Derivation (mathematics)

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

Let R be a ring (mathematics) 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

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

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 $\sum _{i}x_{i}dy_{i}$. An exact differential is of the form $dy$ for some y in A. The exact differentials form a submodule of Ω.

References

• David M. Goldschmidt (2003). Algebraic Functions and Projective Curves. Springer-Verlag, 24-30. ISBN 0-387-95432-5. 
• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 746-749. ISBN 0-201-55540-9.