On properly stratified Gorenstein algebras

TL;DR

This paper characterizes when properly stratified algebras are Gorenstein, revealing their homological properties and applications to minimal Auslander-Gorenstein algebras and centralizer algebras of nilpotent matrices.

Contribution

It establishes a criterion for properly stratified algebras to be Gorenstein and explores their homological properties and applications to specific algebra classes.

Findings

Properly stratified Gorenstein algebras have coinciding characteristic tilting and cotilting modules.

All Gorenstein projective modules in such algebras are properly stratified.

Endomorphism rings of standard modules are Frobenius algebras.

Abstract

We show that a properly stratified algebra is Gorenstein if and only if the characteristic tilting module coincides with the characteristic cotilting module. We further show that properly stratified Gorenstein algebras $A$ enjoy strong homological properties such as all Gorenstein projective modules being properly stratified and all endomorphism rings $\operatorname{End}_A(\Delta(i))$ being Frobenius algebras. We apply our results to the study of properly stratified algebras that are minimal Auslander-Gorenstein algebras in the sense of Iyama-Solberg and calculate under suitable conditions their Ringel duals. This applies in particular to all centraliser algebras of nilpotent matrices.