The Nakayama conjecture on dominant dimension

Let $\Lambda$ be a finite-dimensional algebra over a field with minimal injective resolution $0\to\Lambda\to I^0\to I^1\to\cdots$ . Its dominant dimension is infinite when every $I^j$ is projective. The classical Nakayama conjecture asserts that infinite dominant dimension forces $\Lambda$ to be self-injective. Its relatives are the generalized Nakayama conjecture of Auslander and Reiten (every indecomposable injective occurs as a summand of some $I^j$ ), the strong Nakayama conjecture (a module $V$ with $\mathrm{Ext}^i(V,\Lambda)=0$ for all $i\ge0$ is zero), the Auslander-Gorenstein conjecture and the Wakamatsu tilting conjecture; the finitistic dimension conjecture implies the classical one. Must a finite-dimensional algebra of infinite dominant dimension be self-injective?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Representation theory of algebras; homological conjectures
Posed by: Tadasi Nakayama (classical Nakayama conjecture); formulations recalled from Y. Zhang (2018) in the manuscript
Year posed: —
Years open: —
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 50 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims Corollary 1.2: for the symmetric pair $(A,M)$ of the Tachikawa counterexample and every field extension $K$ of $\mathbb F_2(q,H_1,H_2)$ , the finite-dimensional algebra $\Gamma_K=\mathrm{End}(A_K\oplus M_K)^{op}$ has all injective terms projective but infinite self-injective dimension, has a nonzero simple $S$ with $\mathrm{Ext}^i(S,\Gamma_K)=0$ for all $i\ge0$ , and has a projective-injective Wakamatsu tilting module that is not tilting. Hence the classical, generalized and strong Nakayama conjectures, the Auslander-Gorenstein conjecture and the Wakamatsu tilting conjecture all fail. Corollary 1.3: $\Gamma_K$ has infinite little and big finitistic dimension. It does NOT give characteristic-zero examples.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The Nakayama-type counterexamples are Corollary 1.2 of the Tachikawa manuscript, obtained from its symmetric counterexample through the classical Mueller correspondence between pairs (A, M) and algebras of large dominant dimension.

Verification

No independent mathematician has checked this yet. Corollary 1.2 of the Tachikawa manuscript was read against the classical Nakayama conjecture: for $\Gamma_K=\mathrm{End}_{A_K}(A_K\oplus M_K)^{op}$ , every term of the minimal injective resolution is projective while the injective dimension of $\Gamma_K$ is infinite, so $\Gamma_K$ has infinite dominant dimension and is not self-injective. This consequence has no Lean formalisation: lean/docs/199.md says the endomorphism-algebra and related homological consequences are not included; the formalised Tachikawa statement covers only the symmetric counterexample. The release README warns that some unformalised results could have issues. Readers should know that a counterexample here also gives infinite finitistic dimension (Corollary 1.3), contradicting the finitistic dimension conjecture; the release treats that in a separate family (An algebra of infinite little finitistic dimension).

The Nakayama conjecture on dominant dimension

What was actually shown

What the AI did

Verification

Sources

Discussion