Cohen-Macaulay truncated path and Nakayama algebras

Abstract

This paper characterizes Cohen-Macaulay algebras in the sense of Auslander and Reiten [Progr. Math. 95(1991)] within the following two classes of finite dimensional algebras: (i) truncated path algebras and (ii) Nakayama algebras. For the former, our characterization in terms of ordinary quivers shows that if a Cohen-Macaulay truncated path algebra $Λ$ is connected, then $Λ$ or $D(Λ)$ gives a dualizing module, where $D$ denotes the standard duality. For the latter, we show that every Cohen-Macaulay Nakayama algebra $Λ$ is Iwanaga-Gorenstein. In particular, $Λ$ always gives a dualizing module. Consequently, there is no non-trivial Cohen-Macaulay algebra among truncated path and Nakayama algebras.