Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 199-01. An explicit counterexample to the Auslander-Reiten conjecture
Manuscript map · lines 5955–5964
OpenAI@2026-10-07#rem:oai-p-199-01
Authors: OpenAI. Family 199.
Title: An explicit counterexample to the Auslander-Reiten conjecture.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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