Symmetry of self-injective dimensions under the Auslander condition

Let $A$ be a module-finite algebra over a commutative Noetherian ring. We give a local criterion for the $n$-Gorenstein condition on $A$. As an application, we prove that if $A$ satisfies the Auslander condition,then finiteness of the self-injective dimension on one side impliesfiniteness on the other side.