254:, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.
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in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the
207:. In fact, the fraction fields of the rings of regular functions on any affine open set will be the same, so we define, for any
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195:. This means there is not enough room for a regular function to do anything interesting outside of
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222:) to be the common fraction field of any ring of regular functions on any open affine subset of
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of this field extension. All finite transcendence degree field extensions of
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226:. Alternatively, one can define the function field in this case to be the
430:, that is, dimension 1, it follows that any two non-constant functions
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322:) and whose restriction maps are induced from the restriction maps of
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whose value is the fraction field of the global sections of
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correspond to the rational function field of some variety.
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should determine the behavior of the rational functions on
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but not affine, then any non-empty affine open set will be
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https://www.e-periodica.ch/cntmng?pid=ens-001:1979:25::101
246:is no longer integral. Then it is possible to have
359:is defined, it is possible to study properties of
157:will be a localization of the global sections of
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329:by the universal property of localization. Then
257:The correct solution is to proceed as follows:
153:is affine, the ring of regular functions on
49:, such a sheaf associates to each open set
283:) that are not zero divisors in any stalk
103:In the simplest cases, the definition of
336:is the sheaf associated to the presheaf
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39:function field of an algebraic variety
145:of the ring of regular functions on
463:Kleiman, S., "Misconceptions about
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297:be the presheaf whose sections on
61:on that open set; in other words,
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474:25 (1979), 203–206, available at
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272:be the set of all elements in Γ(
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442:satisfy a polynomial equation
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423:In the particular case of an
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18:sheaf of rational functions
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393:we have a field extension
389:, then over each open set
370:. This is the subject of
33:is the generalization to
242:The trouble starts when
120:affine algebraic variety
110:is straightforward. If
87:does not always give a
412:will be equal to the
363:which depend only on
126:is an open subset of
91:for a general scheme
414:transcendence degree
80:. Despite its name,
408:. The dimension of
372:birational geometry
252:total quotient ring
161:, and consequently
47:algebraic varieties
261:For each open set
59:rational functions
43:algebraic geometry
383:algebraic variety
74:regular functions
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41:in classical
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116:irreducible
228:local ring
149:. Because
122:, and if
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230:of the
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290:. Let
265:, let
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