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 O x of R, where x runs through X (when X = S p e c ( A ) , we denote X ′ by L ( A ) ), X ′ verifies the following three properties
For each M ∈ X ′ , R is the field of fractions of M.There is a finite set of noetherian subrings A i of R so that X ′ = ∪ i L ( A i ) and that, for each pair of indices i,j, the subring A i j of R generated by A i ∪ A j is an A i -algebra of finite type.If M ⊆ 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).
