On bounds of homological dimensions in Nakayama algebras

Abstract

Let A A be a Nakayama algebra with n n simple modules and a simple module S S of even projective dimension. Choose m m minimal such that a simple A A -module with projective dimension 2 m 2m exists. Then we show that the global dimension of A A is bounded by n + m − 1 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 n simple modules is bounded by n 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.