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86:, through which we may define meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions, these take their values in
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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
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and, hence, has a field of fractions. Furthermore, it can be verified that these are all the same, and are all equal to the
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coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere). On a general variety
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consists of such local data as agree on the intersections of open affines. We may define the function field of
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of the affine coordinate ring of any open affine subset, since all such subsets are dense.
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arise in this way from some algebraic variety. These field extensions are also known as
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is just the stalk of its generic point. This point of view is developed further in
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over the complex numbers, the global meromorphic functions are exactly the
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in one variable. This is also the function field of the
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that depend only on the function field are studied in
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184:is defined as the ratio of two polynomials in the
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113:{\displaystyle \mathbb {C} \cup \{\infty \}}
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34:consists of objects that are interpreted as
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58:and their higher-dimensional analogues; in
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16:Mathematical concept in algebraic geometry
778:Algebraic Functions and Projective Curves
758:Learn how and when to remove this message
192:, and that a rational function on all of
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707:"Function field of an algebraic variety"
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313:{\displaystyle {\mathcal {O}}_{X}(U)}
82:, on which we have a local notion of
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240:, then for every open affine subset
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563:. Its function field is the field
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785:. Vol. 215. Springer-Verlag.
444:of the variety. All extensions of
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208:Generalization to arbitrary scheme
168:Construction in algebraic geometry
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153:{\displaystyle \mathbb {P} ^{1}}
74:Definition for complex manifolds
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647:Function field (scheme theory)
408:Geometry of the function field
394:function field (scheme theory)
372:. Thus the function field of
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630:{\displaystyle y^{2}=x^{5}+1}
556:{\displaystyle y^{2}=x^{5}+1}
515:affine algebraic plane curve
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494:is isomorphic to the field
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463:Properties of the variety
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48:ratios of polynomials
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773:David M. Goldschmidt
692:improve this article
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432:of the ground field
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649:: a generalization
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801:Hartshorne, Robin
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385:{\displaystyle X}
365:{\displaystyle X}
333:{\displaystyle U}
273:{\displaystyle X}
253:{\displaystyle U}
229:{\displaystyle X}
29:algebraic variety
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42:. In classical
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200:to be the
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442:dimension
105:∞
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837:13348052
803:(1977),
775:(2002).
641:See also
475:Examples
127:For the
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400: (
348:of the
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396:. See
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27:of an
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587:over
502:) of
456:over
346:stalk
122:field
50:; in
833:OCLC
815:ISBN
787:ISBN
711:news
579:and
402:1977
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