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' the set of subrings O x of R, where x runs through X, 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 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 Ai-algebra of finite type.
  • Two sister elements M,N of X' are identical.

Originally, Chevalley also supposed that R was an extension of finite type of a field K and that the Ais were K-algebras of finite type too (this simplifies the second condition above)

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