  We show that a finite dimensional algebra $A$ has dominant dimension at least
$n \geq 2$ if and only if the regular bimodule $A$ is $n$-torsionfree if and
only if $A \cong \Omega^{n}(\text{Tr}(\Omega^{n-2}(V)))$ as $A$-bimodules,
where $V=\text{Hom}_A(D(A),A)$ is the canonical $A$-bimodule in the sense of
\cite{FKY}. We apply this to give new formulas for the Hochschild homology and
cohomology for algebras with dominant dimension at least two and show a new
relation between the first Tachikawa conjecture, the Nakayama conjecture and
Gorenstein homological algebra.
