A new formula for the classical dominant dimension using bimodules

Abstract

We show that a faithful projective-injective module over a finite-dimensional algebra $A$ has the double centraliser property if and only if $A$ as a bimodule is reflexive. More generally, we provide a new characterisation of the classical dominant dimension by showing that having dominant dimension at least $n$ is equivalent to the bimodule $A$ being $n$-torsion-free. This allows us to find new connections between the classical Tachikawa and Nakayama conjectures and Gorenstein homological algebra. Furthermore, we use our results to give new interpretations of Hochschild (co)homology of finite-dimensional algebras using higher Auslander-Reiten translates and the canonical bimodule in the sense of Fang, Kerner and Yamagata.