The Nakayama conjecture: must every Artin algebra of infinite dominant dimension be self-injective?

The Nakayama conjecture (1958) asserts that every Artin algebra whose dominant dimension is infinite is self-injective; equivalently, that every self-orthogonal generator-cogenerator over an Artin algebra is projective, and equivalently that Tachikawa's first and second conjectures both hold. The starting point is the preprint of Chen and Xi on virtually Gorenstein algebras of infinite dominant dimension (arXiv:2509.04990), which proves the conjecture for algebras that are virtually Gorenstein and satisfy an additional Ext-vanishing condition, restates the companion conjecture that all Artin algebras are compactly Gorenstein, and explicitly leaves the general conjectures open. More recent work (Li, 2026) settles the Nakayama conjecture, Tachikawa's first conjecture, and several stronger conjectures for the entire class of virtually Gorenstein Artin algebras, but not every algebra is virtually Gorenstein, and the conjecture for arbitrary Artin algebras — the famous original problem — remains open, as does Tachikawa's second conjecture even for symmetric algebras.

Background

An Artin algebra is an algebra that is finitely generated as a module over a commutative Artin ring; finite-dimensional algebras over a field are the main example, and all modules below may be taken finite-dimensional. The dominant dimension of an Artin algebra $A$, denoted $\operatorname{domdim}(A)$, is the largest $n$ (or $\infty$) such that in a minimal injective coresolution $0 \to {}_A A \to I_0 \to I_1 \to \cdots$ of the regular module ${}_A A$, all terms $I_i$ are projective for $0 \le i < n$. The algebra is self-injective if the regular module is injective, i.e. projective and injective modules coincide. Nakayama introduced dominant dimension in 1958 and conjectured that infinite dominant dimension forces self-injectivity; Müller (1968) showed the conjecture is equivalent to a module statement: a finitely generated module $M$ is self-orthogonal if $\operatorname{Ext}^i_A(M,M)=0$ for all $i \ge 1$, and is a generator-cogenerator if every indecomposable projective and every indecomposable injective module occurs as a direct summand of $M$; the conjecture says every self-orthogonal generator-cogenerator is projective. By the Morita–Tachikawa correspondence (a classical result identifying algebras of dominant dimension at least $2$ with endomorphism algebras of generator-cogenerators) and Müller's theorem, the two formulations coincide. Tachikawa (1973) split the conjecture into two now-famous parts: (TC1) if $\operatorname{Ext}^n_A(D(A),A)=0$ for all $n \ge 1$, where $D$ is the standard duality of Artin algebras, then $A$ is self-injective; and (TC2) over a self-injective Artin algebra, every finitely generated self-orthogonal module is projective. Tachikawa proved that the Nakayama conjecture holds for all Artin algebras if and only if both TC1 and TC2 do. The conjecture sits inside the network of homological conjectures collected by Auslander, Reiten and Smalø: it is implied by the generalized Nakayama conjecture (1975), by the strong Nakayama conjecture, and by the finitistic dimension conjecture (which asserts that the supremum of projective dimensions of finite-dimensional modules of finite projective dimension is finite, and which itself remains open). Modern attacks proceed through Gorenstein homological algebra. A module is Gorenstein projective if it is a syzygy of an acyclic complex of projective modules; an Artin algebra is virtually Gorenstein (Beligiannis, 2005) if, for every module $X$, the functor $\operatorname{Ext}^i_A(X,-)$ vanishes on all Gorenstein injective modules exactly when $\operatorname{Ext}^i_A(-,X)$ vanishes on all Gorenstein projective modules. The class contains all Gorenstein algebras and all algebras of finite representation type, and Beligiannis proved it satisfies the Gorenstein symmetry conjecture; however, Beligiannis and Krause (2008) gave a counterexample showing not every algebra is virtually Gorenstein, and characterized the property by the contravariant (equivalently covariant) finiteness of the thick subcategory generated by the projective and injective modules. To go further, Chen and Xi introduced compactly Gorenstein algebras: call a module compactly filtered if it is a union of a filtration whose subquotients are finitely generated of finite projective dimension, and compactly Gorenstein-projective if it is a compact object (an object whose covariant Hom-functor commutes with arbitrary coproducts) in the stable category of Gorenstein projective modules; the algebra is compactly Gorenstein if every compactly filtered, compactly Gorenstein-projective module is projective, and they conjectured that all Artin algebras should be compactly Gorenstein. Within this circle, Chen–Xi (arXiv:2509.04990) proved: if $\operatorname{domdim}(A)=\infty$ and $\operatorname{Ext}^n_A(D(A),A)=0$ for all $n \ge 1$ and $A$ is virtually Gorenstein, then $A$ is self-injective; and a self-orthogonal generator-cogenerator $M$ with $\operatorname{Ext}^n_A(M \oplus \nu_A(M),M)=0$ for all $n \ge 1$ (where $\nu_A = D(A) \otimes_A -$ is the Nakayama functor) and virtually Gorenstein endomorphism algebra must be projective. This paper, like its companion, explicitly leaves the general Nakayama and Tachikawa conjectures open.

Chen and Xi record that resolving the compactly Gorenstein conjecture would settle the Nakayama conjecture for all strongly Morita (in particular gendo-symmetric) algebras.

Problem Statement

The Nakayama conjecture: prove or refute that every Artin algebra $A$ with infinite dominant dimension — that is, such that all terms $I_i$ of the minimal injective coresolution $0 \to {}_A A \to I_0 \to I_1 \to \cdots$ are projective — is self-injective. By Müller's theorem this is equivalent to: every self-orthogonal generator-cogenerator $M$ over an Artin algebra $A$ (meaning $\operatorname{Ext}^i_A(M,M)=0$ for all $i \ge 1$ and every indecomposable projective and injective $A$-module is a direct summand of $M$) is projective. By Tachikawa's theorem it is equally equivalent to the conjunction of: (TC1) every Artin algebra $A$ with $\operatorname{Ext}^n_A(D(A),A)=0$ for all $n \ge 1$, where $D$ is the standard duality of Artin algebras, is self-injective; and (TC2) every finitely generated self-orthogonal module over a self-injective Artin algebra is projective. The problem is to be answered for the class of all Artin algebras with no restriction; a solution may equally resolve any of the equivalent formulations.

References

1. T. Nakayama, 'On algebras with complete homology', Abh. Math. Sem. Univ. Hamburg 22 (1958) 300-307. DOI: 10.1007/BF02941960, https://doi.org/10.1007/BF02941960
2. B. Müller, 'The classification of algebras by dominant dimension', Canad. J. Math. 20 (1968) 398-409. DOI: 10.4153/CJM-1968-037-9, https://doi.org/10.4153/CJM-1968-037-9
3. H. Tachikawa, 'Quasi-Frobenius rings and generalizations, QF-3 and QF-1 rings', Lecture Notes in Mathematics 351, Springer, Berlin, 1973. DOI: 10.1007/BFb0059997, ISBN 978-3-540-06501-2, https://doi.org/10.1007/BFb0059997
4. M. Auslander and I. Reiten, 'On a generalized version of the Nakayama conjecture', Proc. Amer. Math. Soc. 52 (1975) 69-74. DOI: 10.1090/S0002-9939-1975-0389977-6, https://doi.org/10.1090/S0002-9939-1975-0389977-6
5. M. Auslander, I. Reiten and S. O. Smalø, 'Representation Theory of Artin Algebras', Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, 1995. DOI: 10.1017/CBO9780511623608, https://doi.org/10.1017/CBO9780511623608
6. A. Beligiannis, 'Cohen-Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras', J. Algebra 288 (2005) 137-211. DOI: 10.1016/j.algebra.2005.02.022, https://doi.org/10.1016/j.algebra.2005.02.022
7. A. Beligiannis and H. Krause, 'Thick subcategories and virtually Gorenstein algebras', Illinois J. Math. 52 (2008) 551-562. DOI: 10.1215/IJM/1248355349, https://doi.org/10.1215/IJM/1248355349
8. H. X. Chen, M. Fang and C. C. Xi, 'Tachikawa's second conjecture, derived recollements, and gendo-symmetric algebras', Compos. Math. 160 (2024) 2704-2734. DOI: 10.1112/S0010437X24007395, https://doi.org/10.1112/S0010437X24007395
9. H. X. Chen and C. C. Xi, 'Homological theory of self-orthogonal modules', Trans. Amer. Math. Soc. 378 (2025) 7287-7335. DOI: 10.1090/tran/9426, https://doi.org/10.1090/tran/9426; preprint arXiv:2208.14712, https://arxiv.org/abs/2208.14712
10. H. X. Chen and C. C. Xi, 'Virtually Gorenstein algebras of infinite dominant dimension', preprint arXiv:2509.04990 (2025), https://arxiv.org/abs/2509.04990
11. T. Cruz and R. Marczinzik, 'A new formula for the classical dominant dimension using bimodules', preprint arXiv:2508.18398 (2025), https://arxiv.org/abs/2508.18398
12. W. Li, 'Virtually Gorenstein Artin algebras are weakly Gorenstein', preprint arXiv:2608.10049 (2026), https://arxiv.org/abs/2608.10049