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universal derivation
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(Definition)
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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:
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 \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 $A$ -module $\Omega_{A/R}$ is often called the module of Kähler differentials.
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"universal derivation" is owned by pbruin.
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See Also: derivation
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Kähler differentials |
Pronunciation (guide):
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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 2499 times total.
Classification:
| AMS MSC: | 13N05 (Commutative rings and algebras :: Differential algebra :: Modules of differentials) | | | 13N15 (Commutative rings and algebras :: Differential algebra :: Derivations) |
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Pending Errata and Addenda
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![\begin{displaymath} \xymatrix{ A \ar[r]^{d\ \ } \ar[dr]_\delta & \Omega_{A/R} \ar@![d]^f \cr & M } \end{displaymath}](http://images.planetmath.org:8080/cache/objects/7318/js/img1.png "\begin{displaymath} \xymatrix{ A \ar[r]^{d\ \ } \ar[dr]_\delta & \Omega_{A/R} \ar@![d]^f \cr & M } \end{displaymath}")