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

The Nakayama conjecture (1958) predicts that every Artin algebra whose dominant dimension is infinite is self-injective, i.e. that a minimal injective coresolution consisting entirely of projective terms forces the algebra to be injective as a module over itself. Tachikawa (1973) proved it equivalent to the conjunction of his first and second conjectures on the vanishing of Ext-groups. Despite confirmations for many proper classes - most recently for strongly Morita virtually Gorenstein algebras by Chen and Xi (2025-2026), and for all virtually Gorenstein Artin algebras by Li (2026, preprint) - the conjecture remains open for arbitrary Artin algebras, and no counterexample is known. The problem is to prove or refute the conjecture in this full generality.

Background

An Artin algebra is an algebra that is finitely generated as a module over a commutative artinian ring; the standard example is a finite-dimensional algebra over a field. Its dominant dimension, introduced by Nakayama in 1958, is the largest integer $n$ (or $\infty$) such that in a minimal injective coresolution $0\to{}_{A}A\to E^{0}\to E^{1}\to\cdots$ of the regular module ${}_{A}A$, all terms $E^{i}$ are projective for $0\le i<n$. An algebra is self-injective if ${}_{A}A$ is an injective module over itself, equivalently if every projective module is injective. The dominant dimension measures how far an algebra is from being self-injective: by the Morita-Tachikawa correspondence, algebras of dominant dimension at least 2 are exactly the endomorphism algebras of generator-cogenerator modules (modules containing every indecomposable projective and every indecomposable injective as direct summands), and a lemma of Müller states that the endomorphism algebra of a generator-cogenerator $M$ has dominant dimension at least $n>1$ if and only if $\operatorname{Ext}^{i}_{A}(M,M)=0$ for all $1\le i<n-1$; here $\operatorname{Ext}$ denotes extension groups and $A$-modules are finitely generated left modules. A module $M$ is called self-orthogonal when $\operatorname{Ext}^{i}_{A}(M,M)=0$ for all $i\ge1$. Nakayama conjectured (in a formulation equivalent to the modern one by Müller and Tachikawa) that infinite dominant dimension forces self-injectivity. In module-theoretic form (NC-M): a self-orthogonal generator-cogenerator over an Artin algebra must be projective. Tachikawa split the conjecture into two parts: (TC1) if $\operatorname{Ext}^{n}_{A}(D(A),A)=0$ for all $n\ge1$, 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; he proved that (NC) holds if and only if both (TC1) and (TC2) hold. All of these statements remain open in general, and they sit inside the classical network of homological conjectures for Artin algebras (generalized Nakayama, Auslander-Reiten, Auslander-Gorenstein, Gorenstein symmetry, finitistic dimension), each of which implies or relates to (NC). A modern attack route, developed by Beligiannis and Krause and taken up in [1,7], uses the class of virtually Gorenstein algebras: writing $D$ for the duality, a Gorenstein projective module is one arising as the image of a totally acyclic complex of projective modules (an acyclic complex of projectives that stays exact after applying $\operatorname{Hom}(-,P)$ for every projective $P$), Gorenstein injective modules are defined dually, and an Artin algebra $A$ is virtually Gorenstein when, for every $A$-module $X$, the functor $\operatorname{Ext}^{i}_{A}(X,-)$ vanishes on all Gorenstein injective modules for all $i\ge1$ exactly when $\operatorname{Ext}^{i}_{A}(-,X)$ vanishes on all Gorenstein projective modules for all $i\ge1$. The class contains all Gorenstein algebras (algebras of finite injective dimension on both sides) and all algebras of finite representation type, and is closed under derived equivalence and stable equivalence of Morita type; however, not every algebra is virtually Gorenstein, since Beligiannis and Krause produced a counterexample. A further enlargement, the class of compactly Gorenstein algebras, was introduced by Chen and Xi (2025), who also proved the Nakayama conjecture for their Gorenstein-Morita algebras; Chen and Xi [1] conjecture that all Artin algebras should be compactly Gorenstein, which would give a potential route toward the general conjecture.

Problem Statement

Prove or refute the Nakayama conjecture in its standard general form: every Artin algebra $A$ with infinite dominant dimension - equivalently, such that all terms $E^{i}$, $i\ge0$, of a minimal injective coresolution of the regular module ${}_{A}A$ are projective - is self-injective. Equivalently, prove or refute either of the following: (NC-M) every self-orthogonal generator-cogenerator over an Artin algebra is projective; or the conjunction of Tachikawa's two conjectures, (TC1) $\operatorname{Ext}^{n}_{A}(D(A),A)=0$ for all $n\ge1$ implies that $A$ is self-injective, and (TC2) over a self-injective Artin algebra every finitely generated self-orthogonal module is projective. The statement concerns arbitrary Artin algebras; a confirmation restricted to a proper class such as the virtually Gorenstein algebras does not answer it, since Artin algebras outside that class are known to exist.

References

1. Hongxing Chen, Changchang Xi, Virtually Gorenstein algebras of infinite dominant dimension, J. Pure Appl. Algebra 230(4) (2026) 108224, DOI 10.1016/j.jpaa.2026.108224 (preprint arXiv:2509.04990), https://doi.org/10.1016/j.jpaa.2026.108224
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