The Generalized Nakayama Conjecture for artin algebras: does every indecomposable injective occur in the minimal injective resolution of the algebra?

The Generalized Nakayama Conjecture (GNC), proposed by Auslander and Reiten in 1975, asserts that for every artin algebra each indecomposable injective module appears as a direct summand of some term of a minimal injective resolution of the algebra. Half a century of work has verified the conjecture only for special classes of algebras (e.g., algebras with truncated radical and representation-finite quotient, centralizer matrix algebras, virtually Gorenstein algebras) and has established transfer results (e.g., from a coefficient ring to its path algebras), while the conjecture for arbitrary artin algebras remains open.

Background

An artin algebra is an algebra over a commutative artinian ring that is finitely generated as a module over its center; the leading example is a finite-dimensional algebra over a field. The representation theory of artin algebras studies the category of finitely generated modules, and a central homological tool is the minimal injective resolution of the algebra: an injective resolution $0\to\Lambda\to I_0\to I_1\to I_2\to\cdots$ of $\Lambda$ as a left module over itself in which each map is an essential extension into an injective envelope (an injective envelope of a module $M$ is a minimal injective module containing $M$; a module is injective if Hom into it is an exact functor). Minimality makes the resolution unique up to isomorphism, so the multiset of indecomposable direct summands of the terms $I_i$ is an invariant of $\Lambda$. Since indecomposable injectives correspond bijectively to simple modules, the resolution 'sees' only some of the injectives in general, and it is a priori possible that an indecomposable injective is never a summand of any $I_i$.

The conjecture originates in a 1958 conjecture of Nakayama, which results of Müller showed equivalent to the statement that if every term of the minimal injective resolution of a finite-dimensional algebra is projective then the algebra is self-injective (an algebra is self-injective if it is injective as a module over itself). Auslander and Reiten proposed the generalized version in 1975: every indecomposable injective module over an artin algebra occurs as a direct summand of some term of the minimal injective resolution of the algebra. The conjecture sits inside a much-studied family of open homological conjectures: the Strong Nakayama Conjecture of Colby and Fuller (1990) implies GNC, GNC implies Nakayama's original conjecture, and GNC is also equivalent to the vanishing criterion that a simple module $S$ with $\operatorname{Ext}^i_\Lambda(S,\Lambda)=0$ for all $i\ge 0$ must be zero.

A quiver is a finite directed graph, and the path algebra $R\mathcal{Q}$ of a quiver $\mathcal{Q}$ over a ring $R$ has as basis the directed paths in $\mathcal{Q}$, multiplied by concatenation; when $\mathcal{Q}$ is finite and acyclic, $R\mathcal{Q}$ is again an artin algebra whenever $R$ is. The paper of Asadollahi, Hafezi, and Keshavarz on path algebras computes the terms of the minimal injective resolution of $R\mathcal{Q}$ explicitly and, as its Corollary 3.9, proves that the GNC holds for $R\mathcal{Q}$ whenever it holds for $R$. This is a transfer result, not a resolution: the conjecture for arbitrary artin algebras — indeed already for most individual base rings — has remained open since 1975.

Problem Statement

Prove or refute the Generalized Nakayama Conjecture (Auslander–Reiten, 1975): for every artin algebra $\Lambda$, with minimal injective resolution $0\to\Lambda\to I_0\to I_1\to I_2\to\cdots$ of $\Lambda$ as a left $\Lambda$-module, every indecomposable injective left $\Lambda$-module is isomorphic to a direct summand of $I_n$ for at least one $n\ge 0$. Equivalently: for every artin algebra $\Lambda$, a simple left $\Lambda$-module $S$ with $\operatorname{Ext}^i_\Lambda(S,\Lambda)=0$ for all $i\ge 0$ must be zero. The claim must be established or refuted for arbitrary artin algebras; verifications restricted to special classes (such as path algebras over a base ring already satisfying GNC, virtually Gorenstein algebras, or centralizer matrix algebras) do not answer the question.

References

1. M. Auslander, 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
2. T. 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
3. B. 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
4. R. R. Colby, K. R. Fuller, 'A note on the Nakayama conjectures', Tsukuba Journal of Mathematics 14 (1990), 343–352. DOI: 10.21099/tkbjm/1496161457, https://doi.org/10.21099/tkbjm/1496161457
5. I. Dräxler, D. Happel, 'A proof of the generalized Nakayama conjecture for algebras with $J^{2l+1}=0$ and $A/J^l$ representation finite', Journal of Pure and Applied Algebra 78 (1992), 161–164. DOI: 10.1016/0022-4049(92)90093-U, https://doi.org/10.1016/0022-4049(92)90093-U
6. J. Asadollahi, R. Hafezi, M. H. Keshavarz, 'Minimal injective resolutions and Auslander-Gorenstein property for path algebras', Communications in Algebra 45 (2016), 2557–2568; preprint arXiv:1505.04526. DOI: 10.1080/00927872.2016.1233217, https://arxiv.org/abs/1505.04526
7. Y. Zhang, 'Excellent extensions and homological conjectures', preprint arXiv:1702.05902 (2017). arXiv ID: 1702.05902, https://arxiv.org/abs/1702.05902
8. S. Kvamme, R. Marczinzik, 'Co-Gorenstein algebras', preprint arXiv:1805.00274 (2018). arXiv ID: 1805.00274, https://arxiv.org/abs/1805.00274
9. M. Barrios, G. Mata, 'On algebras of $\Omega^n$-finite and $\Omega^\infty$-infinite representation type', preprint arXiv:1911.02325 (2019). arXiv ID: 1911.02325, https://arxiv.org/abs/1911.02325
10. H. X. Chen, C. C. Xi, 'Virtually Gorenstein algebras of infinite dominant dimension', Journal of Pure and Applied Algebra 230 (2026), article 108224; preprint arXiv:2509.04990 (2025). DOI: 10.1016/j.jpaa.2026.108224, https://arxiv.org/abs/2509.04990
11. Z. Chen, C. Xi, 'Singular equivalences and homological conjectures', preprint arXiv:2603.20643 (2026). arXiv ID: 2603.20643, https://arxiv.org/abs/2603.20643
12. W. Li, 'Virtually Gorenstein Artin algebras are weakly Gorenstein', preprint arXiv:2608.10049 (2026). arXiv ID: 2608.10049, https://arxiv.org/abs/2608.10049