Kähler differential

Question

I'm reading section III.2 of "Algebraic Number Theory" by J.Neukirch and I don't know why Kähler differential is a generalization of (usual,analytic) differential.

$\mathbf{Definition}$:Let $B/A$ be an extension of commutative rings and $I$ be the kernel of the morphism $$\mu:B\otimes_AB\to B;x\otimes y\mapsto xy$$We define the module of Kähler differentials as $\Omega^{1}_{B/A}:=I/I^2=I\otimes_{B\otimes_A B}B$.

Answer 1

Not sure this answers exactly your question, but we have a map \begin{align} D: B&\longrightarrow \Omega^1_{B/A}\\ x&\longmapsto 1\otimes x-x\otimes x, \end{align} which is an $A$-derivation, i.e which

1. is $A$-linear,
2. satisfies the relation $\; D(xy)=D(x)y+xD(y)$ for all $x,y\in B$.

Furthermore, from a categorical point of view, this module represents the functor $\operatorname{Der}_A(B,-)$, i.e. for any $B$-module $M$, we have an isomorphism $$\operatorname{Der}_A(B,M)\simeq\operatorname{Hom}_B\bigl(\Omega^1_{B/A},M\bigr).$$