Let $X$ be a complex variety. It is well-known that $X$ is smooth if and only if the sheaf of Kähler differentials $\Omega_X^1$ is locally free (Hartshorne p. 177).

Question: What happens for forms of higher degree? I.e. define $\Omega_X^p := \bigwedge^p \Omega_X^1$ (no reflexive hull or something). For what values of $p$ and under what additional assumptions on $X$ does $\Omega_X^p$ locally free imply $X$ smooth?

Gorenstein varieties(for instance hypersurfaces in $\mathbb{P}^n$, or projective surfaces with only rational double points). $\endgroup$1more comment