An explicit counterexample to the Auslander-Reiten conjecture
Source: github.com/openai/math/tree/main/preprints/An-explicit-coun…
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 .
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After OpenAI solved (it still has to be checked carefully!) the second Tachikawa, most of the important homological conjectures in our field might be wrong after all. Here I suggest some own interesting problems and conjectures related to homological algebra of finite dimensional algebras. (last…
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C.M.Ringel (Bielefeld): New Activitiesmath.uni-bielefeld.deClaus Michael Ringel
A new age has started! (9.10.2026) It is said that AI has exhibited examples which show that all the classical homological conjectures are wrong. Whether these assertions are correct, has still to be verified (like any claim of a mathematician). But at least someone has started to take…