Corollary

Let $A$ be a noetherian ring admitting a dualizing module $\Omega$; let $B$ be a polynomial algebra over $A$ in a finite number of indeterminates. The $B$-module $\Omega_{(B)}$ is dualizing.

Indeed, for every prime ideal $\mathfrak{p}$ of $A$, the ring $\kappa(\mathfrak{p}) \otimes_A A[X]$ is identified with $\kappa(\mathfrak{p})[X]$, which is regular, hence Gorenstein.