= Local freeness of differentials on a smooth variety
{title2=$X\text{ smooth of dimension }n\Longrightarrow\Omega^1_{X/k}\text{ locally free of rank }n$}
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.
Back to article page
