Relative Kähler differentials

For a homomorphism of commutative rings $A\to B$ , the module of relative Kähler differentials $\Omega_{B/A}$ is the $B$ -module equipped with an $A$ -derivation

d:B\longrightarrow \Omega_{B/A}

that is universal: every $A$ -derivation from $B$ to a $B$ -module $M$ factors uniquely through a $B$ -linear map $\Omega_{B/A}\to M$ .

Concrete presentation

The module $\Omega_{B/A}$ is generated by symbols $db$ , subject to

d(b+b')=db+db',\qquad d(bb')=b\,db'+b'\,db,\qquad da=0\quad(a\in A).

For example, $\Omega_{A[x]/A}$ is a free $A[x]$ -module with basis $dx$ . If $B=A[x]/(f)$ , then the relation $df=f'(x)\,dx=0$ is imposed after passing to $B$ .

Geometric meaning

For a morphism of schemes $f:Y\to X$ , these affine modules glue to the sheaf of relative Kähler differentials $\Omega_{Y/X}$ . On an affine piece $\operatorname{Spec}B\to\operatorname{Spec}A$ , it is the sheaf associated to $\Omega_{B/A}$ . Thus $\Omega_{Y/X}=0$ precisely when this sheaf is zero on every affine piece.

The fiber of $\Omega_{Y/X}$ at a point is dual to the relative tangent directions there. Its vanishing is therefore the infinitesimal part of being unramified.