Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

What OpenAI claims

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)

Papers (2)

1. An explicit counterexample to the Auslander-Reiten conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over k=𝔽₂(q,H₁,H₂) and a finite-dimensional nonprojective left module Z such that Ext ⁱ_Λ(Z,Z)=Ext ⁱ_Λ(Z,Λ)=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.

2. A counterexample to Tachikawa's second conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over k=𝔽₂(q,H₁,H₂) 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.