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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.
Manuscripts in this family · 2
An explicit counterexample to the Auslander-Reiten conjecture
Sep 23, 2026·paper.pdfLean
We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over and a finite-dimensional nonprojective left module Z such that 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
Sep 23, 2026·paper.pdfLean
We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over 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.