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

The Nakayama conjecture (1958) predicts that every finite-dimensional algebra over a field whose dominant dimension is infinite must be self-injective; equivalently, no non-self-injective algebra has infinite dominant dimension. The conjecture remains open in general and is one of the classical homological conjectures: by a theorem of Tachikawa, it would follow from the finitistic dimension conjecture, so a counterexample would refute that conjecture as well. The source paper (Pressland–Sauter, Glasgow Math. J. 2022) records the conjecture and derives that any counterexample would carry special (projective-injective-generated) tilting modules of every finite projective dimension. Partial confirmations are known only for restricted classes such as Nakayama/monomial-type, virtually Gorenstein, and centralizer matrix algebras.

Background

Let $k$ be a field and $\Lambda$ a finite-dimensional $k$-algebra (everything below also holds, with the same statements, for Artin algebras — rings that are finitely generated as modules over a commutative artinian base ring). The dominant dimension of $\Lambda$, written $\operatorname{domdim}\Lambda$, is the largest $d$ such that the first $d$ terms of a minimal injective resolution of $\Lambda$ as a left module over itself are projective (it is $\infty$ if all terms are projective, which holds in particular when $\Lambda$ is self-injective); by a result of Müller this invariant is left–right symmetric. Dominant dimension $\ge 1$ means $\Lambda$ has a faithful projective-injective module, the QF-3 property studied by Tachikawa. An algebra is self-injective if it is injective as a (left) module over itself, equivalently if its projective and injective modules coincide; self-injective algebras include group algebras and Frobenius algebras. Nakayama (1958), in work on algebras with 'complete homology', conjectured that infinite dominant dimension forces self-injectivity. The conjecture has resisted solution for over six decades and sits among the classical homological conjectures: Tachikawa (1973) proved that the finitistic dimension conjecture — the still-open statement that the supremum of projective dimensions of all finite-projective-dimensional modules over $\Lambda$ is finite — implies the Nakayama conjecture for any class of algebras; contrapositively, a non-self-injective algebra of infinite dominant dimension would have infinite finitistic dimension and would refute the finitistic dimension conjecture. Auslander and Reiten (1975) formulated a stronger generalized version. A tilting module of projective dimension $k$ is a module $T$ with $\operatorname{pdim} T \le k$ and $\operatorname{Ext}^i_\Lambda(T,T)=0$ for all $i>0$, such that the regular module $\Lambda$ admits an add-$T$ coresolution of length $k$, i.e. an exact sequence $0 \to \Lambda \to T_0 \to \cdots \to T_k \to 0$ with each $T_j$ a direct summand of a finite direct sum of copies of $T$; Pressland and Sauter (2022) studied special tilting modules, i.e. tilting modules generated (and typically also cogenerated) by projective-injective modules. For an algebra $\Lambda$ with $\operatorname{domdim}\Lambda \ge k$ they construct canonical $k$-shifted tilting modules $T_k$, and their Proposition 2.7 shows that if $\Lambda$ is not self-injective then $T_k$ has projective dimension exactly $k$ for every $0 \le k \le \operatorname{domdim}\Lambda$. Consequently (their Remark 2.8), any counterexample to the Nakayama conjecture would admit special tilting modules of every finite projective dimension — a reformulation essentially equivalent to Tachikawa's argument. Despite substantial activity around dominant dimension (upper bounds for special classes, behaviour under derived equivalence, flat-dominant dimension for Frobenius extensions), the general conjecture is open: it has been confirmed only within restricted classes — Nakayama algebras (algebras whose indecomposable modules are uniserial, i.e. have a unique composition series) and related monomial-type algebras (Marczinzik), strongly Morita virtually Gorenstein algebras (Chen–Xi 2025), virtually Gorenstein Artin algebras (Li 2026), and centralizer matrix algebras (Chen–Xi 2026) — where an Artin algebra is virtually Gorenstein if its class of Gorenstein-projective modules (modules appearing as syzygies of an acyclic complex of projectives that stays exact after applying $\operatorname{Hom}_\Lambda(-,Q)$ for every projective module $Q$) is contravariantly finite, and a centralizer matrix algebra is the algebra of linear operators commuting with a single fixed matrix — and no non-self-injective algebra of infinite dominant dimension is known.

Problem Statement

Does every finite-dimensional algebra over a field of infinite dominant dimension have to be self-injective? Equivalently: does there exist a finite-dimensional algebra $\Lambda$ that is not self-injective yet whose minimal injective resolution of $\Lambda$ as a left module over itself consists entirely of projective modules ($\operatorname{domdim}\Lambda=\infty$)? This is the Nakayama conjecture (1958). The analogous, stronger question — not the question asked here — is whether every Artin algebra of infinite dominant dimension is self-injective; a positive answer for Artin algebras implies the field-algebra version, while no reduction in the converse direction is known. In the equivalent tilting-theoretic form recorded in the source paper (Pressland–Sauter, Proposition 2.7 and Remark 2.8), a negative answer is the same as producing a non-self-injective algebra of infinite dominant dimension, which would necessarily admit special tilting modules $T_k$ (generated and cogenerated by projective-injectives) of projective dimension exactly $k$ for every integer $k \ge 0$, and would also be a counterexample to the finitistic dimension conjecture; a positive answer proves the Nakayama conjecture in full generality, with no restriction to special classes (e.g. virtually Gorenstein, monomial, or centralizer matrix algebras).

References

1. Matthew Pressland, Julia Sauter, 'Special tilting modules for algebras with positive dominant dimension', Glasgow Mathematical Journal 64 (2022) 79–105. DOI: 10.1017/S0017089520000609, arXiv:1705.03367. https://doi.org/10.1017/S0017089520000609
2. Tadasi Nakayama, 'On algebras with complete homology', Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 22 (1958). DOI: 10.1007/bf02941960. https://doi.org/10.1007/bf02941960
3. Hiroyuki Tachikawa, 'Quasi-Frobenius Rings and Generalizations. QF-3 and QF-1 Rings', Lecture Notes in Mathematics 351, Springer, 1973. DOI: 10.1007/bfb0059997. https://doi.org/10.1007/bfb0059997
4. Maurice Auslander, Idun 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
5. René Marczinzik, 'Upper bounds for the dominant dimension of Nakayama and related algebras', Journal of Algebra 496 (2018) 216–241. DOI: 10.1016/j.jalgebra.2017.10.025, arXiv:1605.09634. https://doi.org/10.1016/j.jalgebra.2017.10.025
6. Hongxing Chen, Changchang Xi, 'Dominant dimensions, derived equivalences and tilting modules', arXiv:1503.02385 (2015). DOI: 10.1007/s11856-016-1327-4. https://arxiv.org/abs/1503.02385
7. Changchang Xi, 'Frobenius bimodules and flat-dominant dimensions', arXiv:1903.07921 (2019). DOI: 10.1007/s11425-018-9519-2. https://arxiv.org/abs/1903.07921
8. Hongxing Chen, Changchang Xi, 'Virtually Gorenstein algebras of infinite dominant dimension', arXiv:2509.04990 (2025), Journal of Pure and Applied Algebra 230 (2026), article 108224. DOI: 10.1016/j.jpaa.2026.108224. https://doi.org/10.1016/j.jpaa.2026.108224
9. Weiqing Li, 'Virtually Gorenstein Artin algebras are weakly Gorenstein', arXiv:2608.10049 (2026). https://arxiv.org/abs/2608.10049
10. Zhenxian Chen, Changchang Xi, 'Singular equivalences and homological conjectures', arXiv:2603.20643 (2026). https://arxiv.org/abs/2603.20643