Tachikawa's second conjecture implies the Auslander-Reiten conjecture

RESUMO

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.