A Note on Stable Equivalence and Nakayama Algebras

Abstract

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