Weakly-Gorenstein Symmetry Conjecture for Artin algebras: does left weakly Gorenstein imply right weakly Gorenstein?

The Weakly-Gorenstein Symmetry Conjecture (WGSC) predicts that an Artin algebra is left weakly Gorenstein if and only if it is right weakly Gorenstein, where a ring is left weakly Gorenstein when every finitely generated left module that is Ext-orthogonal to the ring (a so-called semi-Gorenstein-projective module) is Gorenstein projective. The question was posed by Marczinzik within the framework of Ringel and Zhang, who showed that WGSC implies the classical Gorenstein Symmetry Conjecture (GSC) on finiteness of self-injective dimensions. Huang (2024) proved WGSC for algebras satisfying two-sided Auslander-type conditions and observed that it remains open even for left quasi Auslander Artin algebras, where GSC is already known. As of August 2026 the conjecture is open in general: recent work adds new positive classes (virtually Gorenstein Artin algebras; tensor and Morita algebras via transfer theorems), and the only claimed 2026 proof of the weaker GSC was withdrawn by its author. The problem asks for a proof of WGSC for arbitrary Artin algebras, or an explicit counterexample.

Background

An Artin algebra is a ring $R$ that is finitely generated as a module over a commutative artinian subring of its center (finite-dimensional algebras over a field are the special case most used in representation theory); one studies the category $\mathrm{mod}\,R$ of finitely generated left modules and the corresponding category of right modules, which need not behave symmetrically. A finitely generated left $R$-module $M$ is called Gorenstein projective if it occurs as a syzygy in a totally acyclic complex of finitely generated projective modules, that is, an acyclic complex remaining exact after applying $\mathrm{Hom}_R(-,P)$ for every projective module $P$; this generalizes the notion of projective module and is the basic object of Gorenstein homological algebra. A weaker property is semi-Gorenstein-projectivity (also called self-orthogonality): $\mathrm{Ext}^i_R(M,R)=0$ for all $i\geq 1$. Every Gorenstein projective module is semi-Gorenstein-projective, and the inclusion can be strict. Following Ringel and Zhang, $R$ is left weakly Gorenstein if every semi-Gorenstein-projective module in $\mathrm{mod}\,R$ is Gorenstein projective, with the right-sided notion defined symmetrically; and $R$ is Gorenstein if its left and right self-injective dimensions are both finite. Two long-standing conjectures organize this area. The Gorenstein Symmetry Conjecture (GSC) asserts that for an Artin algebra the left self-injective dimension is finite if and only if the right one is. The Weakly-Gorenstein Symmetry Conjecture (WGSC), isolated as a question by Marczinzik and stated as a conjecture by Huang, asserts that an Artin algebra is left weakly Gorenstein if and only if it is right weakly Gorenstein. Ringel and Zhang proved that WGSC implies GSC, so WGSC is the stronger statement: a proof of WGSC would settle GSC, and any counterexample to GSC (an algebra finite on one side and infinite on the other) would in particular refute WGSC, while a counterexample to WGSC would refute GSC itself only if, in addition, exactly one of its two self-injective dimensions were finite; GSC is itself related to the Nakayama conjecture (every indecomposable injective module appears in the minimal injective resolution of the regular module) and to the Auslander–Reiten conjecture that an Artin algebra satisfying the Auslander condition is Gorenstein. The relevant hypotheses of the source paper are Auslander-type conditions on modules: a module $M$ satisfies $G_\infty(m)$ for $m\geq 0$ if the flat dimension of the $i$-th term of a minimal injective resolution of $M$ is at most $i+m$ for every $i\geq 0$ (a hierarchy introduced by Huang, building on the Auslander-type conditions introduced by Huang and Iyama as noncommutative analogues of commutative Gorenstein rings). The Auslander condition is $G_\infty(0)$, and $R$ is left quasi Auslander if the left regular module ${}_R R$ satisfies $G_\infty(1)$; each of these is a one-sided condition, so symmetry questions are meaningful. Huang (2024) proved that if both sides satisfy such conditions, ${}_R R\in G_\infty(m)$ and $R_R\in G_\infty(m')^{op}$, then WGSC and GSC both hold, and that over a left quasi Auslander Artin algebra GSC holds; however, he explicitly notes that over a left quasi Auslander Artin algebra it is not known whether WGSC holds. Known one-sided positive results (monomial algebras, that is algebras presented by a quiver whose relations are paths; endomorphism rings over representation-finite algebras, where representation-finite means having only finitely many indecomposable finitely generated modules up to isomorphism; and algebras with representation-finite subcategories of self-orthogonal syzygies; due to Ringel–Zhang and Marczinzik) supply algebras that are weakly Gorenstein on each side simultaneously, and Marczinzik constructed the first systematic families of non-weakly-Gorenstein algebras, which fail the property on both sides; none of these addresses asymmetry between the two sides.

Problem Statement

Prove or disprove the Weakly-Gorenstein Symmetry Conjecture: for an Artin algebra $R$ (a ring finitely generated as a module over a commutative artinian subring of its center), if every finitely generated left $R$-module $M$ with $\mathrm{Ext}^i_R(M,R)=0$ for all $i\geq 1$ is Gorenstein projective — that is, $M$ occurs as a syzygy of a totally acyclic complex of finitely generated projective left $R$-modules, a complex that is acyclic and remains acyclic after applying $\mathrm{Hom}_R(-,P)$ for every projective left $R$-module $P$ (in which case $R$ is left weakly Gorenstein) — must the analogous statement hold for finitely generated right $R$-modules ($R$ right weakly Gorenstein), and conversely? A resolution must settle this for arbitrary Artin algebras, with no auxiliary hypothesis such as an Auslander-type condition; in particular, the already-open special case of left quasi Auslander Artin algebras (where the left regular module ${}_R R$ has a minimal injective resolution whose $i$-th term has flat dimension at most $i+1$ for every $i\geq 0$, the Gorenstein Symmetry Conjecture is known to hold, and Huang explicitly leaves WGSC open) would constitute progress.

References

1. 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
2. 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://arxiv.org/abs/1808.01809
3. Rene Marczinzik, "Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra", arXiv:1608.04212 (2016, revised 2021), https://arxiv.org/abs/1608.04212
4. Nan Gao, Pu Zhang, and Shijie Zhu, "Weakly Gorensteinness of tensor algebras and Morita algebras", arXiv:2512.16072 (2025), https://arxiv.org/abs/2512.16072
5. Souvik Dey, Jian Liu, and Xue-Song Lu, "Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules", arXiv:2603.01424 (2026), https://arxiv.org/abs/2603.01424
6. Weiqing Li, "Virtually Gorenstein Artin algebras are weakly Gorenstein", arXiv:2608.10049 (2026), https://arxiv.org/abs/2608.10049
7. Weiqing Li, "On a proof of the Gorenstein Symmetry Conjecture", arXiv:2607.28011 (2026, withdrawn by the author), https://arxiv.org/abs/2607.28011
8. Kaito Kimura, Yuki Mifune, Yuya Otake, and Ryo Takahashi, "On strongly G-regular rings", arXiv:2608.08228 (2026), https://arxiv.org/abs/2608.08228
9. Víctor Becerril, "Strongly FP-injective dimensions and Gorenstein projective precovers", arXiv:2602.00932 (2026), https://arxiv.org/abs/2602.00932