Grade, dominant dimension and Gorenstein algebras

We first give precise connections between Auslander–Bridger's grade, double centraliser properties and dominant dimension, and apply these to homological conjectures. Then we introduce gendo-d-Gorenstein algebras as correspondents of Gorenstein algebras under a Morita–Tachikawa correspondence. We characterise these algebras by homological properties and derive several of their properties, including higher Auslander correspondence.