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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.
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| 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.
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| ==Definition==
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| Suppose that ''R'' is a commutative [[local ring]], with the maximal ideal ''m''. Then the '''residue field''' is the quotient ring ''R''/''m''.
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| 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'' ''':
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| :''k''(''x'') := ''A''<sub>''p''</sub> / ''p''·''A''<sub>''p''</sub>.
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| 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>
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| A point is called [[rational point|''K''-rational]] for a certain field ''K'', if ''k''(''x'') ⊂ ''K''.
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| ==Example==
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| 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
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| *(''t'' − ''a''), ''a'' ∈ ''k''
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| *(0), the zero-ideal.
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| The residue fields are
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| *<math>k[t]_{(t-a)}/(t-a)k[t]_{(t-a)} \cong k</math>
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| *<math>k[t]_{(0)} \cong k(t)</math>, the function field over ''k'' in one variable.
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| 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'''.
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| ==Properties==
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| * 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''.
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| * A morphism Spec(''K'') → ''X'', ''K'' some field, is equivalent to giving a point ''x'' ∈ ''X'' and an [[field extension|extension]] ''K''/''k''(''x'').
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| * 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.
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * {{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
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| [[Category:Algebraic geometry|*]]
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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.
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