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| A '''Chevalley scheme''' in [[algebraic geometry]] was a precursor notion of [[scheme theory]].
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| Let ''X'' be a separated integral [[noetherian scheme]], ''R'' its [[function field of an algebraic variety|function field]]. If we denote by <math>X'</math> the set of subrings <math>\mathcal O_x</math> of ''R'', where ''x'' runs through ''X'' (when <math>X=\mathrm{Spec}(A)</math>, we denote <math>X'</math> by <math>L(A)</math>), <math>X'</math> verifies the following three properties
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| * For each <math>M\in X' </math>, ''R'' is the field of fractions of ''M''.
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| * There is a finite set of noetherian subrings <math>A_i</math> of ''R'' so that <math>X'=\cup_i L(A_i) </math> and that, for each pair of indices ''i,j'', the subring <math>A_{ij} </math> of ''R'' generated by <math> A_i \cup A_j </math> is an <math>A_i</math>-algebra of finite type.
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| * If <math>M\subseteq N</math> in <math>X'</math> are such that the maximal ideal of ''M'' is contained in that of ''N'', then ''M=N''.
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| Originally, [[Chevalley]] also supposed that R was an extension of finite type of a field K and that the <math> A_i </math>'s were algebras of finite type over a field too (this simplifies the second condition above).
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| ==Bibliography==
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| *{{cite journal
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| | last = Grothendieck
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| | first = Alexandre
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| | authorlink = Alexandre Grothendieck
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| | coauthors = [[Jean Dieudonné]]
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| | year = 1960
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| | title = [[Éléments de géométrie algébrique]]
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| | volume = I. Le langage des schémas
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| | journal = [[Publications Mathématiques de l'IHÉS]]
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| | pages = I.8
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| <!-- Please do not add url field as the template cannot handle a title with both an external link and a wikilink. -->
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| }} [http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960__4_ Online]
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| [[Category:Scheme theory]]
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