For an Artin algebra Λ\Lambda (for example, a finite-dimensional algebra over a field), it is conjectured — the Gorenstein Symmetry Conjecture, raised by Auslander and Reiten in 1991 — that if the injective dimension of Λ\Lambda as a left module over itself is finite, then the injective dimension of Λ\Lambda as a right module over itself is also finite, and conversely; an algebra with both sides finite is called Gorenstein. The conjecture is open: it has been proved only for algebras whose ring and opposite ring both satisfy Auslander's vanishing condition (AC) (together with an Artin or dualizing-complex hypothesis), by Christensen and Holm (2010).

Background

An Artin algebra is a ring Λ\Lambda that is finitely generated as a module over its center, the latter being a commutative artinian ring; the leading examples are finite-dimensional algebras over a field, which can be presented combinatorially by a quiver (directed graph) with relations. Homological invariants of such an algebra are measured by the Ext\mathrm{Ext} functors, the right derived functors of Hom\mathrm{Hom}, which detect how far modules are from being projective or injective. The injective dimension idΛΛ\mathrm{id}_\Lambda \Lambda of Λ\Lambda as a left module over itself is the length of the shortest resolution of Λ\Lambda by injective left modules; idΛ∘Λ\mathrm{id}_{\Lambda^\circ} \Lambda denotes the same for the opposite ring Λ∘\Lambda^\circ (the ring with multiplication reversed), i.e. for Λ\Lambda as a right module over itself. An Artin algebra is called Gorenstein (or an Iwanaga–Gorenstein algebra) in representation theory when both of these numbers are finite; when one side is finite, the algebra is called left- or right-Gorenstein. By a theorem of Iwanaga recorded in this literature, when both injective dimensions are finite they are equal.

The question whether one-sided finiteness forces the other side to be finite — so that "Gorenstein" could be checked from either side — was raised by Auslander and Reiten (1991) and is known as the Gorenstein Symmetry Question or Conjecture; it is one of the long-standing homological conjectures for finite-dimensional algebras, alongside the Nakayama, Auslander–Reiten, and Tachikawa conjectures, and it makes sense for arbitrary two-sided noetherian rings. It is trivially true for commutative rings (a ring equals its opposite) but is open already for finite-dimensional algebras over any field. The condition is subtle because a minimal injective resolution on one side carries no visible information about the other side.

Christensen and Holm (2010) approached symmetry through Auslander's condition (AC): a left-noetherian ring AA satisfies (AC) if for every finitely generated left module MM there is a bound bMb_M such that, for every finitely generated NN, vanishing of ExtAi(M,N)\mathrm{Ext}^i_A(M,N) for all i≫0i \gg 0 implies its vanishing for all i>bMi > b_M. Auslander had conjectured that every Artin algebra satisfies (AC); Jorgensen and Şega (2004) disproved this, which made the class of AC rings a meaningful object of study. Christensen and Holm proved (Theorem B) that a two-sided noetherian ring AA such that both AA and A∘A^\circ satisfy (AC), and which is either an Artin algebra or admits a dualizing complex — a complex of bimodules, in the sense of Christensen–Frankild–Holm after Yekutieli–Zhang, with finite injective dimensions on both sides, a finite resolution by bimodules of finite projective dimension, and isomorphic homothety maps — has idAA<∞\mathrm{id}_A A < \infty if and only if idA∘A<∞\mathrm{id}_{A^\circ} A < \infty. Their proof needs (AC) on both sides, and they record that they do not know whether (AC) is itself left–right symmetric, nor whether every Artin algebra admits a dualizing complex (every finite-dimensional algebra over a field does, via D=Homk(Λ,k)D = \mathrm{Hom}_k(\Lambda, k)). Partial symmetry results on adjacent invariants are known: Iyama (2003) proved symmetry statements for nn-Gorenstein rings in his sense, Christensen–Estrada–Thompson (2021) proved the Gorenstein weak global dimension is symmetric, and Cummings (2024) showed that left–right symmetry of finiteness of the finitistic dimension (the supremum of projective dimensions of modules of finite projective dimension) for all finite-dimensional algebras is equivalent to the Finitistic Dimension Conjecture — itself open. None of these settles the self-injective-dimension symmetry asked here.

Problem Statement

Let Λ\Lambda be an Artin algebra; the leading examples are finite-dimensional algebras over a field, and for that class the question below is already open. Is it true that

idΛΛ<∞⟺idΛ∘Λ<∞,\mathrm{id}_{\Lambda} \Lambda < \infty \quad\Longleftrightarrow\quad \mathrm{id}_{\Lambda^{\circ}} \Lambda < \infty,

i.e., is every left-Gorenstein Artin algebra right-Gorenstein (and conversely)? The same question, due to Auslander and Reiten (1991) and known as the Gorenstein Symmetry Question or Conjecture, is open for the class of two-sided noetherian rings, where the known partial results (Christensen–Holm's Theorem B, Wei's Proposition 3.4) require Auslander's vanishing condition (AC) on both sides plus an Artin or dualizing-complex hypothesis. For the full class of Artin algebras, with no extra hypothesis: does one-sided finiteness of the self-injective dimension always imply finiteness on the other side, or is there an Artin algebra whose self-injective dimension is finite on exactly one side?

References

  1. L. W. Christensen and H. Holm, 'Algebras that satisfy Auslander's condition on vanishing of cohomology', Mathematische Zeitschrift 265 (2010) 21–40. DOI: 10.1007/s00209-009-0500-4, https://arxiv.org/abs/0801.0401
  2. M. Auslander and I. Reiten, 'Applications of contravariantly finite subcategories', Advances in Mathematics 86 (1991) 111–152. DOI: 10.1016/0001-8708(91)90037-8, https://doi.org/10.1016/0001-8708(91)90037-8
  3. 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, https://doi.org/10.1090/S0002-9939-1975-0389977-6
  4. D. A. Jorgensen and L. M. Şega, 'Nonvanishing cohomology and classes of Gorenstein rings', Advances in Mathematics 188 (2004) 470–490. DOI: 10.1016/j.aim.2003.11.003, https://doi.org/10.1016/j.aim.2003.11.003
  5. C. Huneke and D. A. Jorgensen, 'Symmetry in the vanishing of Ext over Gorenstein rings', Mathematica Scandinavica 93 (2003) 161–184. DOI: 10.7146/math.scand.a-14418, https://doi.org/10.7146/math.scand.a-14418
  6. A. Yekutieli and J. J. Zhang, 'Rings with Auslander dualizing complexes', Journal of Algebra 213 (1999) 1–51. DOI: 10.1006/jabr.1998.7657, https://doi.org/10.1006/jabr.1998.7657
  7. L. W. Christensen, A. Frankild, and H. Holm, 'On Gorenstein projective, injective and flat dimensions — A functorial description with applications', Journal of Algebra 302 (2006) 231–279. DOI: 10.1016/j.jalgebra.2005.12.007, https://doi.org/10.1016/j.jalgebra.2005.12.007
  8. Jiaqun Wei, 'Auslander bounds and homological conjectures', Revista Matemática Iberoamericana 27 (2011) 871–884. DOI: 10.4171/RMI/655, https://doi.org/10.4171/RMI/655
  9. Osamu Iyama, 'Symmetry and duality on n-Gorenstein rings', Journal of Algebra 269 (2003) 528–535. DOI: 10.1016/s0021-8693(03)00419-8, https://doi.org/10.1016/s0021-8693(03)00419-8
  10. L. W. Christensen, S. Estrada and R. Thompson, 'Gorenstein weak global dimension is symmetric', Mathematische Nachrichten 294 (2021) 2121–228. DOI: 10.1002/mana.202100141, https://doi.org/10.1002/mana.202100141
  11. Charley Cummings, 'Left–right symmetry of finite finitistic dimension', Bulletin of the London Mathematical Society 56 (2024) 624–633. DOI: 10.1112/blms.12954, https://doi.org/10.1112/blms.12954
  12. Yongyun Qin and Yang Han, 'Reducing homological conjectures by n-recollements', Algebras and Representation Theory 19 (2016) 377–395. DOI: 10.1007/s10468-015-9578-z, https://doi.org/10.1007/s10468-015-9578-z
  13. Yingying Zhang, 'Excellent extensions and homological conjectures', Algebra Colloquium 25 (2018) 619–626. DOI: 10.1142/S1005386718000433, https://doi.org/10.1142/S1005386718000433
  14. Yongyun Qin and Dawei Shen, 'Eventually homological isomorphisms and Gorenstein projective modules', Science China Mathematics 67 (2024) 1719–1734. DOI: 10.1007/s11425-022-2146-5, https://doi.org/10.1007/s11425-022-2146-5
  15. Liran Shaul (with an appendix by Tsutomu Nakamura and Peder Thompson), 'Acyclic complexes of injectives and finitistic dimensions', arXiv:2303.08756 (2023), https://arxiv.org/abs/2303.08756
  16. Andrew J. Soto Levins, 'The Auslander bound for complexes', Archiv der Mathematik 126 (2026) 29–39. DOI: 10.1007/s00013-025-02179-x, https://doi.org/10.1007/s00013-025-02179-x
  17. Víctor Becerril, 'Strongly FP-injective dimensions and Gorenstein projective precovers', arXiv:2602.00932 (2026), https://arxiv.org/abs/2602.00932
  18. Weiqing Li, 'On a proof of the Gorenstein Symmetry Conjecture', arXiv:2607.28011 (2026), withdrawn by the author ('Since there is an error in the proof, I hereby withdraw the article'), https://arxiv.org/abs/2607.28011
  19. Weiqing Li, 'Virtually Gorenstein Artin algebras are weakly Gorenstein', arXiv:2608.10049 (2026), https://arxiv.org/abs/2608.10049
  20. Zhaoyong Huang, 'On Auslander-Type Conditions of Modules', Publications of the Research Institute for Mathematical Sciences 59 (2023) 57–88. DOI: 10.4171/prims/59-1-2, https://doi.org/10.4171/prims/59-1-2