The Wakamatsu Tilting Conjecture (WTC) predicts that, over an artin algebra — and equally in the extension to arbitrary associative rings — every Wakamatsu tilting module of finite projective dimension is already a (Miyashita) tilting module, i.e. that self-orthogonality plus existence of a long Ext\mathrm{Ext}-exact core of the regular module forces that core to terminate in finitely many steps. The conjecture sits in the web of classical homological conjectures: the finitistic dimension conjecture implies it, and it implies both the Generalized Nakayama Conjecture and the Gorenstein Symmetry Conjecture. Despite partial positive results (Gorenstein artin algebras, left artinian local rings, finite group rings over commutative artinian rings, centralizer matrix algebras) and equivalent reformulations via higher Morita–Tachikawa correspondence, the conjecture remains open in general as of 2026.

Background

Tilting theory is a central tool in the representation theory of algebras. An artin algebra is an algebra over a commutative artinian ring that is finitely generated as a module over its center; finite-dimensional algebras over a field are the main example. A (Miyashita) tilting module TT over a ring RR is a finitely presented module of finite projective dimension such that ExtRi(T,T)=0\mathrm{Ext}^i_R(T,T)=0 for all i>0i>0 (self-orthogonality) and such that the regular module RR admits a finite coresolution 0→R→T0→T1→⋯→Tn→00\to R\to T_0\to T_1\to\cdots\to T_n\to 0 with each TiT_i a direct summand of a finite direct sum of copies of TT (the class of such summands is denoted add(T)\mathrm{add}(T)). Tilting modules induce derived equivalences and control much of the homological behavior of the algebra.

Wakamatsu (1988), studying modules with trivial self-extensions over artin algebras, introduced a broader class: a module TT is Wakamatsu tilting if it satisfies the self-orthogonality and add(T)\mathrm{add}(T) conditions but the coresolution of RR is allowed to be infinite — an exact sequence 0→R→T0→T1→T2→⋯0\to R\to T_0\to T_1\to T_2\to\cdots with each Ti∈add(T)T_i\in\mathrm{add}(T) that stays exact after applying HomR(−,T)\mathrm{Hom}_R(-,T) — with no bound on projective dimension. Wakamatsu (2004) extended the definition and the associated equivalence theory between module categories to arbitrary associative rings, via bimodules STR_S T_R with R≅End(ST)opR\cong\mathrm{End}(_S T)^{\mathrm{op}} and S≅End(TR)opS\cong\mathrm{End}(T_R)^{\mathrm{op}} that are self-orthogonal on both sides (equivalently, semidualizing bimodules of finite type). Every tilting module is Wakamatsu tilting; the difficulty is the converse when the projective dimension happens to be finite.

The conjecture that finite projective dimension collapses the distinction — every Wakamatsu tilting module of finite projective dimension is tilting — was formulated in print by Beligiannis and Reiten (2007, Chapter III) and is standardly called the Wakamatsu Tilting Conjecture. It occupies a distinguished node in the network of long-standing homological conjectures for artin algebras: the Finitistic Dimension Conjecture (FDC, asserting that projective dimensions of finitely generated modules of finite projective dimension are uniformly bounded) implies the WTC (Mantese–Reiten), and the WTC in turn implies the Generalized Nakayama Conjecture (GNC: every indecomposable injective appears in the minimal injective resolution of the regular module) and the Gorenstein Symmetry Conjecture (GSC: the injective dimensions of RR as a left and as a right module are simultaneously finite or infinite); the GNC further implies the classical Nakayama Conjecture. Recently, the Generalized Auslander–Reiten Conjecture (GARC) was shown to imply the WTC, although GARC is itself known to fail in general, so this route does not settle the question.

Zhu and Ding (2009, Forum Mathematicum) proved that, over a left coherent ring RR (coherence: finitely generated left ideals are finitely presented) with a Wakamatsu tilting module ω\omega such that S=End(Rω)S=\mathrm{End}(_R\omega) is right coherent, and using FP-injective dimension (a weak injective dimension adapted to finitely presented modules, introduced for coherent rings), the FP-injective dimensions of ω\omega on the two sides coincide whenever both are finite, and they established the existence of special precovers and preenvelopes under these hypotheses. Their results remain conditional on these finiteness and coherence assumptions.

Problem Statement

Wakamatsu Tilting Conjecture. Over an artin algebra Λ\Lambda (and, in the extension to arbitrary associative rings with identity introduced by Wakamatsu in 2004): if ΛT_\Lambda T is a Wakamatsu tilting module — a finitely generated module with ExtΛi(T,T)=0\mathrm{Ext}^i_\Lambda(T,T)=0 for all i>0i>0, admitting an exact core 0→Λ→T0→T1→T2→⋯0\to\Lambda\to T_0\to T_1\to T_2\to\cdots with each Ti∈add(T)T_i\in\mathrm{add}(T) that remains exact after applying HomΛ(−,T)\mathrm{Hom}_\Lambda(-,T) — and TT has finite projective dimension, must TT be a tilting module in the sense of Miyashita, i.e. must there exist such a coresolution of Λ\Lambda of finite length nn (equivalently, TT is nn-tilting for some nn)? Equivalently: does finite projective dimension force the infinite Ext\mathrm{Ext}-exact core of a self-orthogonal equivalence module to terminate? The conjecture is open in general, over artin algebras as well as over arbitrary associative rings.

References

  1. Haiyan Zhu and Nanqing Ding, "Wakamatsu tilting modules with finite FP-injective dimension", Forum Mathematicum 21(1) (2009). DOI: 10.1515/forum.2009.005. https://doi.org/10.1515/forum.2009.005
  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. Yoichi Miyashita, "Tilting modules of finite projective dimension", Mathematische Zeitschrift 193 (1986) 113–146. DOI: 10.1007/BF01163359. https://doi.org/10.1007/BF01163359
  5. Apostolos Beligiannis and 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
  6. 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
  7. Zhaoyong Huang, "Generalized tilting modules with finite injective dimension", Journal of Algebra 311(2) (2007) 619–634. DOI: 10.1016/j.jalgebra.2006.11.025. https://doi.org/10.1016/j.jalgebra.2006.11.025
  8. Kamran Divaani-Aazar, Ali Mahin Fallah, and Massoud Tousi, "On the Wakamatsu tilting conjecture", Archiv der Mathematik 125 (2025) 291–302. DOI: 10.1007/s00013-025-02147-5. arXiv:2407.06353. https://doi.org/10.1007/s00013-025-02147-5
  9. 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
  10. Haruhisa Enomoto, "Maximal self-orthogonal modules and a new generalization of tilting modules", preprint (2023). arXiv:2301.13498. https://arxiv.org/abs/2301.13498
  11. Zhenxian Chen and Changchang Xi, "Singular equivalences and homological conjectures", preprint (2026). arXiv:2603.20643. https://arxiv.org/abs/2603.20643