For an artin algebra Λ\Lambda, let Ωn(mod-Λ)\Omega^n(\mathrm{mod}\text{-}\Lambda) denote the class of nn-th syzygies of finitely generated right Λ\Lambda-modules. The Co-Gorenstein conjecture (Conjecture 1 of Kvamme–Marczinzik 2019, isolating a claim of Beligiannis 2000) asserts that if Ωn(mod-Λ)\Omega^n(\mathrm{mod}\text{-}\Lambda) is extension closed for every n≥1n\ge 1, then Λ\Lambda is right Co-Gorenstein — equivalently, every module that embeds in an infinite acyclic complex of finitely generated projectives is Gorenstein projective, i.e. Ω∞(mod-Λ)=GP(Λ)\Omega^\infty(\mathrm{mod}\text{-}\Lambda)=\mathcal{GP}(\Lambda), or Ω∞(mod-Λ)⊆⊥Λ\Omega^\infty(\mathrm{mod}\text{-}\Lambda)\subseteq{}^\perp\Lambda. The conjecture sits between the Generalized Nakayama conjecture, which implies it, and the classical Nakayama conjecture of 1958, which it implies. Despite partial structural results for algebras of Ωn\Omega^n-finite representation type and for truncated path algebras (Barrios–Mata), the conjecture remains open for arbitrary artin algebras.

Background

An artin algebra is an algebra Λ\Lambda that is finitely generated as a module over a commutative artinian ring RR; the standard example is a finite-dimensional algebra over a field, presented by a quiver with relations. Write mod-Λ\mathrm{mod}\text{-}\Lambda for the category of finitely generated right Λ\Lambda-modules. For a module MM, choose a projective cover P→MP\to M; its kernel ΩM\Omega M is the first syzygy of MM, defined up to projective summands. Iterating gives the classes Ωn(mod-Λ)={M∣there is an exact sequence 0→M→P1→⋯→Pn with all Pi finitely generated projective}\Omega^n(\mathrm{mod}\text{-}\Lambda)=\{M \mid \text{there is an exact sequence } 0\to M\to P_1\to\cdots\to P_n\text{ with all } P_i \text{ finitely generated projective}\} of nn-th syzygies. A full subcategory is extension closed if every short exact sequence 0→X→Y→Z→00\to X\to Y\to Z\to 0 with XX and ZZ in the subcategory also has YY in it. Whether Ωn(mod-Λ)\Omega^n(\mathrm{mod}\text{-}\Lambda) is extension closed is a strong and rare property: by a theorem of Auslander–Reiten recorded as Theorem 16 of Kvamme–Marczinzik, it holds for 1≤k≤n1\le k\le n if and only if inj.dim Pk(DΛ)≤k+1\mathrm{inj.dim}\,P_k(D\Lambda)\le k+1 for 0≤k<n0\le k<n, where D=HomR(−,I)D=\mathrm{Hom}_R(-,I) is the standard duality with II an injective cogenerator of RR and Pk(DΛ)P_k(D\Lambda) are the terms of the minimal projective resolution of DΛD\Lambda. Two further classes matter. First, Ω∞(mod-Λ)\Omega^\infty(\mathrm{mod}\text{-}\Lambda) consists of the modules MM admitting an infinite exact sequence 0→M→P0→P1→⋯0\to M\to P_0\to P_1\to\cdots with all PiP_i finitely generated projective — the modules realized in the costabilization of the projectively stable category in the sense of Beligiannis. Second, GP(Λ)\mathcal{GP}(\Lambda) is the class of Gorenstein projective (also called totally reflexive) modules: those MM for which the natural map to the double Λ\Lambda-dual M→M∗∗M\to M^{**}, with M∗=HomΛ(M,Λ)M^{*}=\mathrm{Hom}_\Lambda(M,\Lambda), is an isomorphism and ExtΛi(M,Λ)=ExtΛi(M∗,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=\mathrm{Ext}^i_\Lambda(M^{*},\Lambda)=0 for all i≥1i\ge 1; equivalently, the modules occurring as kernels of complete projective resolutions (acyclic bi-infinite complexes of projectives remaining exact after applying HomΛ(−,projectives)\mathrm{Hom}_\Lambda(-,\text{projectives})). One always has GP(Λ)⊆Ω∞(mod-Λ)\mathcal{GP}(\Lambda)\subseteq\Omega^\infty(\mathrm{mod}\text{-}\Lambda), and the inclusion can be strict. Beligiannis introduced Co-Gorenstein categories via the costabilization; for an artin algebra this yields, by Definition 14 and Corollary 11 of Kvamme–Marczinzik, that Λ\Lambda is right Co-Gorenstein if and only if Ω∞(mod-Λ)⊆⊥Λ={M∣ExtΛi(M,Λ)=0 for all i≥1}\Omega^\infty(\mathrm{mod}\text{-}\Lambda)\subseteq{}^\perp\Lambda=\{M\mid \mathrm{Ext}^i_\Lambda(M,\Lambda)=0 \text{ for all } i\ge 1\}, if and only if Ω∞(mod-Λ)=GP(Λ)\Omega^\infty(\mathrm{mod}\text{-}\Lambda)=\mathcal{GP}(\Lambda). Every Iwanaga–Gorenstein algebra (both injective dimensions of Λ\Lambda over itself finite) is right Co-Gorenstein, and finite right injective dimension of Λ\Lambda already suffices; the converse fails — the radical-square-zero local algebra k[x,y]/(x2,xy,yx,y2)k[x,y]/(x^2,xy,yx,y^2) of Kvamme–Marczinzik, Example 15, is right Co-Gorenstein but not Iwanaga–Gorenstein. The conjecture belongs to the web of classical homological conjectures for artin algebras: the finitistic dimension conjecture (FDC, 1960) asserts that the supremum of the finite projective dimensions of modules is finite; the Nakayama conjecture (NC, 1958) asserts that if the dominant dimension of Λ\Lambda — the number of initial injective terms in a minimal injective resolution of Λ\Lambda — is infinite, then Λ\Lambda is self-injective; the Generalized Nakayama conjecture (GNC, Auslander–Reiten) asserts that every indecomposable projective right module PP occurs as a direct summand of some term Pn(DΛ)P_n(D\Lambda) of the minimal projective resolution of DΛD\Lambda. It is known that FDC implies GNC and that GNC implies NC, and all three remain open. Kvamme–Marczinzik proved that GNC implies the Co-Gorenstein conjecture (Proposition 21, through the intermediate conclusion inj.dim ΛΛ<∞\mathrm{inj.dim}\,\Lambda_\Lambda<\infty) and that the Co-Gorenstein conjecture implies NC (Proposition 18); hence a proof of the Co-Gorenstein conjecture would unconditionally settle the 1958 Nakayama conjecture without settling GNC. Historically, the statement appears as Lemma 6.19(3) in Beligiannis's 2000 paper, where it is claimed to follow from earlier results; Kvamme–Marczinzik state that this derivation is not clear to them and therefore pose it as a conjecture. Structural knowledge is partial: Barrios–Mata proved that for algebras of Ωn\Omega^n-finite representation type (finitely many indecomposable nn-th syzygies for some nn), a module lies in Ω∞(mod-Λ)\Omega^\infty(\mathrm{mod}\text{-}\Lambda) if and only if it is projective or Ω\Omega-periodic, and they characterized by quiver geometry when a truncated path algebra (a path algebra modulo all paths of a fixed length) is Co-Gorenstein — but for arbitrary artin algebras no structure theory of Ω∞\Omega^\infty is available, and the conjecture stands.

Problem Statement

Let Λ\Lambda be an artin algebra and mod-Λ\mathrm{mod}\text{-}\Lambda the category of finitely generated right Λ\Lambda-modules. Suppose that Ωn(mod-Λ)\Omega^n(\mathrm{mod}\text{-}\Lambda) is extension closed for every n≥1n\ge 1, where Ωn(mod-Λ)\Omega^n(\mathrm{mod}\text{-}\Lambda) is the class of nn-th syzygies, i.e. of modules MM admitting an exact sequence 0→M→P1→⋯→Pn0\to M\to P_1\to\cdots\to P_n with each PiP_i finitely generated projective, and where a class of modules is extension closed if every short exact sequence 0→X→Y→Z→00\to X\to Y\to Z\to 0 with XX and ZZ in the class also has YY in the class. Must Λ\Lambda be right Co-Gorenstein, in any of the following equivalent forms: every module MM admitting an infinite exact sequence 0→M→P0→P1→⋯0\to M\to P_0\to P_1\to\cdots of finitely generated projectives satisfies ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=0 for all i≥1i\ge 1; equivalently Ω∞(mod-Λ)⊆⊥Λ\Omega^\infty(\mathrm{mod}\text{-}\Lambda)\subseteq{}^\perp\Lambda, where ⊥Λ{}^\perp\Lambda denotes the modules MM with ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=0 for all i≥1i\ge 1; equivalently Ω∞(mod-Λ)=GP(Λ)\Omega^\infty(\mathrm{mod}\text{-}\Lambda)=\mathcal{GP}(\Lambda), every infinite syzygy is Gorenstein projective (totally reflexive: isomorphic to its double Λ\Lambda-dual via the natural map, with Λ\Lambda-dual M∗=HomΛ(M,Λ)M^*=\mathrm{Hom}_\Lambda(M,\Lambda), and with ExtΛi(M,Λ)=ExtΛi(M∗,Λ)=0\mathrm{Ext}^i_\Lambda(M,\Lambda)=\mathrm{Ext}^i_\Lambda(M^*,\Lambda)=0 for all i≥1i\ge 1)? By the results of Kvamme–Marczinzik, an affirmative answer unconditionally implies the classical Nakayama conjecture.

References

  1. Sondre Kvamme, René Marczinzik, 'Co-Gorenstein Algebras', Applied Categorical Structures 27(3), pp. 277–287 (2019). DOI: 10.1007/s10485-018-09554-6; arXiv:1805.00274, https://arxiv.org/abs/1805.00274
  2. Marcos Barrios, Gustavo Mata, 'On algebras of Ω^n-finite and Ω^∞-infinite representation type', Journal of Algebra and Its Applications 23(11) (published online 2023; issue dated 2024). DOI: 10.1142/s0219498824501834; arXiv:1911.02325, https://arxiv.org/abs/1911.02325
  3. Apostolos Beligiannis, 'The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and (co-)stabilization', Communications in Algebra 28(10), pp. 4547–4596 (2000). DOI: 10.1080/00927870008827105, https://doi.org/10.1080/00927870008827105
  4. Tiago Cruz, René Marczinzik, 'A new formula for the classical dominant dimension using bimodules', arXiv:2508.18398 (2025, revised 2026), https://arxiv.org/abs/2508.18398
  5. Xiaojin Zhang, 'A Note on Gorenstein Projective Conjecture II', arXiv:1211.5624 (2012, revised 2013), https://arxiv.org/abs/1211.5624