On bounds of homological dimensions in Nakayama algebras

TL;DR

This paper establishes bounds on the global and finitistic dimensions of Nakayama algebras based on simple modules' projective dimensions, generalizing previous results and providing new proofs.

Contribution

It introduces a new bound on the global dimension of Nakayama algebras using minimal simple module projective dimensions, extending prior work and generalizing to stratified cases.

Findings

Global dimension bounded by n + m - 1 for certain Nakayama algebras

Provides a new proof of Brown's bound for quasi-hereditary Nakayama algebras

Generalizes bounds to standardly stratified Nakayama algebras

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 \cite{Gus} and Madsen \cite{Mad}. In \cite{Bro}, 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.