On stable modules that are not Gorenstein projective

Abstract

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.