Higher torsion-free Auslander-Reiten sequences and the dominant dimension of algebras

We generalise a theorem of Tachikawa about reflexive Auslander-Reiten sequences. We apply this to give a new characterisation of the dominant dimension of gendo-symmetric algebras. We also generalise a formula due to Reiten about the dominant dimension of an algebra $A$ and grades of torsion $A$-modules.