As the title says, when is the module of Kahler differentials a free module? In particular, are there known conditions or criterions that could be met that ensures that it will be free?

For example, if one has a finitely generated algebra $S=k[x_0,\cdots,x_n]/(f_1,\cdots, f_l)$ over a field $k$, then one could require that the generators induced from the $f_i$ for $\Omega_{S/k}$ be linearly independent. However, this is a very naive approach. I was curious if there something more interesting. For example, if the ring $S=k[x_0,\cdots,x_n]/(f_1,\cdots, f_l)$ (where $l<n$) has the property that the determinant of the matrix $(\frac{\partial f_i}{\partial x_j})_{i,j=1}^l$ is a unit of $S$. I am not entirely sure if that is accurate on the top of my head, but something along those lines.

Another question is, when is the module of differentials reflexive?

Kähler differentials, Theorem 8.1. $\endgroup$1more comment