On stable modules that are not Gorenstein projective

In \cite{AB}, Auslander and Bridger introduced Gorenstein projective modules and only about 40 years after their introduction a finite dimensional algebra $A$ was found in \cite{JS} where the subcategory of Gorenstein projective modules did not coincide with $^{\perp}A$, the category of stable modules. The example in \cite{JS} is a commutative local algebra. We explain why it is of interest to find such algebras that are non-local with regard to the homological conjectures. We then give a first systematic construction of algebras where the subcategory of Gorenstein projective modules does not coincide with $^{\perp}A$ using the theory of gendo-symmetric algebras. We use Liu-Schulz algebras to show that our construction works to give examples of such non-local algebras with an arbitrary number of simple modules

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 $m$. 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