The Gorenstein bimodule conjecture: is a finite-dimensional algebra selfinjective exactly when its regular bimodule is Gorenstein projective?

For a finite-dimensional algebra $A$ over a field, the Gorenstein bimodule conjecture, posed by Marczinzik (2020), predicts that $A$ is selfinjective if and only if the regular bimodule $_A A_A$, regarded as a module over the enveloping algebra $A^e = A \otimes_K A^{\mathrm{op}}$, is Gorenstein projective. The source paper proves that for each fixed algebra this conjecture is equivalent to the disjunction of the Nakayama conjecture (1958) and the first Tachikawa conjecture (1973), two of the classical open homological conjectures. It is known to hold for algebras of finite finitistic dimension, for Iwanaga-Gorenstein algebras (Shen 2019), for left weakly Gorenstein algebras (Marczinzik 2019), for right weakly Gorenstein algebras and for virtually Gorenstein algebras (Chen–Xi 2026), but it remains open for arbitrary finite-dimensional algebras as of August 2026. A counterexample to the conjecture is an explicit finite-dimensional algebra that is not selfinjective while its regular bimodule is Gorenstein projective.

Background

In the representation theory of associative algebras one studies a finite-dimensional algebra $A$ over a field $K$ through the homological properties of its module category. The algebra $A$ is called selfinjective if projective and injective left $A$-modules coincide, or equivalently if the left regular module $_A A$ is injective. The dominant dimension of $A$, written $\mathrm{domdim}(A)$, is the largest $n$ (or $\infty$) such that in a minimal injective coresolution $0 \to A \to I_0 \to I_1 \to \cdots$ of the regular module the terms $I_0, \dots, I_{n-1}$ are projective; Mueller (1968) showed that this invariant may equivalently be computed for $A$ as a bimodule over its enveloping algebra $A^e = A \otimes_K A^{\mathrm{op}}$, whose module category is identified with the category of $A$-bimodules. The Nakayama conjecture (Nakayama 1958) asserts that $A$ is selfinjective if and only if $\mathrm{domdim}(A) = \infty$; writing $D(A) = \mathrm{Hom}_K(A,K)$ for the standard duality, the first Tachikawa conjecture (Tachikawa 1973) asserts that $A$ is selfinjective if and only if $\mathrm{Ext}_A^i(D(A),A) = 0$ for all $i \geq 1$. Both conjectures remain open in general, and the finitistic dimension conjecture (every finite-dimensional algebra bounds the projective dimensions of its finite-dimensional modules) implies the Nakayama conjecture.

Gorenstein homological algebra relativizes these questions. Following Auslander and Bridger (1969), a finitely generated module $M$ over an algebra $R$ is Gorenstein projective if $\mathrm{Ext}_R^i(M,R) = 0 = \mathrm{Ext}_R^i(\mathrm{Tr}(M),R)$ for all $i \geq 1$, where $\mathrm{Tr}(M)$ is the transpose of $M$, the cokernel of the dual of a minimal projective presentation; equivalently, $M$ occurs as a syzygy of an acyclic complex of finitely generated projective $R$-modules that stays acyclic after applying $\mathrm{Hom}_R(-,R)$. An algebra $R$ is Iwanaga-Gorenstein when the left and right injective dimensions of the regular module are equal and finite; over such rings Gorenstein projective modules behave like projectives, and over a selfinjective ring every module is Gorenstein projective. An algebra $A$ is left weakly Gorenstein (terminology of Ringel and Zhang 2020) when every module $M$ with $\mathrm{Ext}_A^i(M,A)=0$ for all $i>0$ is Gorenstein projective, and virtually Gorenstein (Beligiannis–Reiten) when the subcategory of Gorenstein projective modules and its dual determine each other orthogonally — a condition satisfied by all Iwanaga-Gorenstein algebras and all algebras of finite representation type (algebras having, up to isomorphism, only finitely many indecomposable finite-dimensional modules), but not by all algebras.

Marczinzik (2020) proved that $\mathrm{domdim}(A) \geq n \geq 2$ if and only if the regular bimodule $A$ is $n$-torsionfree over $A^e$, i.e. $\mathrm{Ext}_{A^e}^i(\mathrm{Tr}(A),A^e)=0$ for $1 \leq i \leq n$, and used this to compute the Hochschild homology and cohomology of algebras of dominant dimension at least two — the standard (co)homology groups $HH_*(A)$ and $HH^*(A)$ of an algebra, defined from the bar resolution of $A$ as an $A$-bimodule. Motivated by the analogy with the classical equality $\mathrm{pd}_{A^e}(A) = \mathrm{gldim}(A)$ for the bimodule $A$ (over algebraically closed fields), where $\mathrm{gldim}(A)$ denotes the global dimension of $A$ (the supremum of the projective dimensions of all $A$-modules), the paper poses the Gorenstein bimodule conjecture and proves that, for each fixed algebra $A$, the conjecture holds for $A$ if and only if the Nakayama conjecture or the first Tachikawa conjecture holds for $A$. Consequently the conjecture holds for all algebras of finite finitistic dimension (in particular for Iwanaga-Gorenstein, monomial and local algebras), it was verified for Iwanaga-Gorenstein algebras by Shen (2019) via Gorenstein projective modules over tensor products, for left weakly Gorenstein algebras by Marczinzik (2019), for right weakly Gorenstein algebras by Cruz and Marczinzik, and for virtually Gorenstein algebras by Chen and Xi (2026), who proved that a virtually Gorenstein algebra with infinite dominant dimension and $\mathrm{Ext}_A^i(D(A),A)=0$ for all $i \geq 1$ is selfinjective. No proof or counterexample is known for an arbitrary finite-dimensional algebra.

Problem Statement

Let $A$ be a finite-dimensional algebra over a field $K$, and let $A^e = A \otimes_K A^{\mathrm{op}}$ be its enveloping algebra, so that $A$-bimodules are exactly left $A^e$-modules. The Gorenstein bimodule conjecture asks: is it true that $A$ is selfinjective if and only if the regular bimodule $_A A_A$, regarded as an $A^e$-module, is Gorenstein projective, that is, $\mathrm{Ext}_{A^e}^i(A, A^e) = 0 \quad \text{and} \quad \mathrm{Ext}_{A^e}^i(\mathrm{Tr}(A), A^e) = 0 \quad \text{for all } i \geq 1,$ where $\mathrm{Tr}(A)$ denotes the transpose of $A$ over $A^e$? The implication from selfinjectivity to Gorenstein projectivity is standard, since $A$ selfinjective makes $A^e$ selfinjective and every module over a selfinjective artin algebra is Gorenstein projective; the open substantive direction is: if $A$ is Gorenstein projective as a bimodule, must $A$ be selfinjective? The question is whether this holds for every finite-dimensional algebra over every field; a counterexample is an explicit finite-dimensional algebra that is not selfinjective while its regular bimodule is Gorenstein projective. By the results in the source, a positive answer for all algebras is equivalent to establishing, for every finite-dimensional algebra, the Nakayama conjecture or the first Tachikawa conjecture, and a counterexample would simultaneously refute both conjectures for that algebra.

References

1. Rene Marczinzik, A bimodule approach to dominant dimension, arXiv preprint (2020), arXiv:2005.08656, DOI 10.48550/arXiv.2005.08656, https://arxiv.org/abs/2005.08656
2. Tiago Cruz, René Marczinzik, A new formula for the classical dominant dimension using bimodules, arXiv preprint (2025, v2 August 2026), arXiv:2508.18398, DOI 10.48550/arXiv.2508.18398, https://arxiv.org/abs/2508.18398
3. Hongxing Chen, Changchang Xi, Virtually Gorenstein algebras of infinite dominant dimension, Journal of Pure and Applied Algebra 230(4), Paper No. 108224 (2026), DOI 10.1016/j.jpaa.2026.108224, arXiv:2509.04990, https://arxiv.org/abs/2509.04990
4. Tiago Cruz, René Marczinzik, Higher torsion-free Auslander-Reiten sequences and the dominant dimension of algebras, Journal of Algebra 665 (2025) 282-297, DOI 10.1016/j.jalgebra.2024.11.004, arXiv:2404.02274, https://arxiv.org/abs/2404.02274
5. Dawei Shen, A description of Gorenstein projective modules over the tensor products of algebras, Communications in Algebra 47(7) (2019) 2753-2765, DOI 10.1080/00927872.2018.1539172, https://doi.org/10.1080/00927872.2018.1539172
6. René Marczinzik, On a new formula for the Gorenstein dimension, Journal of Algebra 526 (2019) 104-111, DOI 10.1016/j.jalgebra.2019.02.013, https://doi.org/10.1016/j.jalgebra.2019.02.013
7. Claus Michael 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
8. Tadasi Nakayama, On algebras with complete homology, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 22 (1958) 300-307, DOI 10.1007/BF02941960, https://doi.org/10.1007/BF02941960
9. Bruno J. Mueller, The classification of algebras by dominant dimension, Canadian Journal of Mathematics 20 (1968) 398-409, DOI 10.4153/cjm-1968-037-9, https://doi.org/10.4153/cjm-1968-037-9
10. Hiroyuki Tachikawa, Quasi-Frobenius Rings and Generalizations: QF-3 and QF-1 Rings, Lecture Notes in Mathematics 351, Springer, Berlin, 1973, ISBN 978-3-540-06501-2, DOI 10.1007/BFb0059997, https://doi.org/10.1007/BFb0059997
11. Maurice Auslander, Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society 94, American Mathematical Society, Providence, 1969, DOI 10.1090/memo/0094, https://doi.org/10.1090/memo/0094
12. Hongxing Chen, Changchang Xi, Homological theory of self-orthogonal modules, Transactions of the American Mathematical Society 378 (2025) 7287-7335, DOI 10.1090/tran/9426, https://doi.org/10.1090/tran/9426