Derivation (mathematics)

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This editable Main Article is under development and subject to a disclaimer.

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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 ( a b ) = a ⋅ D ( b ) + D ( a ) ⋅ b . {\displaystyle 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[edit]

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

## Universal derivation[edit]

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

Der R ⁡ ( A , M ) = Hom A ⁡ ( Ω , M ) {\displaystyle \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)

μ : A ⊗ A -> A {\displaystyle \mu :A\otimes A\rightarrow A\,}

defined by  μ : a ⊗ b ↦ a b {\displaystyle \mu :a\otimes b\mapsto ab} . Let J be the kernel of μ. We define the module of differentials

Ω A / R = J / J 2 {\displaystyle \Omega _{A/R}=J/J^{2}\,}

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

d : a ↦ 1 ⊗ a − a ⊗ 1 ( mod J 2 ) . {\displaystyle d:a\mapsto 1\otimes a-a\otimes 1{\pmod {J^{2}}}.\,} .

This is the universal derivation.

## Kähler differentials[edit]

A Kähler differential, or formal differential form, is an element of the universal derivation space Ω, hence of the form Σi xi dyi. An exact differential is of the form  d y {\displaystyle dy} for some y in A. The exact differentials form a submodule of Ω.