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en>Niceguyedc: WPCleaner v1.33 - Repaired 2 links to disambiguation pages - (You can help) - Involution, Subfield

2014-08-17T19:27:28Z

WPCleaner v1.33 - Repaired 2 links to disambiguation pages - (You can help) - Involution, Subfield

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	~~In [[mathematics]], the '''residue field''' is a basic construction in [[commutative algebra]]~~. ~~If ''R''~~ is ~~a [[commutative ring]]~~ and ~~''m'' is a [[maximal ideal]], then the residue field is the [[quotient ring]] ''k'' = ''R''/''m'', which is a [[field (mathematics)\|field]]~~. ~~Frequently, ''R'' is a [[local ring]]~~ and ~~''m'' is then its unique maximal ideal.~~		Friends contact her Claude Gulledge. Managing people is how she tends to make cash and she will not alter it anytime soon. The factor I adore most flower arranging and now I have time to consider on new things. Arizona has usually been my residing place but my wife desires us to move.<br><br>Feel free to visit my web blog [http://www.shownetbook.com/enone/xe/?document_srl=3029968 www.shownetbook.com]

	~~This construction is applied in [[algebraic geometry]], where~~ to ~~every point ''x'' of a [[scheme (mathematics)\|scheme]] ''X'' one associates its '''residue field''' ''k''(''x'')~~. ~~One can say a little loosely that the residue field of a point of an abstract [[algebraic variety]] is the 'natural domain' for the coordinates of the point.~~

	~~==Definition==~~
	~~Suppose that ''R'' is a commutative [[local ring]], with the maximal ideal ''m''. Then the '''residue field''' is the quotient ring ''R''/''m''.~~

	Now suppose that ''X'' is a [[scheme (mathematics)\|scheme]] and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some [[commutative ring]]. Considered in the neighbourhood ''U'', the point ''x'' corresponds to ~~a [[prime ideal]] ''p'' ⊂ ''A'' (see [[Zariski topology]])~~. ~~The ''[[local ring]]'' of ''X'' in ''x'' is by definition the [[localization of a ring\|localization]] ''R'' = ''A~~<~~sub~~>p<~~/sub~~>~~'', with the maximal ideal ''m'' = ''p·A<sub>p</sub>''. Applying the construction above, we obtain the '''residue field of the point ''x'' ''':~~

	~~:''k''(''x'')~~ :~~= ''A''<sub>''p''</sub> / ''p''·''A''<sub>''p''</sub>.~~

	One can prove that this definition does not depend on the choice of the affine neighbourhood ''U''.<ref>Intuitively, the residue field of a point is a local invariant. Axioms of schemes are set up in such a way as to assure the compatibility between various affine open neighborhoods of a point, which implies the statement.</~~ref>~~

	~~A point is called [[rational point\|''K''-rational]] for a certain field ''K'', if ''k''(''x'') ⊂ ''K''.~~

	~~==Example==~~
	~~Consider the [[affine line]] '''A'''<sup>1<~~/~~sup>(''k'') = Spec(''k''[''t'']) over a [[field (mathematics)\|field]] ''k''~~. ~~If ''k'' is [[algebraically closed field\|algebraically closed]], there are exactly two types of prime ideals, namely~~

	*(''t'' − ''a''), ''a'' ∈ ''k''
	*(0), the zero-ideal.

	~~The residue fields are~~

	*<math>k[t]_{(t-a)}/~~(t-a)k[t]_{(t-a)} \cong k<~~/~~math>~~
	*<math>k[t]_{(0)} \cong k(t)</~~math>, the function field over ''k'' in one variable.~~

	~~If ''k'' is not algebraically closed, then more types arise, for example if ''k''~~ = ~~'''R''', then the prime ideal (''x''<sup>2</sup> + 1) has residue field isomorphic to '''C'''~~.

	~~==Properties==~~
	* For a scheme locally of [[morphism of finite type\|finite type]] over a field ''k'', a point ''x'' is closed if and only if ''k''(''x'') is a finite extension of the base field ''k''. This is a geometric formulation of [[Hilbert's Nullstellensatz]]. In the above example, the points of the first kind are closed, having residue field ''k'', whereas the second point is the [[generic point]], having [[transcendence degree]] 1 over ''k''.
	* A morphism Spec(''K'') → ''X'', ''K'' some field, is equivalent to giving a point ''x'' ∈ ''X'' and an [[field extension\|extension]] ''K''/''k''(''x'').
	* The [[Krull dimension\|dimension]] of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.

	~~== Notes ==~~

	~~{{reflist}}~~

	~~==References==~~
	* {{Citation \| last1=Hartshorne \| first1=Robin \| author1-link = Robin Hartshorne \| title=[[Algebraic Geometry (book)\|Algebraic Geometry]] \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90244-9 \| id={{MathSciNet \| id = 0463157}} \| year=1977}}, section II.2

	~~[[Category:Algebraic geometry\|*]~~]

en>ClueBot NG: Reverting possible vandalism by 94.20.43.208 to version by The Disambiguator. False positive? Report it. Thanks, ClueBot NG. (1649336) (Bot)

2013-05-23T23:56:58Z

Reverting possible vandalism by 94.20.43.208 to version by The Disambiguator. False positive? Report it. Thanks, ClueBot NG. (1649336) (Bot)

← Older revision		Revision as of 01:56, 24 May 2013
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	~~Friends contact him Royal Cummins~~. ~~The factor I adore most bottle tops collecting~~ and ~~now I have time~~ to ~~take on new issues~~. ~~Her family members life~~ in ~~Idaho~~. ~~My occupation~~ is a ~~messenger~~.<br><br>~~my site~~ ... [~~http:~~/~~/www~~.~~frag5~~.~~com~~/~~index~~.~~php?mod~~=~~users~~&~~action~~=~~view&id~~=~~13770 http://www~~.~~frag5~~.~~com~~/~~index~~.~~php?mod~~=~~users&action~~=~~view&~~id=~~13770~~]		In [[mathematics]], the '''residue field''' is a basic construction in [[commutative algebra]]. If ''R'' is a [[commutative ring]] and ''m'' is a [[maximal ideal]], then the residue field is the [[quotient ring]] ''k'' = ''R''/''m'', which is a [[field (mathematics)\|field]]. Frequently, ''R'' is a [[local ring]] and ''m'' is then its unique maximal ideal.

			This construction is applied in [[algebraic geometry]], where to every point ''x'' of a [[scheme (mathematics)\|scheme]] ''X'' one associates its '''residue field''' ''k''(''x''). One can say a little loosely that the residue field of a point of an abstract [[algebraic variety]] is the 'natural domain' for the coordinates of the point.

			==Definition==
			Suppose that ''R'' is a commutative [[local ring]], with the maximal ideal ''m''. Then the '''residue field''' is the quotient ring ''R''/''m''.

			Now suppose that ''X'' is a [[scheme (mathematics)\|scheme]] and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some [[commutative ring]]. Considered in the neighbourhood ''U'', the point ''x'' corresponds to a [[prime ideal]] ''p'' ⊂ ''A'' (see [[Zariski topology]]). The ''[[local ring]]'' of ''X'' in ''x'' is by definition the [[localization of a ring\|localization]] ''R'' = ''A<sub>p</sub>'', with the maximal ideal ''m'' = ''p·A<sub>p</sub>''. Applying the construction above, we obtain the '''residue field of the point ''x'' ''':

			:''k''(''x'') := ''A''<sub>''p''</sub> / ''p''·''A''<sub>''p''</sub>.

			One can prove that this definition does not depend on the choice of the affine neighbourhood ''U''.<ref>Intuitively, the residue field of a point is a local invariant. Axioms of schemes are set up in such a way as to assure the compatibility between various affine open neighborhoods of a point, which implies the statement.</ref>

			A point is called [[rational point\|''K''-rational]] for a certain field ''K'', if ''k''(''x'') ⊂ ''K''.

			==Example==
			Consider the [[affine line]] '''A'''<sup>1</sup>(''k'') = Spec(''k''[''t'']) over a [[field (mathematics)\|field]] ''k''. If ''k'' is [[algebraically closed field\|algebraically closed]], there are exactly two types of prime ideals, namely

			*(''t'' − ''a''), ''a'' ∈ ''k''
			*(0), the zero-ideal.

			The residue fields are

			*<math>k[t]_{(t-a)}/(t-a)k[t]_{(t-a)} \cong k</math>
			*<math>k[t]_{(0)} \cong k(t)</math>, the function field over ''k'' in one variable.

			If ''k'' is not algebraically closed, then more types arise, for example if ''k'' = '''R''', then the prime ideal (''x''<sup>2</sup> + 1) has residue field isomorphic to '''C'''.

			==Properties==
			* For a scheme locally of [[morphism of finite type\|finite type]] over a field ''k'', a point ''x'' is closed if and only if ''k''(''x'') is a finite extension of the base field ''k''. This is a geometric formulation of [[Hilbert's Nullstellensatz]]. In the above example, the points of the first kind are closed, having residue field ''k'', whereas the second point is the [[generic point]], having [[transcendence degree]] 1 over ''k''.
			* A morphism Spec(''K'') → ''X'', ''K'' some field, is equivalent to giving a point ''x'' ∈ ''X'' and an [[field extension\|extension]] ''K''/''k''(''x'').
			* The [[Krull dimension\|dimension]] of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.

			== Notes ==

			{{reflist}}

			==References==
			* {{Citation \| last1=Hartshorne \| first1=Robin \| author1-link = Robin Hartshorne \| title=[[Algebraic Geometry (book)\|Algebraic Geometry]] \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| isbn=978-0-387-90244-9 \| id={{MathSciNet \| id = 0463157}} \| year=1977}}, section II.2

			[[Category:Algebraic geometry\|*]]

en>Brad7777: −Category:Algebra; +Category:Structures on manifolds using HotCat

2012-02-02T18:17:35Z

−Category:Algebra; +Category:Structures on manifolds using HotCat

New page

Friends contact him Royal Cummins. The factor I adore most bottle tops collecting and now I have time to take on new issues. Her family members life in Idaho. My occupation is a messenger.<br><br>my site ... [http://www.frag5.com/index.php?mod=users&action=view&id=13770 http://www.frag5.com/index.php?mod=users&action=view&id=13770]