Sheaf of Kähler differentials over a field (${\Omega }_{X/k}^1$)

For a variety over $k$, the module of Kähler differentials ${\Omega }_{A/k}$ on each affine chart gives a quasi-coherent sheaf. Compatibility with localization glues these into ${\Omega }_{X/k}^1$. A finite presentation of an algebra gives a finite presentation of a module for its differentials, so this is a coherent sheaf on a variety.

For a smooth algebraic variety of dimension $r$ over a field, the sheaf of Kähler differentials over a field is locally free of rank $r$ and defines its algebraic cotangent bundle. Its dual is the tangent sheaf of a scheme. For a smooth hypersurface of degree $d$ in ${\mathbb{P}}^n$, the Conormal exact sequence for Kähler differentials is the short exact sequence $0\to {\mathcal{O}}_X(-d)\to {\Omega }_{{\mathbb{P}}^n}^1{|}_X\to {\Omega }_X^1\to 0$.

An invertible maximal-rank minor in the Jacobian matrix eliminates generators and gives a local surjection from ${\mathcal{O}}_X^n$ to the Kähler differential sheaf. For an integral smooth variety the map is an isomorphism at the generic point. Its kernel is a submodule of a free module over an integral domain, so it is a torsion-free module; vanishing generically forces it to be zero. This gives local free frames for differentials.