 Kähler differential

For a ring map $A\to B$, the module of Kähler differentials represents $A$-derivations: ${\mathrm{Hom}}_B({\Omega }_{B/A},M)\cong {\mathrm{Der}}_A(B,M)$.

For an algebra $B$ over $A$, the module ${\Omega }_{B/A}$ represents $A$-linear derivations: every derivation $B\to N$ factors uniquely through the universal map $d:B\to {\Omega }_{B/A}$.

Kähler differentials are a concept from algebraic geometry and commutative algebra. They arise in the context of the study of a ring \( R \) and its associated differentials with respect to a base field or a base ring. Specifically, Kähler differentials provide a way to study the infinitesimal behavior of functions and their properties on schemes.