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The Nakayama conjecture on dominant dimension

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.

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Nakayama conjecture on dominant dimension · Nakayama conjecture

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VibeMathed record: The Nakayama conjecture on dominant dimension

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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).