# universal derivation

Let $R$ be a commutative ring, and let $A$ be a commutative $R$-algebra. A universal derivation of $A$ over $R$ is defined to be an $A$-module $\Omega_{A/R}$ together with an $R$-linear derivation $d\colon A\to\Omega_{A/R}$, such that the following universal property holds: for every $A$-module $M$ and every $R$-linear derivation $\delta\colon A\to M$ there exists a unique $A$-linear map $f\colon\Omega_{A/R}\to M$ such that $\delta=f\circ d$.

The universal property can be illustrated by a commutative diagram:

 $\xymatrix{A\ar[r]^{d\ \ }\ar[dr]_{\delta}&\Omega_{A/R}\ar@![d]^{f}\default@cr&M}$

An $A$-module with this property can be constructed explicitly, so $\Omega_{A/R}$ always exists. It is generated as an $A$-module by the set $\{dx:x\in A\}$, with the relations

 $\displaystyle d(ax+by)$ $\displaystyle=$ $\displaystyle a\,dx+b\,dy$ $\displaystyle d(xy)$ $\displaystyle=$ $\displaystyle x\cdot dy+y\,dx$

for all $a,b\in R$ and $x,y\in A$.

The universal property implies that $\Omega_{A/R}$ is unique up to a unique isomorphism. The $A$-module $\Omega_{A/R}$ is often called the module of Kähler differentials.

Title universal derivation UniversalDerivation 2013-03-22 15:27:57 2013-03-22 15:27:57 pbruin (1001) pbruin (1001) 9 pbruin (1001) Definition msc 13N15 msc 13N05 Kähler differentials Derivation