The Gorenstein Symmetric Conjecture (GSC), recorded by Auslander–Reiten–Smalø, predicts that for any artin algebra Λ\Lambda, finiteness of the self-injective dimension on one side implies finiteness on the other: if id ΛΛ<∞\mathrm{id}\,{}_{\Lambda}\Lambda<\infty then id ΛΛ<∞\mathrm{id}\,\Lambda_{\Lambda}<\infty. Huang (2022) proved, via his equivalence theorem for relative homological dimensions over left and right Noetherian rings, that GSC is equivalent to a family of module-level statements: for every artin algebra with id ΛΛ≤n\mathrm{id}\,{}_{\Lambda}\Lambda\le n, the Gorenstein projective (equivalently Gorenstein injective, Gorenstein flat, strongly Gorenstein flat, or projectively coresolved Gorenstein flat) dimension of every module is uniformly bounded by nn on at least one side. Despite substantial partial results (weakly Gorenstein and monomial algebras, reductions via recollements and excellent extensions, Auslander-type conditions) and one claimed proof withdrawn by its author in July 2026, the conjecture remains open for arbitrary artin algebras.

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; the model case is a finite-dimensional algebra over a field, typically given by a quiver with relations — a directed graph whose paths, taken modulo the imposed relations, form a basis of the algebra. The left and right finitely generated module categories of Λ\Lambda are linked by a standard duality, but homological properties of Λ\Lambda as a left versus right module over itself need not coincide. The injective dimension id M\mathrm{id}\,M of a module is the minimal length of an injective resolution of MM, and the self-injective dimensions of Λ\Lambda are id ΛΛ\mathrm{id}\,{}_{\Lambda}\Lambda and id ΛΛ\mathrm{id}\,\Lambda_{\Lambda}. The algebra is called Gorenstein (or Iwanaga–Gorenstein) when both are finite, and nn-Gorenstein when both are at most nn; a classical lemma of Auslander–Reiten shows that when both self-injective dimensions are finite they are equal. Gorenstein algebras are the natural home of Gorenstein homological algebra, where the role of projectives is played by Gorenstein projective modules — modules appearing as cycles of a totally acyclic complex of projective modules, i.e. an acyclic complex of projectives that stays acyclic after applying Hom(−,P)\mathrm{Hom}(-,P) for every projective PP — dually for Gorenstein injective modules, and analogously for Gorenstein flat modules (cycles of acyclic complexes of flat modules staying acyclic after tensoring with any injective), strongly Gorenstein flat modules (cycles of acyclic complexes of projectives staying acyclic after tensoring with any injective, in the sense of Ding–Li–Mao), and projectively coresolved Gorenstein flat modules (cycles of acyclic complexes of projectives staying acyclic after applying Hom(−,F)\mathrm{Hom}(-,F) for every flat FF). Each class yields a relative dimension of an arbitrary module, namely the minimal length of a resolution by objects of the class. The Gorenstein Symmetric Conjecture, recorded on p. 410 of Auslander–Reiten–Smalø's book 'Representation Theory of Artin Algebras' (1995), states that for any artin algebra Λ\Lambda, if id ΛΛ<∞\mathrm{id}\,{}_{\Lambda}\Lambda<\infty then id ΛΛ<∞\mathrm{id}\,\Lambda_{\Lambda}<\infty. It is one of the long-standing homological conjectures for artin algebras, studied alongside the Nakayama conjecture — which predicts that every non-self-injective artin algebra has finite dominant dimension, meaning only finitely many leading terms in a minimal injective resolution of the regular module are projective — and it underlies whether 'Gorensteinness' of an artin algebra can be detected from a single side. Huang's paper 'Homological dimensions relative to preresolving subcategories II' proves that for a left and right Noetherian ring RR, being nn-Gorenstein is equivalent to each of the one-sided conditions that the Gorenstein projective (equivalently Gorenstein injective, Gorenstein flat, strongly Gorenstein flat, or projectively coresolved Gorenstein flat) dimension of every left RR-module is at most nn (Theorem 1.2 = Theorems 4.9, 4.11, 4.13), and observes as a consequence that GSC is equivalent to these module-level uniform-bound implications for artin algebras of finite one-sided self-injective dimension.

Problem Statement

Prove or refute the Gorenstein Symmetric Conjecture: for every artin algebra Λ\Lambda, if the injective dimension of Λ\Lambda as a left module over itself is finite, id ΛΛ<∞\mathrm{id}\,{}_{\Lambda}\Lambda<\infty, then the injective dimension of Λ\Lambda as a right module over itself is also finite, id ΛΛ<∞\mathrm{id}\,\Lambda_{\Lambda}<\infty (by symmetry the same implication with left and right interchanged is part of the conjecture; when both sides are finite they are known to be equal). Equivalently, by Huang's Theorem 1.2 (arXiv:2112.12977), establish or refute the following module-level statement: for every artin algebra Λ\Lambda and every n≥0n\ge 0 with id ΛΛ≤n\mathrm{id}\,{}_{\Lambda}\Lambda\le n, the Gorenstein projective dimension of every left Λ\Lambda-module is at most nn — equivalently, the Gorenstein injective, the Gorenstein flat, the strongly Gorenstein flat, or the projectively coresolved Gorenstein flat dimension of every left Λ\Lambda-module is at most nn, or the corresponding statement holds for right Λ\Lambda-modules (equivalently, left modules over Λop\Lambda^{\mathrm{op}}). A refutation consists of an artin algebra with finite self-injective dimension on exactly one side, which would also falsify every equivalent module-level implication.

References

  1. Zhaoyong Huang, "Homological dimensions relative to preresolving subcategories II", Forum Mathematicum 34 (2022), no. 2, 507–530, DOI 10.1515/forum-2021-0136, arXiv:2112.12977, https://arxiv.org/abs/2112.12977
  2. Maurice Auslander, Idun Reiten, Sverre O. Smalø, "Representation Theory of Artin Algebras", Cambridge Studies in Advanced Mathematics 36, Cambridge University Press (1995), DOI 10.1017/CBO9780511623608, https://doi.org/10.1017/CBO9780511623608
  3. Xiao-Wu Chen, "Gorenstein Homological Algebra of Artin Algebras", arXiv:1712.04587 (2017), https://arxiv.org/abs/1712.04587
  4. Rene Marczinzik, "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra", arXiv:1608.04212, https://arxiv.org/abs/1608.04212
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  9. Zhaoyong Huang, "Auslander-type conditions and weakly Gorenstein algebras", Bulletin of the London Mathematical Society 56 (2024), no. 11, 3382–3399, DOI 10.1112/blms.13138, arXiv:2408.05468, https://arxiv.org/abs/2408.05468
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  12. Weiqing Li, "On a proof of the Gorenstein Symmetry Conjecture", arXiv:2607.28011 (2026, withdrawn by the author), https://arxiv.org/abs/2607.28011
  13. Víctor Becerril, "Strongly FP-injective dimensions and Gorenstein projective precovers", arXiv:2602.00932 (2026), https://arxiv.org/abs/2602.00932