  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.

Representation Theory