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| In [[algebraic geometry]], the '''function field''' of an [[algebraic variety]] ''V'' consists of objects which are interpreted as rational functions on ''V''. In classical [[algebraic geometry]] they are [[Rational function|ratios of polynomials]]; in [[analytic variety|complex algebraic geometry]] these are [[meromorphic function]]s and their higher-dimensional analogues; in [[scheme (mathematics)|modern algebraic geometry]] they are elements of some quotient ring's [[field of fractions]].
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| ==Definition for complex manifolds==
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| More precisely, in complex algebraic geometry the objects of study are complex [[analytic varieties]], on which we have a local notion of [[complex analysis]], through which we may define meromorphic functions. The function field is then the set of all meromorphic functions on the variety. For the [[Riemann sphere]], which is the variety '''P'''<sup>1</sup> over the complex numbers, the global meromorphic functions are exactly the [[rational function]]s (that is, the ratios of complex polynomial functions). In any case, the meromorphic functions form a field, the function field.
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| ==Construction in algebraic geometry==
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| In classical algebraic geometry, we generalize the second point of view. For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an [[affine space|affine]] coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety ''V'', we say that a rational function on an open affine subset ''U'' is defined as the ratio of two polynomials in the [[affine variety|affine coordinate ring]] of ''U'', and that a rational function on all of ''V'' consists of such local data which agree on the intersections of open affines. We have defined the rational functions on ''V'' to be the [[field of fractions]] of the affine coordinate ring of any open affine subset, since all such subsets are dense.
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| ==Generalization to arbitrary scheme==
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| In the most general setting, that of modern [[scheme theory]], we take the latter point of view above as a point of departure. Namely, if ''X'' is an integral [[Scheme (mathematics)|scheme]], then every open affine subset ''U'' is an integral domain and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the [[local ring]] of the [[generic point]] of ''X''. Thus the function field of ''X'' is just the local ring of its generic point. This point of view is developed further in [[function field (scheme theory)]].
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| ==Geometry of the function field==
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| If ''V'' is a variety over a field ''K'', then the function field ''K''(''V'') is a field extension of the ground field ''K'' over which ''V'' is defined; its [[transcendence degree]] is equal to the [[dimension of an algebraic variety|dimension]] of the variety. All extensions of ''K'' that are finitely-generated as fields arise in this way from some algebraic variety.
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| Properties of the variety ''V'' that depend only on the function field are studied in [[birational geometry]].
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| == Examples ==
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| The function field of a point over ''K'' is ''K''.
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| The function field of the affine line over ''K'' is isomorphic to the field ''K''(''t'') of [[rational function]]s in one variable. This is also the function field of the projective line.
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| Consider the affine plane curve defined by the equation <math>y^2 = x^5 + 1</math>. Its function field is the field ''K''(''x'',''y''), generated by [[transcendental element]]s satisfying this algebraic relation.
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| == See also ==
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| * [[Function field (scheme theory)]]: a generalization
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| * [[Algebraic function field]]
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| * [[Cartier divisor]]
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| ==References==
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| {{refimprove|date=September 2008}}
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| * {{cite book | title=Algebraic Functions and Projective Curves | series=Graduate Texts in Mathematics | volume=215 | author=David M. Goldschmidt | publisher=Springer-Verlag | year=2002 | isbn=0-387-95432-5 }}
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| [[Category:Algebraic varieties]]
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