Conormal sheaf (${\mathcal{C}}_{W/V}={\mathcal{I}}_W/{\mathcal{I}}_W^2$)

For a closed immersion, this quasi-coherent sheaf measures the first-order equations of the subvariety. Its local module is $I/I^2$, and its map to the restricted Kähler differential sheaf sends the class of $f$ to $df$. This is the definition in Stacks Project, Section 29.32.

Let $W$ be an integral effective Cartier divisor in an integral variety $V$ over a perfect field, and suppose $W$ is not contained in the singular locus of $V$. On a dense open subset both varieties are smooth. The Zariski tangent space of $W$ there has codimension one in that of $V$, so the differential of the local defining equation is nonzero. Thus the conormal sheaf map is injective generically. Its kernel is a subsheaf of a line bundle on the integral variety $W$ and is therefore a torsion-free sheaf; generic vanishing implies zero everywhere. Right exactness is the Conormal exact sequence for Kähler differentials.