Английская Википедия:Chevalley scheme

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A Chevalley scheme in algebraic geometry was a precursor notion of scheme theory.

Let X be a separated integral noetherian scheme, R its 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

• For each <math>M\in X' </math>, R is the field of fractions of M.
• 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.
• 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.

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).

Bibliography

• Шаблон:Cite journal Online Шаблон:Webarchive