The Auslander-Reiten conjecture for Artin algebras

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The Nakayama conjecture on dominant dimension evidence for this event

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Lean comparator statement: Auslander-Reiten counterexample evidence for this event

Lean: Auslander-Reiten counterexample (formalization.yaml file) evidence for this event

Auslander and Reiten (1975), On a generalized version of the Nakayama conjecture evidence for this event

VibeMathed record: The Auslander-Reiten conjecture for Artin algebras evidence for this event

The equivalence of Tachikawa's second conjecture and Auslander-Reiten evidence for this event

Claims Theorem 1.1: over $k=\mathbb F_2(q,H_1,H_2)$ there are a finite-dimensional triangular algebra $\Lambda$ with eight simple modules and $\mathrm{rad}(\Lambda)^4\ne0$, and a finite-dimensional nonprojective Gorenstein-projective module $Z$ with $\mathrm{Ext}^i(Z,Z)=\mathrm{Ext}^i(Z,\Lambda)=0$ for all $i\ge1$, persisting under every field extension (so also over an algebraically closed field of characteristic two). This disproves the Auslander-Reiten and Gorenstein-projective conjectures; $Z\oplus\Lambda$ also refutes the generator form. It does NOT touch the commutative Noetherian form of the conjecture, and gives no example in characteristic zero. parent of this event

The Auslander-Reiten conjecture for Artin algebras parent of this event

Companion: An explicit counterexample to the Auslander-Reiten conjecture evidence for this event