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

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).


Universal derivation

There is a universal derivation (Ω,d) with a universal property. Given a derivation D:AM, 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 Σi xi dyi. An exact differential is of the form dy for some y in A. The exact differentials form a submodule of Ω.

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