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 $$X'$$ the set of subrings $$\mathcal O_x$$ of R, where x runs through X (when $$X=\mathrm{Spec}(A)$$, we denote $$X'$$ by $$L(A)$$), $$X'$$ verifies the following three properties
 * For each $$M\in X' $$, R is the field of fractions of M.
 * There is a finite set of noetherian subrings $$A_i$$ of R so that $$X'=\cup_i L(A_i) $$ and that, for each pair of indices i,j, the subring $$A_{ij} $$ of R generated by $$ A_i \cup A_j $$ is an $$A_i$$-algebra of finite type.
 * If $$M\subseteq N$$ in $$X'$$ 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 $$ A_i $$'s were algebras of finite type over a field too (this simplifies the second condition above).