Derivation (mathematics): Difference between revisions
imported>Richard Pinch (new entry, expanding redirect) |
mNo edit summary |
||
(8 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
{{subpages}} | |||
In [[mathematics]], a '''derivation''' is a map which has formal algebraic properties generalising those of the [[derivative]]. | 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 of a ring|centre]]). A derivation is an ''R''-linear map ''D'' with the property that | Let ''R'' be a [[ring (mathematics)|ring]] and ''A'' an ''R''-algebra (''A'' is a ring containing a copy of ''R'' in the [[centre of a ring|centre]]). A derivation is an ''R''-linear map ''D'' from ''A'' to some ''A''-module ''M'' with the property that | ||
:<math>D(ab) = a | :<math>D(ab) = a \cdot D(b) + D(a) \cdot b .\,</math> | ||
The ''constants'' of ''D'' are the elements mapped to zero. The constants include the copy of ''R'' inside ''A''. | The ''constants'' of ''D'' are the elements mapped to zero. The constants include the copy of ''R'' inside ''A''. | ||
Line 13: | Line 14: | ||
==Examples== | ==Examples== | ||
* The [[zero map]] is a derivation. | * The [[zero map]] is a derivation. | ||
* The [[formal derivative]] is a derivation on the polynomial ring ''R''[''X''] with constants ''R''. | * The [[formal derivative]] is a derivation on the [[polynomial ring]] ''R''[''X''] with constants ''R''. | ||
==Universal derivation== | ==Universal derivation== | ||
There is a ''universal'' derivation Ω such that | 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 | ||
:<math> \operatorname{Der}_R(A,M) = \operatorname{Hom}_A(\Omega,M) \,</math> | :<math> \operatorname{Der}_R(A,M) = \operatorname{Hom}_A(\Omega,M) \,</math> | ||
as a [[functor]]ial isomorphism. | as a [[functor]]ial isomorphism. | ||
= | Consider the multiplication map μ on the [[tensor product]] (over ''R'') | ||
:<math>\mu : A \otimes A \rightarrow A \,</math> | |||
defined by <math>\mu : a \otimes b \mapsto ab</math>. Let ''J'' be the [[kernel]] of μ. We define the module of ''differentials'' | |||
:<math>\Omega_{A/R} = J/J^2 \,</math> | |||
as an ideal in <math>(A \otimes A)/J^2</math>, where the ''A''-module structure is given by ''A'' acting on the first factor, that is, as <math>A \otimes 1</math>. We define the map ''d'' from ''A'' to Ω by | |||
:<math>d : a \mapsto 1 \otimes a - a \otimes 1 \pmod{J^2} .\,</math>. | |||
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 Σ<sub>''i''</sub> ''x<sub>i</sub>'' ''dy<sub>i</sub>''. An ''exact'' differential is of the form <math>dy</math> for some ''y'' in ''A''. The exact differentials form a submodule of Ω.[[Category:Suggestion Bot Tag]] |
Latest revision as of 12:00, 6 August 2024
![](http://s9.addthis.com/button1-share.gif)
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
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·f. Hence
as a functorial isomorphism.
Consider the multiplication map μ on the tensor product (over R)
defined by . Let J be the kernel of μ. We define the module of differentials
as an ideal in , where the A-module structure is given by A acting on the first factor, that is, as . We define the map d from A to Ω by
- .
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 for some y in A. The exact differentials form a submodule of Ω.