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

• For each ${\displaystyle M\in X'}$, R is the field of fractions of M.
• There is a finite set of noetherian subrings ${\displaystyle A_{i}}$ of R so that ${\displaystyle X'=\cup _{i}L(A_{i})}$ and that, for each pair of indices i,j, the subring ${\displaystyle A_{ij}}$ of R generated by ${\displaystyle A_{i}\cup A_{j}}$ is an ${\displaystyle A_{i}}$-algebra of finite type.
• If ${\displaystyle M\subseteq N}$ in ${\displaystyle X'}$ 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 ${\displaystyle A_{i}}$'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