For a semiperfect noetherian ring AA with Jacobson radical JJ, Chen, Iyengar, and Marczinzik proved inj dim⁡AJ=gl dim⁡A\operatorname{inj\,dim}_A J=\operatorname{gl\,dim}A and raised three open questions: (2.9) whether the Gorenstein-injective dimension of JJ equals the injective dimension of AA, Ginj dim⁡AJ=inj dim⁡AA\operatorname{Ginj\,dim}_A J=\operatorname{inj\,dim}_A A; (2.10) whether Ginj dim⁡AJ=Ginj dim⁡AopJ\operatorname{Ginj\,dim}_A J=\operatorname{Ginj\,dim}_{A^{\mathrm{op}}}J is left-right symmetric; and (2.11) whether every nonzero syzygy Ωn(A0)\Omega^n(A_0) of the top A0=A/JA_0=A/J has injective dimension equal to the global dimension of AA. These are known in the commutative semilocal case, when the global dimension is finite, and for several special classes (including commutative local rings and minimal Auslander-Gorenstein algebras), but remain open for general semiperfect noetherian rings; in the artinian case they follow, via the same paper's Proposition 4.3, from the left-right symmetry of the self-injective dimension, the subject of the still-open Gorenstein symmetry conjecture.

Background

Throughout, AA is an associative ring with unit and JJ its Jacobson radical, the intersection of all left (equivalently right) maximal ideals. The ring is assumed noetherian (left and right) and semiperfect, meaning every finitely generated module has a projective cover, a minimal surjection from a projective module; this class contains local rings, Artin algebras (finite-dimensional algebras over a commutative artinian center), and algebras finite over commutative noetherian complete semilocal rings. The top of AA is the semisimple quotient A0=A/JA_0=A/J; since projective covers exist, the nn-th syzygy Ωn(A0)\Omega^n(A_0) — the kernel iterated nn times in a minimal projective resolution of A0A_0 — is well-defined up to isomorphism. Note J=Ω1(A0)J=\Omega^1(A_0).

For a module MM, its injective dimension inj dim⁡AM\operatorname{inj\,dim}_A M is the minimal length of an injective resolution, and the global dimension gl dim⁡A\operatorname{gl\,dim}A is the supremum of projective (equivalently injective) dimensions of all modules. Projective invariants of JJ are classical and elementary: proj dim⁡AJ=gl dim⁡A−1\operatorname{proj\,dim}_A J=\operatorname{gl\,dim}A-1 when AA is not semisimple. Injective invariants are subtler, and the ground was laid only recently: Marczinzik (2020) proved inj dim⁡AJ=gl dim⁡A\operatorname{inj\,dim}_A J=\operatorname{gl\,dim}A for several classes of Artin algebras and conjectured it in general, and Chen–Iyengar–Marczinzik (arXiv 2022; Proc. Amer. Math. Soc. Ser. B 2024) confirmed it for every semiperfect noetherian ring, deducing it from the characterization Ext⁡Ai(M,J)=0\operatorname{Ext}^i_A(M,J)=0 for i≫0i\gg 0 iff proj dim⁡AM<∞\operatorname{proj\,dim}_A M<\infty.

Gorenstein homological algebra refines these invariants when injective dimensions are infinite. A module is Gorenstein injective if it arises as a cycle in a totally acyclic complex of injective modules (a complex of injectives whose Hom against any injective remains exact); over Iwanaga-Gorenstein rings — noetherian rings of finite injective dimension on both sides — these play the role injectives play over regular rings. The Gorenstein injective dimension Ginj dim⁡AM\operatorname{Ginj\,dim}_A M is the minimal length of a resolution of MM by Gorenstein injective modules; it vanishes exactly on Gorenstein injective modules and may be finite even when inj dim⁡AM=∞\operatorname{inj\,dim}_A M=\infty. Since gl dim⁡A=gl dim⁡Aop\operatorname{gl\,dim}A=\operatorname{gl\,dim}A^{\mathrm{op}} for the opposite ring AopA^{\mathrm{op}} (the ring with multiplication reversed), the theorem above makes the injective dimension of JJ left-right symmetric, and it is natural to ask which other homological invariants of JJ share this symmetry.

Chen–Iyengar–Marczinzik established: equality of Gorenstein injective and ordinary injective dimension of JJ, and its left-right symmetry, for commutative semilocal noetherian rings (Proposition 3.6) and whenever gl dim⁡A<∞\operatorname{gl\,dim}A<\infty (Proposition 2.8), the latter giving Ginj dim⁡AJ=inj dim⁡AA=inj dim⁡AopA=Ginj dim⁡AopJ\operatorname{Ginj\,dim}_A J=\operatorname{inj\,dim}_A A=\operatorname{inj\,dim}_{A^{\mathrm{op}}}A=\operatorname{Ginj\,dim}_{A^{\mathrm{op}}}J; the Gorenstein projective analogue of the symmetry for JJ over any semilocal Noether algebra (Theorem 3.2), with finiteness equivalent to AA being Iwanaga-Gorenstein; and, for Question 2.11, the cases n=0n=0 (a 1956 theorem of Eilenberg: inj dim⁡AA0=gl dim⁡A\operatorname{inj\,dim}_A A_0=\operatorname{gl\,dim}A), n=1n=1 (their main theorem), commutative local rings (Ghosh–Gupta–Puthenpurakal 2018), and minimal Auslander-Gorenstein algebras — Gorenstein algebras with inj dim⁡A≤dom dim⁡A\operatorname{inj\,dim}A\le\operatorname{dom\,dim}A, where dominant dimension measures how far the regular module's minimal injective coresolution starts with projective-injective terms (Theorem 4.9). Computer experiments with the GAP package QPA (Quivers and Path Algebras, software for path-algebra computations) verified the syzygy equality on several thousand quiver algebras. The remaining cases are genuinely tied to major open conjectures: by Proposition 4.3 of the same paper, for an Artin algebra, (2.9) together with its opposite-ring analogue Ginj dim⁡AopJ=inj dim⁡AopA\operatorname{Ginj\,dim}_{A^{\mathrm{op}}}J=\operatorname{inj\,dim}_{A^{\mathrm{op}}}A holds if and only if inj dim⁡AA=inj dim⁡AopA\operatorname{inj\,dim}_A A=\operatorname{inj\,dim}_{A^{\mathrm{op}}}A, the left-right symmetry of the self-injective dimension, and when these hold (2.10) follows as well; the assertion that one-sided finiteness of the self-injective dimension implies the other is the Gorenstein symmetry conjecture, which remains open (a claimed proof posted in July 2026 was withdrawn by its author the following day). For Question 2.11 with n≥2n\ge 2, the work of Gélinas (2022) on the delooping level — a homological invariant bounding the big finitistic injective dimension, the supremum of injective dimensions of all (not necessarily finitely generated) modules of finite injective dimension — shows the question is intertwined with open finiteness problems for Artin algebras.

Problem Statement

Let AA be a semiperfect noetherian ring (not semisimple) with Jacobson radical JJ, top A0=A/JA_0=A/J, opposite ring AopA^{\mathrm{op}}, and let Ωn(A0)\Omega^n(A_0) denote the nn-th syzygy of A0A_0 in a minimal projective resolution (Ω0(A0)=A0\Omega^0(A_0)=A_0, Ω1(A0)=J\Omega^1(A_0)=J). Let Ginj dim⁡\operatorname{Ginj\,dim} denote Gorenstein injective dimension. The joint objective, raised as Questions (2.9), (2.10), and (2.11) of Chen–Iyengar–Marczinzik, is to determine the Gorenstein-injective dimension of the Jacobson radical and the injective dimensions of the syzygies of the top over such rings, by establishing or refuting, for every semiperfect noetherian ring AA:

Ginj dim⁡AJ=inj dim⁡AA    (2.9),\operatorname{Ginj\,dim}_{A}J=\operatorname{inj\,dim}_{A}A\;\;\text{(2.9)},

Ginj dim⁡AJ=Ginj dim⁡AopJ    (2.10),\operatorname{Ginj\,dim}_{A}J=\operatorname{Ginj\,dim}_{A^{\mathrm{op}}}J\;\;\text{(2.10)},

and, for every n≥0n\ge 0 with Ωn(A0)≠0\Omega^{n}(A_0)\neq 0,

inj dim⁡AΩn(A0)=gl dim⁡A    (Question 2.11).\operatorname{inj\,dim}_{A}\Omega^{n}(A_0)=\operatorname{gl\,dim}A\;\;\text{(Question 2.11)}.

All quantities may take the value ∞\infty, and the equalities are to be read in the extended integers. The three questions are scientifically linked: each generalizes the same source theorem inj dim⁡AJ=gl dim⁡A\operatorname{inj\,dim}_A J=\operatorname{gl\,dim}A, and in the artinian case the left-right symmetry of the self-injective dimension implies both (2.9) and (2.10) (Proposition 4.3 of the source).

References

  1. Xiao-Wu Chen, Srikanth B. Iyengar, René Marczinzik, "Homological dimensions of the Jacobson radical", Proc. Amer. Math. Soc. Ser. B 11 (2024), 211–223. DOI: 10.1090/bproc/187, https://doi.org/10.1090/bproc/187; preprint arXiv:2210.08691, https://arxiv.org/abs/2210.08691
  2. René Marczinzik, "On the injective dimension of the Jacobson radical", Proc. Amer. Math. Soc. 148 (2020), 1481–1485. DOI: 10.1090/proc/14911, https://doi.org/10.1090/proc/14911; preprint arXiv:1801.05674, https://arxiv.org/abs/1801.05674
  3. Samuel Eilenberg, "Homological Dimension and Syzygies", Ann. of Math. (2) 64 (1956), 328–336. DOI: 10.2307/1969977, https://doi.org/10.2307/1969977
  4. Dipankar Ghosh, Anjan Gupta, Tony J. Puthenpurakal, "Characterizations of regular local rings via syzygy modules of the residue field", J. Commut. Algebra 10 (2018), no. 3, 327–337. DOI: 10.1216/jca-2018-10-3-327, https://doi.org/10.1216/jca-2018-10-3-327
  5. Vincent Gélinas, "The depth, the delooping level and the finitistic dimension", Adv. Math. 394 (2022), 108052. DOI: 10.1016/j.aim.2021.108052, https://doi.org/10.1016/j.aim.2021.108052
  6. Viktória Klász, René Marczinzik, "A survey on Auslander-Gorenstein algebras", preprint (2025). arXiv:2508.19079, https://arxiv.org/abs/2508.19079
  7. Tiago Cruz, René Marczinzik, "An Auslander-Buchsbaum formula for higher Auslander algebras and applications", preprint (2025). arXiv:2502.08422, https://arxiv.org/abs/2502.08422
  8. Weiqing Li, "On a proof of the Gorenstein Symmetry Conjecture", preprint (2026), withdrawn by the author on 2026-07-31 with the note "Since there is an error in the proof, I hereby withdraw the article". arXiv:2607.28011, https://arxiv.org/abs/2607.28011