Chevalley scheme

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]\displaystyle{ X' }[/math] the set of subrings [math]\displaystyle{ \mathcal O_x }[/math] of R, where x runs through X (when [math]\displaystyle{ X=\mathrm{Spec}(A) }[/math], we denote [math]\displaystyle{ X' }[/math] by [math]\displaystyle{ L(A) }[/math]), [math]\displaystyle{ X' }[/math] verifies the following three properties

• For each [math]\displaystyle{ M\in X' }[/math], R is the field of fractions of M.
• There is a finite set of noetherian subrings [math]\displaystyle{ A_i }[/math] of R so that [math]\displaystyle{ X'=\cup_i L(A_i) }[/math] and that, for each pair of indices i,j, the subring [math]\displaystyle{ A_{ij} }[/math] of R generated by [math]\displaystyle{ A_i \cup A_j }[/math] is an [math]\displaystyle{ A_i }[/math]-algebra of finite type.
• If [math]\displaystyle{ M\subseteq N }[/math] in [math]\displaystyle{ 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]\displaystyle{ A_i }[/math]'s were algebras of finite type over a field too (this simplifies the second condition above).

Bibliography

• Grothendieck, Alexandre; Jean Dieudonné (1960). "Éléments de géométrie algébrique". Publications Mathématiques de l'IHÉS I. Le langage des schémas: I.8.  Online