# 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^{}:

$$\text{xymatrix}A\text{ar}{[r]}^{d\mathit{\hspace{1em}}}\text{ar}{[dr]}_{\delta}\mathrm{\&}{\mathrm{\Omega}}_{A/R}\text{ar}\mathrm{@}!{[d]}^{f}\mathit{}\mathrm{\&}M$$ |

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 |