A note on stable equivalence and Nakayama algebras

Reiten, Idun

doi:10.1090/S0002-9939-1978-0500481-7

A note on stable equivalence and Nakayama algebras
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by Idun Reiten

Proc. Amer. Math. Soc. 71 (1978), 157-163

DOI: https://doi.org/10.1090/S0002-9939-1978-0500481-7

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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.

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Algebras stably equivalent to Nakayama algebras of Loewy length at most 4