Derivation (mathematics)

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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

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

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 Der_R(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

${\displaystyle \operatorname {Der} _{R}(A,M)=\operatorname {Hom} _{A}(\Omega ,M)\,}$

as a functorial isomorphism.

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

${\displaystyle \mu :A\otimes A\rightarrow A\,}$

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

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

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

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

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 Σ_i x_i dy_i. An exact differential is of the form ${\displaystyle dy}$ for some y in A. The exact differentials form a submodule of Ω.