Sheaf of differentials

Remarks

If $U = \operatorname{Spec} B \subseteq X$ maps into $V = \operatorname{Spec} A \subseteq Y$, then $\Omega_{X/Y}|_U$ is the quasi-coherent sheaf associated to the module of Kähler differentials $\Omega_{B/A}$. There is a universal derivation $d \colon \mathcal{O}_X \to \Omega_{X/Y}$, and $\Omega_{X/Y}$ is generated by the sections $dg$. If $X$ is a variety over a field $k$, one writes $\Omega_X = \Omega_{X/k}$, the cotangent sheaf.

Example: for $X = \mathbb{A}^n_k = \operatorname{Spec} k[x_1, \dots, x_n]$, $\Omega_{X/k}$ is free of rank $n$ with basis $dx_1, \dots, dx_n$. This matches $\Omega_{B/k}$ for the polynomial ring $B$.

Non-example of local freeness: for the cuspidal curve $X = \operatorname{Spec} k[x, y]/(y^2 - x^3)$ with $\operatorname{char} k \ne 2, 3$, $\Omega_{X/k}$ is generated by $dx, dy$ with the relation $2y \, dy = 3x^2 \, dx$. Its fiber at the origin has dimension $2$, while the generic rank is $1$, so $\Omega_{X/k}$ is not locally free there. An irreducible variety $X$ over a perfect field $k$ is smooth over $k$ if and only if $\Omega_{X/k}$ is locally free of rank $\dim X$.