Derivation (mathematics)

From Citizendium, the Citizens' Compendium

Jump to: navigation, search

Main Article

Talk

Related Articles ^[?]

Bibliography ^[?]

External Links ^[?]

This is a draft article, under development and not meant to be cited but you can help to improve it. These unapproved articles are subject to a disclaimer.

[edit intro]

In mathematics, a derivation is a map which has formal algebraic properties generalising those of the derivative.

Let R be a ring and A an R-algebra (A is a ring containing a copy of R in the centre). A derivation is an R-linear map D from A to some A-module M with the property that

$D(ab) = a \cdot D(b) + D(a) \cdot b .\,$

The constants of D are the elements mapped to zero. The constants include the copy of R inside A.

A derivation "on" A is a derivation from A to A.

Linear combinations of derivations are again derivations, so the derivations from A to M form an R-module, denoted Der_R(A,M).

Examples

The zero map is a derivation.
The formal derivative is a derivation on the polynomial ring R[X] with constants R.

Universal derivation

There is a universal derivation (Ω,d) with a universal property. Given a derivation D:A → M, there is a unique A-linear f:Ω → M such that D = d·f. Hence

$\operatorname{Der}_R(A,M) = \operatorname{Hom}_A(\Omega,M) \,$

as a functorial isomorphism.

Consider the multiplication map μ on the tensor product (over R)

$\mu : A \otimes A \rightarrow A \,$

defined by $\mu : a \otimes b \mapsto ab$ . Let J be the kernel of μ. We define the module of differentials

$\Omega_{A/R} = J/J^2 \,$

as an ideal in $(A \otimes A)/J^2$ , where the A-module structure is given by A acting on the first factor, that is, as $A \otimes 1$ . We define the map d from A to Ω by

$d : a \mapsto 1 \otimes a - a \otimes 1 \pmod{J^2} .\,$ .

This is the universal derivation.

Kähler differentials

A Kähler differential, or formal differential form, is an element of the universal derivation space Ω, hence of the form Σ_i x_i dy_i. An exact differential is of the form $dy$ for some y in A. The exact differentials form a submodule of Ω.