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 ΩA/R together with an R-linear derivation
d:A→ΩA/R, such that the following universal property
holds:
for every A-module M and every R-linear derivation
δ:A→M there exists a unique A-linear map f:ΩA/R→M such that δ=f∘d.
The universal property can be illustrated by a commutative diagram:
\xymatrixA\ar[r]d |
An -module with this property can be constructed explicitly, so
always exists. It is generated as an -module by
the set , with the relations
for all and .
The universal property implies that is unique up to
a unique isomorphism. The -module 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 |