For an Artin algebra Λ\Lambda, call a finitely generated module MM semi-Gorenstein-projective if ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M, \Lambda) = 0 for all i≥1i \geq 1, and Gorenstein projective if MM occurs in a complete projective resolution. Ringel and Zhang ask (Algebra & Number Theory 14 (2020), Question 9.2; restated as Question 1.4 in Li, arXiv:2608.10049): if every finitely generated Gorenstein projective Λ\Lambda-module is projective (Λ\Lambda is CM-free, or G-regular), must Λ\Lambda be weakly Gorenstein, i.e., must every finitely generated semi-Gorenstein-projective module be Gorenstein projective? The question is confirmed open as of August 2026, including for commutative Artinian rings. It is known affirmatively for virtually Gorenstein algebras (unconditionally, by Li's theorem that virtually Gorenstein Artin algebras are weakly Gorenstein) and for algebras whose possibly infinitely generated Gorenstein projective modules are all projective (Li's Corollary 3.12).

Background

An Artin algebra is a ring Λ\Lambda that is finitely generated as a module over a commutative Artinian ring (for example, a finite-dimensional algebra over a field, given by a quiver with relations — a directed graph whose path algebra is quotiented by chosen linear relations on paths); one studies its finitely generated left and right modules. For a finitely generated Λ\Lambda-module MM, ExtΛi(M,Λ)\mathrm{Ext}^i_\Lambda(M, \Lambda) denotes the ii-th higher extension group of MM against the regular module Λ\Lambda. A module MM is called Gorenstein projective (also totally reflexive, or maximal Cohen–Macaulay) if it is the syzygy of a complete projective resolution: an acyclic complex of finitely generated projective modules, unbounded in both directions, that stays exact after applying HomΛ(−,P)\mathrm{Hom}_\Lambda(-, P) for every projective PP. A weaker condition was isolated by Ringel and Zhang: MM is semi-Gorenstein-projective if ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M, \Lambda) = 0 for all i≥1i \geq 1. Every Gorenstein projective module is semi-Gorenstein-projective, and Ringel–Zhang built an explicit 6-dimensional algebra (dimension over its base field) with a semi-Gorenstein-projective module that is not Gorenstein projective, so the inclusion is strict. Following Ringel–Zhang, Λ\Lambda is left (resp. right) weakly Gorenstein if every semi-Gorenstein-projective left (resp. right) module is Gorenstein projective, and weakly Gorenstein if both sides hold. Separately, an Artin algebra is called CM-free (equivalently G-regular) if every finitely generated Gorenstein projective module is projective — for instance, by a theorem of Xiao-Wu Chen, every connected radical-square-zero algebra (one whose Jacobson radical RR satisfies R2=0R^2 = 0) that is not self-injective is CM-free. A larger and actively studied class is the virtually Gorenstein algebras (defined by the equality of orthogonality classes GP⊥=⊥GI\mathrm{GP}^{\perp} = {}^{\perp}\mathrm{GI} among finitely generated modules, where GP and GI are the Gorenstein projective and Gorenstein injective modules); the class is left–right symmetric and contains all Iwanaga–Gorenstein algebras, i.e., Artin algebras of finite injective dimension on both sides. The question sits inside the classical network of homological conjectures for Artin algebras: the Gorenstein symmetry conjecture (GSC) predicts that finite injective dimension of the left regular module forces finite injective dimension of the right regular module, and Ringel–Zhang proved that if weak Gorensteinness were always left–right symmetric (their adjacent Question 9.3), then GSC would follow. Within their 2020 paper, Ringel–Zhang showed that torsionless-finite algebras (those with only finitely many indecomposable torsionless modules — modules embeddable in a projective) are both left and right weakly Gorenstein, but their methods do not touch the CM-free case, which is the subject of this question.

Problem Statement

Let Λ\Lambda be an Artin algebra — a ring that is finitely generated as a module over a commutative Artinian ring (for example, a finite-dimensional algebra over a field, given by a quiver with relations). Call a finitely generated right Λ\Lambda-module MM semi-Gorenstein-projective if ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M, \Lambda) = 0 for all i≥1i \geq 1, where ExtΛi(−,Λ)\mathrm{Ext}^i_\Lambda(-, \Lambda) denotes the ii-th higher extension group against the regular module; call MM Gorenstein projective (equivalently, totally reflexive) if it occurs as a syzygy in a complete projective resolution — an acyclic complex of finitely generated projective Λ\Lambda-modules, unbounded in both directions, that stays exact after applying HomΛ(−,P)\mathrm{Hom}_\Lambda(-, P) for every projective module PP. Say that Λ\Lambda is CM-free (equivalently, G-regular) if every finitely generated Gorenstein projective Λ\Lambda-module is projective; that Λ\Lambda is right weakly Gorenstein if every finitely generated semi-Gorenstein-projective right Λ\Lambda-module is Gorenstein projective; and that Λ\Lambda is weakly Gorenstein if both Λ\Lambda and its opposite algebra Λop\Lambda^{\mathrm{op}} are right weakly Gorenstein (equivalently, if the same property holds for left modules). Ringel–Zhang's Question 9.2 (2020), restated as Question 1.4 in Li (2026), asks: if Λ\Lambda is CM-free, must Λ\Lambda be weakly Gorenstein? Concretely, in the right-module formulation: if all finitely generated right Gorenstein projective Λ\Lambda-modules are projective, is every finitely generated right Λ\Lambda-module MM with ExtΛi(M,Λ)=0\mathrm{Ext}^i_\Lambda(M, \Lambda) = 0 for all i≥1i \geq 1 itself Gorenstein projective — which, under this hypothesis, is equivalent to being projective? The left-module version is the identical question for the opposite algebra; a complete answer addresses both sides. No restriction to virtually Gorenstein algebras, commutative rings, finite representation type, or bounded dimension is permitted — those are known or potential special cases, not part of the question. A resolution is either a proof that every CM-free (G-regular) Artin algebra is weakly Gorenstein, or an explicit Artin algebra that is CM-free but not weakly Gorenstein (exhibited by a finitely generated semi-Gorenstein-projective module that is not Gorenstein projective).

References

  1. Claus Michael Ringel and Pu Zhang, Gorenstein-projective and semi-Gorenstein-projective modules, Algebra & Number Theory 14 (2020), no. 1, 1–36. DOI: 10.2140/ant.2020.14.1, arXiv:1808.01809, https://doi.org/10.2140/ant.2020.14.1
  2. Weiqing Li, Virtually Gorenstein Artin algebras are weakly Gorenstein, preprint (2026). arXiv:2608.10049, https://arxiv.org/abs/2608.10049
  3. Kaito Kimura, Yuki Mifune, Yuya Otake, Ryo Takahashi, On strongly G-regular rings, preprint (2026). arXiv:2608.08228, https://arxiv.org/abs/2608.08228
  4. Xiao-Wu Chen, Gorenstein Homological Algebra of Artin Algebras, survey preprint (2017). arXiv:1712.04587, https://arxiv.org/abs/1712.04587