A Wakamatsu tilting module over an artin algebra is a self-orthogonal module that behaves like a tilting module except that its projective dimension may be infinite and the regular module need only admit an infinite coresolution by it, remaining exact after applying Hom against the module. The Wakamatsu tilting conjecture (WTC) asks whether finite projective dimension forces such a module to be an honest (Miyashita) tilting module. The conjecture, formulated by Beligiannis and Reiten, implies both the Gorenstein symmetry conjecture and the generalized Nakayama conjecture and is implied by the finitistic dimension conjecture and the generalized Auslander–Reiten conjecture. The conjecture is open in general: it is proved only in restricted classes (representation-finite algebras, Gorenstein artin algebras, tensorly faithful modules, i.e. modules TT with T⊗Λ−T\otimes_\Lambda- faithful, left Artinian local rings, finite-group group rings) and no counterexample is known.

Background

Throughout, Λ\Lambda denotes an artin algebra (a ring that is finitely generated as a module over its artinian center; for example a finite-dimensional algebra over a field), and mod Λ\mathrm{mod}\,\Lambda its category of finitely generated right modules. A module TT is called self-orthogonal if ExtΛi(T,T)=0\mathrm{Ext}^i_\Lambda(T,T)=0 for all i>0i>0. A classical (Miyashita) tilting module is a self-orthogonal module TT of finite projective dimension for which the regular module Λ\Lambda admits a finite coresolution 0→Λ→T0→T1→⋯→Tn→00\to\Lambda\to T_0\to T_1\to\cdots\to T_n\to 0 with each Ti∈add TT_i\in\mathrm{add}\,T (direct summands of finite direct sums of TT); tilting modules of finite projective dimension are the basic objects of tilting theory and induce derived equivalences. Wakamatsu, in his study of modules with trivial self-extensions and of tilting-type modules of possibly infinite projective dimension, introduced the following generalization: a self-orthogonal module TT is Wakamatsu tilting if there is a long exact sequence 0→Λ→T0→T1→T2→⋯0\to\Lambda\to T_0\to T_1\to T_2\to\cdots with all Ti∈add TT_i\in\mathrm{add}\,T that stays exact after applying HomΛ(−,T)\mathrm{Hom}_\Lambda(-,T). Equivalently, in the formulation of Auslander and Reiten, TT is Wakamatsu tilting when the injective cogenerator DΛD\Lambda (with DD the standard duality, for artin algebras, between finitely generated left and right modules) lies in the subcategory YT\mathcal{Y}_T of modules admitting such a TT-coresolution inside the Ext-perpendicular category T⊥={X∣ExtΛi(T,X)=0, i>0}T^\perp=\{X\mid\mathrm{Ext}^i_\Lambda(T,X)=0,\ i>0\}. Every tilting module is Wakamatsu tilting, and the containment can be strict; Wakamatsu tilting modules also arise canonically as the injective cogenerators of small exact categories (additive categories equipped with a distinguished class of conflations, i.e. admissible short exact sequences, satisfying the exact-category axioms). The conjecture asking whether finite projective dimension collapses the two notions was formulated in this generality by Beligiannis and Reiten, and the problem sits at the center of a dense web of the classical homological conjectures for artin algebras: the finitistic dimension conjecture (FDC: the supremum of finite projective dimensions of finite modules is finite) implies WTC (Mantese–Reiten); WTC in turn implies the Gorenstein symmetry conjecture (GSC: id ΛΛ<∞\mathrm{id}\,\Lambda_\Lambda<\infty iff id ΛΛ<∞\mathrm{id}\,{}_{\Lambda}\Lambda<\infty, where id denotes injective dimension) and the generalized Nakayama conjecture (GNC: every indecomposable injective appears in the minimal injective resolution of Λ\Lambda); and the generalized Auslander–Reiten conjecture (GARC: Exti(M,M⊕Λ)=0\mathrm{Ext}^i(M,M\oplus\Lambda)=0 for all i>ni>n implies pd M≤n\mathrm{pd}\,M\le n) also implies WTC (Divaani-Aazar–Fallah–Tousi). Enomoto (2023) recast the conjecture in exact-category terms: WTC is equivalent to the statement that in every Hom-finite Krull–Schmidt exact category with a progenerator PP and an injective cogenerator II, finite projective dimension of II forces finite injective dimension of PP; and he proved WTC for representation-finite algebras (algebras with finitely many indecomposable finitely generated modules), where the Wakamatsu tilting, the maximal self-orthogonal (a self-orthogonal module that is not a proper direct summand of a strictly larger self-orthogonal module), and the self-orthogonal modules with the same number of non-isomorphic indecomposable direct summands as Λ\Lambda all coincide. Known partial cases beyond representation-finite algebras include all Gorenstein artin algebras (Mantese–Reiten); tensorly faithful modules, i.e. modules ΛT{}_{\Lambda}T for which the functor T⊗Λ−T\otimes_\Lambda- is faithful, equivalently T⊗ΛM≠0T\otimes_\Lambda M\neq 0 for every nonzero finitely generated left Λ\Lambda-module MM (Divaani-Aazar–Fallah–Tousi); left Artinian local rings; and group rings of finite groups over commutative Artinian rings. Cruz characterized WTC through a higher Morita–Tachikawa correspondence. Enomoto's paper further shows that WTC implies that over Iwanaga–Gorenstein algebras (algebras of finite injective dimension on both sides) every Wakamatsu tilting module is tilting, a statement related to the still-open conjecture that self-orthogonal modules over Iwanaga–Gorenstein algebras have finite projective dimension, itself a common generalization of the Tachikawa conjecture (over a self-injective algebra, every self-orthogonal module is projective).

Problem Statement

The Wakamatsu tilting conjecture. Let Λ\Lambda be an artin algebra and let TT be a Wakamatsu tilting Λ\Lambda-module, i.e. a self-orthogonal module (ExtΛi(T,T)=0\mathrm{Ext}^i_\Lambda(T,T)=0 for all i>0i>0) admitting a long exact sequence 0→Λ→T0→T1→T2→⋯0\to\Lambda\to T_0\to T_1\to T_2\to\cdots with each Ti∈add TT_i\in\mathrm{add}\,T that remains exact after applying HomΛ(−,T)\mathrm{Hom}_\Lambda(-,T). If TT has finite projective dimension, must TT be a classical tilting module, i.e. must Λ\Lambda admit a finite coresolution 0→Λ→T0→⋯→Tn→00\to\Lambda\to T_0\to\cdots\to T_n\to 0 with each Ti∈add TT_i\in\mathrm{add}\,T? Equivalently (Enomoto 2023, Lemma 4.12): in every Hom-finite Krull–Schmidt exact category with a progenerator PP and an injective cogenerator II, does pd I<∞\mathrm{pd}\,I<\infty imply id P<∞\mathrm{id}\,P<\infty (with pd\mathrm{pd} and id\mathrm{id} denoting projective and injective dimension)? The problem is posed for all artin algebras; the representation-finite, Gorenstein-artin, tensorly faithful (T⊗Λ−T\otimes_\Lambda- faithful), left-Artinian-local, and finite-group group-ring cases are already settled affirmatively, and the general case (affirmative answer, or an explicit counterexample algebra and module) is what is asked.

References

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  2. Takayoshi Wakamatsu, 'Tilting modules and Auslander's Gorenstein property', Journal of Algebra 275 (2004), no. 1, 3–39. DOI: 10.1016/j.jalgebra.2003.12.008, https://doi.org/10.1016/j.jalgebra.2003.12.008
  3. Francesca Mantese and Idun Reiten, 'Wakamatsu tilting modules', Journal of Algebra 278 (2004), no. 2, 532–552. DOI: 10.1016/j.jalgebra.2004.03.023, https://doi.org/10.1016/j.jalgebra.2004.03.023
  4. Apostolos Beligiannis and Idun Reiten, 'Homological and homotopical aspects of torsion theories', Memoirs of the American Mathematical Society 188 (2007), no. 883. DOI: 10.1090/memo/0883, https://doi.org/10.1090/memo/0883
  5. Haruhisa Enomoto, 'Maximal self-orthogonal modules and a new generalization of tilting modules' (2023). DOI: 10.48550/arXiv.2301.13498, https://arxiv.org/abs/2301.13498
  6. Tiago Cruz, 'Higher Morita–Tachikawa correspondence', Bulletin of the London Mathematical Society 56 (2024), no. 8, 2647–2660. DOI: 10.1112/blms.13090, https://doi.org/10.1112/blms.13090
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