universal derivation

Let R be a commutative ring, and let A be a commutativePlanetmathPlanetmathPlanetmathPlanetmath R-algebraMathworldPlanetmathPlanetmath. A universal derivation of A over R is defined to be an A-module ΩA/R together with an R-linear derivationPlanetmathPlanetmath d:AΩA/R, such that the following universal propertyMathworldPlanetmath holds: for every A-module M and every R-linear derivation δ:AM there exists a unique A-linear map f:ΩA/RM such that δ=fd.

The universal property can be illustrated by a commutative diagramMathworldPlanetmath:

\xymatrixA\ar[r]d\ar[dr]δ&ΩA/R\ar@![d]f &M

An A-module with this property can be constructed explicitly, so ΩA/R always exists. It is generated as an A-module by the set {dx:xA}, with the relationsMathworldPlanetmathPlanetmathPlanetmath

d(ax+by) = adx+bdy
d(xy) = xdy+ydx

for all a,bR and x,yA.

The universal property implies that ΩA/R is unique up to a unique isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The A-module ΩA/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