On bounds of homological dimensions in Nakayama algebras

Madsen, Dag; Marczinzik, René

doi:10.1090/bproc/36

On bounds of homological dimensions in Nakayama algebras
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by Dag Oskar Madsen and René Marczinzik HTML | PDF

Proc. Amer. Math. Soc. Ser. B 5 (2018), 40-49

Abstract:

Let $A$ be a Nakayama algebra with $n$ simple modules and a simple module $S$ of even projective dimension. Choose $m$ minimal such that a simple $A$-module with projective dimension $2m$ exists. Then we show that the global dimension of $A$ is bounded by $n+m-1$. This gives a combined generalisation of results of Gustafson [J. Algebra 97 (1985), pp. 14–16] and Madsen [Projective dimensions and Nakayama algebras, Amer. Math. Soc., Providence, RI, 2005]. In [Comm. Algebra 22 (1994), pp. 1271–1280], Brown proved that the global dimension of quasi-hereditary Nakayama algebras with $n$ simple modules is bounded by $n$. Using our result on the bounds of global dimensions of Nakayama algebras, we give a short new proof of this result and generalise Brown’s result from quasi-hereditary to standardly stratified Nakayama algebras, where the global dimension is replaced with the finitistic dimension.

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