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 ${\mathrm{\Omega }}_{A/R}$ together with an $R$-linear derivation $d:A\to {\mathrm{\Omega }}_{A/R}$, such that the following universal property holds: for every $A$-module $M$ and every $R$-linear derivation $\delta :A\to M$ there exists a unique $A$-linear map $f:{\mathrm{\Omega }}_{A/R}\to M$ such that $\delta =f\circ d$.

The universal property can be illustrated by a commutative diagram:

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

$d(ax+by)$ $=$ $adx+bdy$

$d(xy)$ $=$ $x\cdot dy+ydx$

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

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

Title	universal derivation
Canonical name	UniversalDerivation
Date of creation	2013-03-22 15:27:57
Last modified on	2013-03-22 15:27:57
Owner	pbruin (1001)
Last modified by	pbruin (1001)
Numerical id	9
Author	pbruin (1001)
Entry type	Definition
Classification	msc 13N15
Classification	msc 13N05
Synonym	Kähler differentials
Related topic	Derivation