  Let $A$ be a Noetherian domain and $R$ be a finitely generated $A$-algebra.
We study several features regarding the generic freeness over $A$ of an
$R$-module. For an ideal $I \subset R$, we show that the local cohomology
modules ${\rm H}_I^i(R)$ are generically free over $A$ under certain settings
where $R$ is a smooth $A$-algebra. By utilizing the theory of Gr\"obner bases
over arbitrary Noetherian rings, we provide an effective method to make
explicit the generic freeness over $A$ of a finitely generated $R$-module.
