PlanetMath: universal derivation

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universal derivation

(Definition)

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

$\begin{displaymath}\begin{xy} *!C\xybox{ \xymatrix{ A \ar[r]^{d\ \ } \ar[dr]_\delta & \Omega_{A/R} \ar@![d]^f \cr & M } } \end{xy}\end{displaymath}$

-module with this property can be constructed explicitly, so $\Omega_{A/R}$ always exists. It is generated as an

-module by the set $\{dx:x\in A\}$ , with the relations

$\begin{eqnarray*} d(ax+by)&=&a\,dx+b\,dy \ d(xy)&=&x\cdot dy+y\,dx \end{eqnarray*}$

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 -module $\Omega_{A/R}$ is often called the module of Kähler differentials.

"universal derivation" is owned by pbruin.

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See Also: derivation

Other names:

Kähler differentials

Keywords:

derivation

Pronunciation (guide):

Kahler: /kay''''ler/

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Cross-references: module, implies, relations, property, commutative diagram, map, universal property, derivation, commutative, commutative ring

This is version 6 of universal derivation, born on 2005-08-12, modified 2005-08-15.
Object id is 7318, canonical name is UniversalDerivation.
Accessed 1891 times total.

Classification:

AMS MSC:	13N05 (Commutative rings and algebras :: Differential algebra :: Modules of differentials)
	13N15 (Commutative rings and algebras :: Differential algebra :: Derivations)

Pending Errata and Addenda

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