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In [[mathematics]], '''Kähler differentials''' provide an adaptation of [[differential form]]s to arbitrary [[commutative ring]]s or [[scheme (mathematics)|scheme]]s.
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==Presentation==
The idea was introduced by [[Erich Kähler]] in the 1930s. It was adopted as standard, in [[commutative algebra]] and [[algebraic geometry]], somewhat later, following the need to adapt methods from geometry over the [[complex number]]s, and the free use of [[calculus]] methods, to contexts where such methods are not available.
 
Let ''R'' and ''S'' be commutative rings and ''φ'':''R'' → ''S'' a [[ring homomorphism]]. An important example is for ''R'' a [[Field (mathematics)|field]] and ''S'' a unital [[associative algebra|algebra]] over ''R'' (such as the [[coordinate ring]] of an [[affine variety]]).
 
An ''R''-linear derivation on ''S'' is a [[morphism]] of ''R''-modules <math>\mathrm d \colon S \to M</math> with ''R'' in its kernel, and satisfying [[Product rule|Leibniz rule]] <math>\mathrm d (fg) = f \mathrm \, \mathrm dg + g \, \mathrm df</math>. The [[module (mathematics)|module]] of Kähler differentials is defined as the ''R''-linear derivation <math>\mathrm d \colon S \to \Omega_{S/R}</math> that factors all others.
 
==Construction==
 
The idea is now to give a [[universal construction]] of a ''[[derivation (abstract algebra)|derivation]]''
:d:''S'' &rarr; &Omega;<sup>1</sup><sub>''S''/''R''</sub>
over ''R'', where Ω<sup>1</sup><sub>''S''/''R''</sub> is an ''S''-[[Module (mathematics)|module]], which is a purely algebraic analogue of the [[exterior derivative]]. This means that d is a homomorphism of ''R''-modules such that
:d(''st'') = ''s'' d''t'' + ''t'' d''s''
for all ''s'' and ''t'' in ''S'', and d is the ''best possible'' such derivation in the sense that any other derivation may be obtained from it by composition with an ''S''-module homomorphism.
 
The actual construction of Ω<sup>1</sup><sub>''S''/''R''</sub> and d can proceed by introducing formal generators d''s'' for ''s'' in ''S'', and imposing the relations
*d''r'' = 0 for ''r'' in ''R'',
*d(''s'' + ''t'') = d''s'' + d''t'',
*d(''st'') = ''s'' d''t'' + ''t'' d''s''
for all ''s'' and ''t'' in ''S''.
 
Another construction proceeds by letting ''I'' be the ideal in the [[tensor product]] <math>S \otimes_R S</math>, defined as the [[Kernel (algebra)|kernel]] of the multiplication map: <math>S \otimes_R S\to S</math>, given by <math>\Sigma s_i \otimes t_i \mapsto \Sigma s_i.t_i</math>. Then the module of Kähler differentials of "S" can be equivalently defined by<ref name=N200>Neukirch (1999) p.200</ref> Ω<sup>1</sup><sub>''S''/''R''</sub> = ''I''/''I''<sup>2</sup>, together with the morphism
 
:<math>\mathrm ds = 1 \otimes s - s \otimes 1. \, </math>
 
To see that this construction is equivalent to the previous one, note that ''I'' is the kernel of the projection <math>S \otimes_R S\to S \otimes_R R</math>, given by <math>\Sigma s_i \otimes t_i \mapsto \Sigma s_i.t_i\otimes 1</math>.  Thus we have:
 
:<math>S \otimes_R S \equiv\,\,{} I \,{} \oplus S \otimes_R R.\, </math>
 
Then <math>S \otimes_R S/ S\otimes_R R </math> may be identified with ''I'', by the map induced by the complimentary projection which is given by <math>\Sigma s_i \otimes t_i \mapsto  \Sigma s_i \otimes t_i-\Sigma s_i.t_i\otimes 1</math>.
 
Thus this map identifies ''I'' with the ''S'' module generated by the formal generators d''s'' for ''s'' in ''S'', subject to the first two relations given above (with the second relation strengthened to demanding that d is ''R''-linear).  The elements set to zero by the final relation map to precisely ''I''<sup>2</sup> in ''I''.
 
==Use in algebraic geometry==
 
Geometrically, in terms of [[affine scheme]]s, ''I'' represents the ''ideal defining the [[diagonal]]'' in the [[fiber product]] of Spec(''S'') with itself over Spec(''S'')&nbsp;→&nbsp;Spec(''R''). This construction therefore has a more geometric flavor, in the sense that the notion of ''first infinitesimal neighbourhood'' of the diagonal is thereby captured, via functions vanishing ''[[modulo (jargon)|modulo]]'' functions vanishing at least to second order (see [[Cotangent space#Alternative definition|cotangent space]] for related notions).
 
For any ''S''-module ''M'', the universal property of Ω<sup>1</sup><sub>''S''/''R''</sub> leads to a natural isomorphism
 
:<math>\operatorname{Der}_R(S,M)\cong \operatorname{Hom}_S(\Omega^1_{S/R},M), \,</math>
 
where the left hand side is the ''S''-module of all ''R''-linear derivations from ''S'' to ''M''. As in the case of [[adjoint functors]] (though this isn't an adjunction), this is more than just an isomorphism of modules; it commutes with S-module homomorphisms ''M'' → ''M''' and hence is an isomorphism of functors.
 
To get Ω<sup>''p''</sup><sub>''S''/''R''</sub>, the Kähler ''p''-forms for ''p'' > 1, one takes the ''R''-module [[exterior power]] of degree ''p''. The behaviour of the construction under [[localization of a ring]] (applied to ''R'' and ''S'') ensures that there is a geometric notion of ''[[sheaf (mathematics)|sheaf]] of (relative) Kähler p-forms'' available for use in algebraic geometry.
 
==Use in algebraic number theory==
In [[algebraic number theory]], the Kähler differentials may be used to study the [[ramification#In algebraic number theory|ramification]] in an extension of [[algebraic number fields]]. If ''L''/''K'' is a finite extension with rings of integers ''O'' and ''o'' respectively then the [[different ideal]] δ<sub>''L''/''K''</sub>, which encodes the ramification data, is the annihilator of the ''O''-module Ω<sup>1</sup><sub>''O''/''o''</sub>:<ref name=N201>Neukirch (1999) p.201</ref>
 
<math>\delta_{L/K} = \{ x \in O : x \mathrm{d} y = 0 \text{ for all } y \in O \} . </math>
 
==References==
{{reflist}}
* {{cite journal | first=J. | last=Johnson | year=1969 | title=Kähler differentials and differential algebra | journal=[[Annals of Mathematics]] | volume=89 | pages=92–98 | zbl=0179.34302 }}
* {{Neukirch ANT}}
* {{cite journal | first=M. | last=Rosenlicht | year=1976 | title=On Liouville's theory of elementary functions | journal=Pacific J. Math. | volume=65 | pages=485–492 | zbl=0318.12107 }}
* {{cite journal | first=G., et al | last=Fu et al.| year=2011 | title=Some remarks on Kähler differentials and and ordinary differentials in nonlinear control systems| journal=[[Systems and Control Letters]] | volume=60 | pages=699–703 }}
 
==External links==
* A [http://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials thread] devoted to the question on [[MathOverflow]]
 
{{DEFAULTSORT:Kahler Differential}}
[[Category:Commutative algebra]]
[[Category:Differential algebra]]
[[Category:Algebraic geometry]]

Latest revision as of 19:07, 10 January 2015

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