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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==
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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.
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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]]).
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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 Leibniz rule <math>\mathrm d (fg) = f \mathrm \, \mathrm dg + g \, \mathrm df</math>. The 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.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Construction==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The idea is now to give a [[universal construction]] of a ''[[derivation (abstract algebra)|derivation]]''
'''source'''
:d:''S'' &rarr; &Omega;<sup>1</sup><sub>''S''/''R''</sub>
:<math forcemathmode="source">E=mc^2</math> -->
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
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
*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
==Demos==


:<math>\mathrm ds = 1 \otimes s - s \otimes 1. \, </math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==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]]'' functions vanishing at least to second order (see [[Cotangent space#Alternative definition|cotangent space]] for related notions).
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


For any ''S''-module ''M'', the universal property of Ω<sup>1</sup><sub>''S''/''R''</sub> leads to a natural isomorphism
==Test pages ==


:<math>\operatorname{Der}_R(S,M)\cong \operatorname{Hom}_S(\Omega^1_{S/R},M), \,</math>
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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.
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
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, over any [[field (mathematics)|field]]&nbsp;''R''.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
==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 }}
 
==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]]
 
[[de:Kähler-Differential]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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