About: 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 the set of subrings of R, where x runs through X (when , we denote by ), verifies the following three properties * For each , R is the field of fractions of M. * There is a finite set of noetherian subrings of R so that and that, for each pair of indices i,j, the subring of R generated by is an -algebra of finite type. * If in are such that the maximal ideal of M is contained in that of N, then M=N.

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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 the set of subrings of R, where x runs through X (when , we denote by ), verifies the following three properties * For each , R is the field of fractions of M. * There is a finite set of noetherian subrings of R so that and that, for each pair of indices i,j, the subring of R generated by is an -algebra of finite type. * If in 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 's were algebras of finite type over a field too (this simplifies the second condition above). (en)
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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 the set of subrings of R, where x runs through X (when , we denote by ), verifies the following three properties * For each , R is the field of fractions of M. * There is a finite set of noetherian subrings of R so that and that, for each pair of indices i,j, the subring of R generated by is an -algebra of finite type. * If in are such that the maximal ideal of M is contained in that of N, then M=N. (en)
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  • Chevalley scheme (en)
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