→M→0P_1\to P_0\to M\to 0, applying (−)ω=Hom⁡R(−,ω)(-)^{\omega}=\operatorname{Hom}_R(-,\omega) yields the ω\omega-transpose Tr⁡ωM=Coker⁡(P0ω→P1ω)\operatorname{Tr}_\omega M=\operatorname{Coker}(P_0^{\omega}\to P_1^{\omega}), a finitely generated right SS-module, the relative analogue of the classical Auslander transpose. Writing ⊥Rω={M∈mod⁡R;Ext⁡Ri(M,ω)=0 for all i≥1}{}^{\perp_R}\omega=\{M\in\operatorname{mod}R;\operatorname{Ext}^i_R(M,\omega)=0\ \text{for all }i\ge 1\} (and ⊥Rnω{}^{\perp^n_R}\omega for the truncated version 1≤i≤n1\le i\le n), a module MM is ω\omega-kk-torsionfree if Tr⁡ωM∈⊥kωS\operatorname{Tr}_\omega M\in{}^{\perp^k}\omega_S; MM is ω\omega-torsionless (resp. ω\omega-reflexive) if the evaluation map M→Hom⁡Sop(Hom⁡R(M,ω),ω)M\to\operatorname{Hom}_{S^{\mathrm{op}}}(\operatorname{Hom}_R(M,\omega),\omega) is a monomorphism (resp. an isomorphism); these are the cases k=1k=1 (resp. k=2k=2). Denote by Tω(R)T_\omega(R) the class of ω\omega-∞\infty-torsionfree modules, i.e., those that are ω\omega-kk-torsionfree for every kk. On the other side, MM has generalized Gorenstein dimension zero with respect to ω\omega, written G-dim⁡ω(M)=0G\text{-}\dim_\omega(M)=0, if MM is ω\omega-reflexive, M∈⊥RωM\in{}^{\perp_R}\omega, and Mω∈⊥ωSM^{\omega}\in{}^{\perp}\omega_S; denote this class by Gω(R)G_\omega(R). For ω=R\omega=R these recover the classical ∞\infty-torsionfree modules and the modules of Gorenstein dimension zero of classical stable module theory.

Three known facts frame the question. First (Zhao–Yin, Lemma 2.3), Gω(R)=⊥Rω∩Tω(R)G_\omega(R)={}^{\perp_R}\omega\cap T_\omega(R), so Gω(R)⊆Tω(R)G_\omega(R)\subseteq T_\omega(R) always, and equality holds exactly when Tω(R)⊆⊥RωT_\omega(R)\subseteq{}^{\perp_R}\omega, i.e., when every ω\omega-∞\infty-torsionfree module is Ext⁡\operatorname{Ext}-orthogonal to ω\omega. Second (Huang 2009, Proposition 2.3, as quoted in the source paper), a full subcategory having the ω\omega-torsionless property means all its modules are ω\omega-torsionless, and ⊥Rω{}^{\perp_R}\omega has the ω\omega-torsionless property if and only if ⊥Rω=Gω(R){}^{\perp_R}\omega=G_\omega(R) — a companion equality whose criterion therefore does not automatically decide Tω(R)=Gω(R)T_\omega(R)=G_\omega(R). Third, in the diagonal case ω=RRR\omega={}_RR_R the question is settled: by Hoshino's theorem, a two-sided noetherian ring RR with equal finite left and right self-injective dimensions at most nn (nn-Gorenstein, injective dimension being the length of a shortest injective resolution) is exactly a ring over which every finitely generated module has Gorenstein dimension at most nn, and Zhao–Yin's Theorem 3.13 shows that ⊥RR{}^{\perp_R}R has the RR-torsionless property if and only if TR(R)=GR(R)T_R(R)=G_R(R), via the equivalent formulations of their Lemma 3.14 (for instance TR(R)⊆⊥RRT_R(R)\subseteq{}^{\perp_R}R).

Under the standing hypotheses of the source paper — RR left noetherian, SS right noetherian, RωS_R\omega_S faithfully balanced and selforthogonal — Zhao and Yin pose, immediately after their Proposition 3.11 and Corollary 3.12 (which establish, among other things, that ⊥Rnω{}^{\perp^n_R}\omega has the ω\omega-torsionless property if and only if ⊥nωS=⊥ωS{}^{\perp^n}\omega_S={}^{\perp}\omega_S, and that the minimal such nn is the same on both sides): the question of when Tω(R)=Gω(R)T_\omega(R)=G_\omega(R) for a general Wakamatsu tilting bimodule.

Problem Statement

Let RR be a left noetherian ring, SS a right noetherian ring, and RωS_R\omega_S a faithfully balanced selforthogonal bimodule (equivalently, Rω_R\omega a Wakamatsu tilting module with S=End⁡(Rω)S=\operatorname{End}(_R\omega)). Let Tω(R)T_\omega(R) be the class of finitely generated left RR-modules that are ω\omega-∞\infty-torsionfree, that is, modules MM whose ω\omega-transpose Tr⁡ωM\operatorname{Tr}_\omega M (a finitely generated right SS-module) satisfies Ext⁡Sopi(Tr⁡ωM,ω)=0\operatorname{Ext}^i_{S^{\mathrm{op}}}(\operatorname{Tr}_\omega M,\omega)=0 for all i≥1i\ge 1; and let Gω(R)G_\omega(R) be the class of finitely generated left RR-modules of generalized Gorenstein dimension zero with respect to ω\omega, that is, modules MM that are ω\omega-reflexive (the evaluation map M→Hom⁡Sop(Hom⁡R(M,ω),ω)M\to\operatorname{Hom}_{S^{\mathrm{op}}}(\operatorname{Hom}_R(M,\omega),\omega) is an isomorphism) and satisfy Ext⁡Ri(M,ω)=0\operatorname{Ext}^i_R(M,\omega)=0 and Ext⁡Sopi(Mω,ω)=0\operatorname{Ext}^i_{S^{\mathrm{op}}}(M^\omega,\omega)=0 for all i≥1i\ge 1, where Mω=Hom⁡R(M,ω)M^\omega=\operatorname{Hom}_R(M,\omega). Since Gω(R)=⊥Rω∩Tω(R)G_\omega(R)={}^{\perp_R}\omega\cap T_\omega(R) holds identically, characterize when

Tω(R)=Gω(R),T_\omega(R)=G_\omega(R),

that is, when every ω\omega-∞\infty-torsionfree finitely generated left RR-module MM satisfies Ext⁡Ri(M,ω)=0\operatorname{Ext}^i_R(M,\omega)=0 for all i≥1i\ge 1. In particular, decide whether the equality admits, for a general Wakamatsu tilting bimodule, a criterion analogous to the settled ring case ω=RRR\omega={}_RR_R, where TR(R)=GR(R)T_R(R)=G_R(R) is equivalent to ⊥RR{}^{\perp_R}R having the RR-torsionless property and to TR(R)⊆⊥RRT_R(R)\subseteq{}^{\perp_R}R (Lemma 3.14 of the source paper). The already-resolved diagonal case ω=RRR\omega={}_RR_R is excluded; the problem is the general bimodule case under the standing hypotheses above.

References

  1. Guoqiang Zhao and Lirong Yin, Wakamatsu tilting modules with finite injective dimension, Czechoslovak Mathematical Journal 63 (2013), 865–876. DOI: 10.1007/s10587-013-0058-5, https://doi.org/10.1007/s10587-013-0058-5 (open-access copy: http://dml.cz/dmlcz/143602)
  2. Zhaoyong Huang, ω\omega-kk-torsionfree modules and ω\omega-left approximation dimension, Science in China Series A: Mathematics 44 (2001), 184–192. DOI: 10.1007/BF02874420, https://doi.org/10.1007/BF02874420
  3. Zhaoyong Huang and Gaohua Tang, Self-orthogonal modules over coherent rings, Journal of Pure and Applied Algebra 161 (2001), 167–176. DOI: 10.1016/S0022-4049(00)00109-2, https://doi.org/10.1016/S0022-4049(00)00109-2
  4. Zhaoyong Huang, Generalized tilting modules with finite injective dimension, Journal of Algebra 311 (2007), 619–634. DOI: 10.1016/j.jalgebra.2006.11.025, https://doi.org/10.1016/j.jalgebra.2006.11.025
  5. Zhaoyong Huang, Selforthogonal modules with finite injective dimension III, Algebras and Representation Theory 12 (2009), 371–384. DOI: 10.1007/s10468-009-9157-2, https://doi.org/10.1007/s10468-009-9157-2
  6. Mitsuo Hoshino, Algebras of finite self-injective dimension, Proceedings of the American Mathematical Society 112 (1991), 619–622. DOI: 10.1090/S0002-9939-1991-1047011-8, https://doi.org/10.1090/S0002-9939-1991-1047011-8
  7. Francesca Mantese and 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
  8. 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
  9. Juxiang Sun and Guoqiang Zhao, Wakamatsu tilting pairs, Journal of Algebra and Its Applications 13 (2014), 1450041. DOI: 10.1142/S0219498814500418, https://doi.org/10.1142/S0219498814500418
  10. Yanhong Bao, Jiafeng Lü, and Zhibing Zhao, Relative torsionfreeness and Frobenius extensions, arXiv:2409.11892 (2024), https://arxiv.org/abs/2409.11892
  11. Juxiang Sun, Guoqiang Zhao, and Junling Zheng, ω\omega-left approximation dimensions under stable equivalence, arXiv:2507.09286 (2025), https://arxiv.org/abs/2507.09286
  12. Zhaoyong Huang, Generalized Gorenstein categories, arXiv:2603.04856 (2026), https://arxiv.org/abs/2603.04856
  13. Kamran Divaani-Aazar, Ali Mahin Fallah, and Massoud Tousi, Comparing four definitions of cotilting modules, arXiv:2507.18860 (2025), https://arxiv.org/abs/2507.18860
  14. Kamran Divaani-Aazar, Ali Mahin Fallah, and Massoud Tousi, On the Wakamatsu tilting conjecture, arXiv:2407.06353 (2024), https://arxiv.org/abs/2407.06353
  15. Donglin He and Yuyan Li, Stability of Gorenstein classes relative to Wakamatsu tilting modules (in Chinese; 关于Wakamatsu倾斜模的Gorenstein类的稳定性), Journal of Hebei Normal University (Natural Science Edition) 43 (2019), no. 6, 465–467. DOI: 10.13763/j.cnki.jhebnu.nse.2019.06.002, https://doi.org/10.13763/j.cnki.jhebnu.nse.2019.06.002