# Kähler differentials

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Let ${\displaystyle A\to B}$ be an algebra. An A differential of B into an ${\displaystyle A}$-module ${\displaystyle M}$ is a map ${\displaystyle D:B\to M}$ such that
1. ${\displaystyle D(a)=0}$ for all ${\displaystyle a\in A}$
2. ${\displaystyle D(b+b')=D(b)+D(b')}$ for ${\displaystyle b,b'\in B}$
3. ${\displaystyle D(bb')=b'D(b)+bD(b')}$
Observe that the set of all such maps ${\displaystyle Der_{A}(B,M)}$ is a ${\displaystyle B}$-module. Moreover, ${\displaystyle Der_{A}(B,-)}$ is a representable functor; we call the representative ${\displaystyle \Omega _{B/A}}$ the module of Kähler differentials. That is, ${\displaystyle \Omega _{B/A}}$ satisfies the following universal property: