Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field. (Lean)

Manuscripts (2)

An explicit counterexample to the Auslander-Reiten conjecture

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026}

We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over k=\mathbb F_2(q,H_1,H_2) and a finite-dimensional nonprojective left module Z such that \mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,Z)=\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,\Lambda)=0 for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.

A counterexample to Tachikawa's second conjecture

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026}

We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over k=\mathbb F_2(q,H_1,H_2) with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.

Result 199 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.