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arXiv · 2609.19172

Tachikawa's second conjecture implies the Auslander-Reiten conjecture

Haruhisa Enomoto
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Abstract

We prove that Tachikawa's second conjecture implies the Auslander-Reiten conjecture for artin algebras over a commutative artinian ring. The proof uses the two-fold trivial extension of an algebra. Together with known implications, it follows that the Auslander-Reiten conjecture, the generalized Nakayama conjecture, the Auslander-Gorenstein conjecture, the Nakayama conjecture, the Gorenstein-projective conjecture, and Tachikawa's second conjecture are all equivalent, and that Tachikawa's second conjecture implies his first.

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Haruhisa Enomoto. 2026-09-14. Tachikawa's second conjecture implies the Auslander-Reiten conjecture. https://arxiv.org/abs/2609.19172

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