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

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

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