Sheaf of differentials

Summary

The sheaf of relative differentials $\Omega_{X/Y}$ of a morphism of schemes $X \to Y$ is the sheaf version of the module of Kähler differentials. On an affine piece $\operatorname{Spec} B \to \operatorname{Spec} A$ it is given by the $B$-module $\Omega_{B/A}$. For a smooth variety it is a vector bundle, the cotangent bundle, whose dual is the tangent bundle. Its exterior powers give the sheaves of differential forms, and the top one is the canonical bundle.