The Strong Nakayama Conjecture (SNC), posed by Colby and Fuller in 1990, asserts that over an Artin algebra Λ\Lambda the regular module Λ\Lambda cogenerates finitely generated modules in the strongest homological sense: if a finitely generated Λ\Lambda-module MM satisfies ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=0 for all i≥0i\geq 0 (including i=0i=0, i.e. HomΛ(M,Λ)=0\mathrm{Hom}_\Lambda(M,\Lambda)=0), then M=0M=0. It strengthens the Auslander–Reiten generalized Nakayama conjecture (1975) and implies Nakayama's 1958 conjecture, and it is a formal consequence of the finitistic dimension conjecture. Despite verifications for expanding classes of algebras (algebras with finite opposite finitistic dimension, virtually Gorenstein algebras — even for infinitely generated modules —, centralizer matrix algebras, and frameworks via preresolving subcategories), the conjecture remains open at its original generality over all Artin algebras, as confirmed by explicit statements in 2025–2026 literature. The problem: prove or refute SNC in its original generality.

Background

An Artin algebra is a ring Λ\Lambda that is finitely generated as a module over its center, which is a commutative artinian ring (equivalently, Λ\Lambda is finitely generated as a module over some commutative artinian ring); the basic example is a finite-dimensional algebra over a field. One studies the category mod-Λ\mathrm{mod}\text{-}\Lambda of finitely generated right Λ\Lambda-modules. The module Λ\Lambda viewed as a right module over itself is called the regular module, and the extension groups ExtΛi(M,Λ)\mathrm{Ext}^i_\Lambda(M,\Lambda) measure how far MM is from being detectable by Λ\Lambda: ExtΛ0(M,Λ)=HomΛ(M,Λ)\mathrm{Ext}^0_\Lambda(M,\Lambda)=\mathrm{Hom}_\Lambda(M,\Lambda), and a module with HomΛ(M,Λ)=0\mathrm{Hom}_\Lambda(M,\Lambda)=0 is in particular one that does not embed in any free module (a non-torsionless module) — such nonzero modules genuinely exist over Artin algebras. A module MM with ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=0 for all i≥1i\geq 1 is called semi-Gorenstein-projective; a Gorenstein projective module is one that occurs as a syzygy of a totally acyclic complex of projective modules (a long exact complex staying exact under Hom(−,P)\mathrm{Hom}(-,P) for every projective PP), and Gorenstein projective modules are in particular torsionless. Ringel and Zhang (2020) constructed finitely generated semi-Gorenstein-projective modules that are not Gorenstein projective, showing these classes differ in general; an algebra where every finitely generated semi-Gorenstein-projective module is Gorenstein projective is called (right) weakly Gorenstein. The conjecture sits in a classical chain. Nakayama (1958) conjectured — in a form equivalent via Müller's work — that if every term EiE_i of the minimal injective resolution (the injective resolution 0→Λ→E0→E1→⋯0\to\Lambda\to E_0\to E_1\to\cdots with each EiE_i an injective hull of the previous cokernel) is projective, then Λ\Lambda is self-injective (projective modules coincide with injective ones). Auslander and Reiten (1975) posed the generalized Nakayama conjecture (GNC): if a simple module SS satisfies ExtΛi(S,Λ)=0\mathrm{Ext}^i_\Lambda(S,\Lambda)=0 for all i≥0i\geq 0 then S=0S=0, equivalently every indecomposable injective module occurs as a summand of some EiE_i. Colby and Fuller (1990) proposed the strong Nakayama conjecture (SNC): the same conclusion for every finitely generated module. GNC is equivalent to the Auslander–Reiten conjecture (ExtΛi(M,M⊕Λ)=0\mathrm{Ext}^i_\Lambda(M,M\oplus\Lambda)=0 for all i≥1i\geq 1 implies MM projective), and the whole family is implied by the finitistic dimension conjecture — the assertion that the supremum of projective dimensions of finitely generated modules of finite projective dimension is finite — which has itself been open since the 1960s. The accepted implication chain is: finitistic dimension conjecture ⇒\Rightarrow SNC ⇒\Rightarrow GNC ⇒\Rightarrow Nakayama conjecture. Note the structural constraint on any counterexample: a module of finite projective dimension always has some nonzero Exti(−,Λ)\mathrm{Ext}^i(-,\Lambda), so a counterexample to SNC must be a nonzero semi-Gorenstein-projective module of infinite projective dimension that is simultaneously non-torsionless — two conditions pulling in opposite directions — and it can only exist over a non-weakly-Gorenstein algebra, which is exactly the territory opened by the Ringel–Zhang examples.

Problem Statement

Prove or refute the Strong Nakayama Conjecture in its original generality (Colby–Fuller, 1990): let Λ\Lambda be an Artin algebra and let MM be a finitely generated right Λ\Lambda-module such that ExtΛi(M,Λ)=0for all i≥0\mathrm{Ext}^i_\Lambda(M,\Lambda)=0\quad\text{for all }i\geq 0 (including degree zero, i.e. HomΛ(M,Λ)=0\mathrm{Hom}_\Lambda(M,\Lambda)=0). Must M=0M=0? An affirmative answer proves the conjecture and thereby settles the generalized Nakayama conjecture and Nakayama's 1958 conjecture; a negative answer consists of an explicit Artin algebra Λ\Lambda and an explicit nonzero finitely generated Λ\Lambda-module with ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=0 for every i≥0i\geq 0. The same question for arbitrary (not necessarily finitely generated) modules over arbitrary Artin algebras is a strictly stronger variant that likewise remains open; the problem here is the original finitely generated question.

References

  1. R. R. Colby and K. R. Fuller, "A note on the Nakayama conjectures", Tsukuba Journal of Mathematics 14 (1990), 343–352. DOI: 10.21099/tkbjm/1496161457. URL: https://doi.org/10.21099/tkbjm/1496161457
  2. M. Auslander and I. Reiten, "On a generalized version of the Nakayama conjecture", Proceedings of the American Mathematical Society 52 (1975), 69–74. DOI: 10.1090/S0002-9939-1975-0389977-6. URL: https://doi.org/10.1090/S0002-9939-1975-0389977-6
  3. T. Nakayama, "On algebras with complete homology", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 22 (1958), 300–307. DOI: 10.1007/BF02941960. URL: https://doi.org/10.1007/BF02941960
  4. Z. Huang, "Homological dimensions relative to preresolving subcategories", Kyoto Journal of Mathematics 54 (2014), no. 4, 727–757. DOI: 10.1215/21562261-2801795. URL: https://doi.org/10.1215/21562261-2801795 (also arXiv:1502.02354)
  5. C. M. Ringel and P. Zhang, "Gorenstein-projective and semi-Gorenstein-projective modules", Algebra & Number Theory 14 (2020), no. 1, 1–36. DOI: 10.2140/ant.2020.14.1. URL: https://doi.org/10.2140/ant.2020.14.1
  6. N. Gao, P. Zhang, and S. Zhu, "Weakly Gorensteinness of tensor algebras and Morita algebras", arXiv:2512.16072 (2025). URL: https://arxiv.org/abs/2512.16072
  7. Z. Chen and C. C. Xi, "Singular equivalences and homological conjectures", arXiv:2603.20643 (2026). URL: https://arxiv.org/abs/2603.20643
  8. W. Li, "Virtually Gorenstein Artin algebras are weakly Gorenstein", arXiv:2608.10049 (2026). URL: https://arxiv.org/abs/2608.10049