Tachikawa’s second conjecture implies the Auslander–Reiten conjecture

Comment

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 following conjectures are all equivalent:

• the Auslander–Reiten conjecture,
• the generalized Nakayama conjecture,
• the Auslander–Gorenstein conjecture,
• the Nakayama conjecture,
• the Gorenstein-projective conjecture,
• Tachikawa’s second conjecture.

It also follows that Tachikawa’s second conjecture implies his first.

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