A note on stable equivalence and Nakayama algebras

Abstract

Two artin algebras Λ \Lambda and Λ ′ \Lambda ’ are said to be stably equivalent if the categories of finitely generated modules modulo projective for Λ \Lambda and Λ ′ \Lambda ’ are equivalent categories. If Λ ′ \Lambda ’ is stably equivalent to a Nakayama (i.e. generalized uniserial) algebra Λ \Lambda , we prove that Λ \Lambda and Λ ′ \Lambda ’ have the same number of nonprojective simple modules. And if Λ \Lambda and Λ ′ \Lambda ’ are stably equivalent indecomposable Nakayama algebras where each indecomposable projective module has length at least 3, then Λ \Lambda and Λ ′ \Lambda ’ have the same admissible sequences.