Is the tensor product $A\otimes_k B$ of left weakly Gorenstein algebras left weakly Gorenstein without a finite global dimension hypothesis?

For finite-dimensional algebras over a field $k$, Gao, Zhang and Zhu (arXiv:2512.16072, 2025) proved that if $\operatorname{gl.dim} B<\infty$ then the tensor product algebra $A\otimes_k B$ is left weakly Gorenstein if and only if $A$ is, and they note that the forward implication ($A\otimes_k B$ left weakly Gorenstein $\Rightarrow$ $A$ and $B$ left weakly Gorenstein) needs no hypothesis at all. They explicitly state that they do not know whether the converse holds in general. This problem asks whether $A\otimes_k B$ is left weakly Gorenstein whenever $A$ and $B$ are left weakly Gorenstein, with no restriction on $\operatorname{gl.dim} B$; the unresolved case is $\operatorname{gl.dim} B=\infty$, where the question is whether there exist a field $k$ and finite-dimensional $k$-algebras $A$ and $B$, each left weakly Gorenstein, such that $A\otimes_k B$ admits a semi-Gorenstein-projective left module that is not Gorenstein-projective.

Background

Throughout, algebras are finite-dimensional (and hence Artinian) algebras over a field $k$, and modules are finitely generated. A left module $M$ over an algebra $A$ is called torsionless if it embeds in a projective module, and reflexive if the canonical map $M\to M^{**}$ into its double $A$-dual $\operatorname{Hom}_A(\operatorname{Hom}_A(M,A),A)$ is an isomorphism. $M$ is semi-Gorenstein-projective if $\operatorname{Ext}^i_A(M,A)=0$ for all $i\ge 1$. $M$ is Gorenstein-projective if it is the kernel of $d^0$ in a complete projective resolution, i.e. an acyclic complex $\cdots\to P^1\to P^0\to P^{-1}\to\cdots$ of finitely generated projective $A$-modules that stays acyclic after applying $\operatorname{Hom}_A(-,P)$ for every projective $P$. Every Gorenstein-projective module is semi-Gorenstein-projective, and the containment can be strict: the independence of the classical defining conditions for Gorenstein-projectivity was a longstanding problem, settled by work of Jorgensen and Šega and of Ringel and Zhang around 2006–2020. Following Ringel and Zhang (Algebra & Number Theory 14, 2020), an algebra $A$ is left weakly Gorenstein if every semi-Gorenstein-projective left $A$-module is Gorenstein-projective (equivalently, every semi-Gorenstein-projective left module is reflexive); the right notion is defined symmetrically. Iwanaga–Gorenstein algebras (those of finite injective dimension on both sides) are left and right weakly Gorenstein, but the converse fails, many algebras fail to be left weakly Gorenstein at all, and it is an older open question, posed by Ringel–Zhang (Subsection 9.3) and Marczinzik, whether left weakly Gorenstein implies right weakly Gorenstein. For tensor products, the central technical difficulty is that a general left $(A\otimes_k B)$-module is not of the form $X\otimes_k Y$ with $X$ an $A$-module and $Y$ a $B$-module, while the Cartan–Eilenberg isomorphism $\operatorname{Ext}^n_{A\otimes_k B}(X\otimes_k Y,N\otimes_k Y')\cong\bigoplus_{i+j=n}\operatorname{Ext}^i_A(X,N)\otimes_k\operatorname{Ext}^j_B(Y,Y')$ controls only modules of that tensor form. The equivalence for $B=kQ$ with $Q$ an acyclic quiver (a path algebra, i.e. the span of all $k$-linear combinations of directed paths subject to no relations) was proved by P. Zhang (2023), and for $B=kQ/I$ with $I$ an admissible ideal generated by monomial relations by Luo and Zhu (2024), both via monic-representation techniques. Gao, Zhang and Zhu (2025) unified and extended these results using the monomorphism category $\operatorname{Mon}(B,{}^{\perp}A)$, the full subcategory of $(A\otimes_k B)$-modules whose structural maps are monomorphisms: if $\operatorname{gl.dim} B<\infty$ (the global dimension, i.e. the supremum of projective dimensions of all modules), then $A\otimes_k B$ is left weakly Gorenstein if and only if $A$ is. Their Remark 3.7(1) records that the implication '$A\otimes_k B$ left weakly Gorenstein $\Rightarrow$ $A$ and $B$ left weakly Gorenstein' holds without any global-dimension hypothesis, and that the authors do not know whether the converse holds in general — the question posed here.

Problem Statement

Let $k$ be a field and let $A$ and $B$ be finite-dimensional $k$-algebras, each left weakly Gorenstein (every semi-Gorenstein-projective left module, i.e. every finitely generated left module $M$ with $\operatorname{Ext}^i_A(M,A)=0$ for all $i\ge 1$, is Gorenstein-projective). With no hypothesis on the global dimension of $B$ — in particular with $\operatorname{gl.dim} B=\infty$ — is the tensor product algebra $A\otimes_k B$ necessarily left weakly Gorenstein? Equivalently, in the language of Theorem 3.6 and Remark 3.7(1) of Gao–Zhang–Zhu: the implication '$A\otimes_k B$ left weakly Gorenstein $\Rightarrow$ $A$ and $B$ left weakly Gorenstein' is unconditional; decide whether the converse holds in general.

References

1. Nan Gao, Pu Zhang, Shijie Zhu, 'Weakly Gorensteinness of tensor algebras and Morita algebras', arXiv preprint (2025), arXiv:2512.16072, DOI: 10.48550/arXiv.2512.16072, https://arxiv.org/abs/2512.16072
2. C. M. Ringel, Pu Zhang, 'Gorenstein-projective and semi-Gorenstein-projective modules', Algebra & Number Theory 14(1) (2020) 1–36, DOI: 10.2140/ant.2020.14.1, https://doi.org/10.2140/ant.2020.14.1
3. René Marczinzik, 'Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra', arXiv preprint (2016, v2 2021), arXiv:1608.04212, https://arxiv.org/abs/1608.04212
4. Pu Zhang, 'Monic modules and semi-Gorenstein-projective modules', Journal of Pure and Applied Algebra 227 (2023) 107181, DOI: 10.1016/j.jpaa.2022.107181, https://doi.org/10.1016/j.jpaa.2022.107181
5. X. H. Luo, S. J. Zhu, 'Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules', Journal of Pure and Applied Algebra 228 (2024) 107586, DOI: 10.1016/j.jpaa.2023.107586, https://doi.org/10.1016/j.jpaa.2023.107586
6. Zhaoyong Huang, 'Auslander-type conditions and weakly Gorenstein algebras', Bulletin of the London Mathematical Society 56(11) (2024) 3382–3399, DOI: 10.1112/blms.13138, https://doi.org/10.1112/blms.13138
7. Souvik Dey, Jian Liu, Xue-Song Lu, 'Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules', arXiv preprint (2026), arXiv:2603.01424, https://arxiv.org/abs/2603.01424