{"id":"openai-math/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture","pdf":"https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026","tags":["representation-theory","auslander-reiten-conjecture"],"score":0,"title":"An explicit counterexample to the Auslander-Reiten conjecture","domain":"Algebra","family":"199","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.RA","summary":"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 .","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures"}