Derivation (mathematics)

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 DerR(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&middot;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 &Sigma;i xi dyi. An exact differential is of the form $$dy$$ for some y in A. The exact differentials form a submodule of Ω.