On bounds of homological dimensions in Nakayama algebras

2017/10/12 by Madsen, Dag Oskar, Marczinzik, Rene
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1710.04420

Abstract

Let A be a Nakayama algebra with n simple modules and a simple module S of even projective dimension m. 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 \citeGus and Madsen \citeMad. In \citeBro, 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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