On bounds of homological dimensions in Nakayama algebras

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.