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In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck. Generic flatness states that if ''Y'' is an integral locally noetherian scheme, is a finite type morphism of schemes, and ''F'' is a coherent ''O''''X''-module, then there is a non-empty open subset ''U'' of ''Y'' such that the restriction of ''F'' to ''u''−1(''U'') is flat over ''U''.〔EGA IV2, Théorème 6.9.1〕 Because ''Y'' is integral, ''U'' is a dense open subset of ''Y''. This can be applied to deduce a variant of generic flatness which is true when the base is not integral.〔EGA IV2, Corollaire 6.9.3〕 Suppose that ''S'' is a noetherian scheme, is a finite type morphism, and ''F'' is a coherent ''O''''X'' module. Then there exists a partition of ''S'' into locally closed subsets ''S''1, ..., ''S''''n'' with the following property: Give each ''S''''i'' its reduced scheme structure, denote by ''X''''i'' the fiber product , and denote by ''F''''i'' the restriction ; then each ''F''''i'' is flat. == Generic freeness == Generic flatness is a consequence of the generic freeness lemma. Generic freeness states that if ''A'' is a noetherian integral domain, ''B'' is a finite type ''A''-algebra, and ''M'' is a finite type ''B''-module, then there exists an element ''f'' of ''A'' such that ''M''''f'' is a free ''A''''f''-module.〔EGA IV2, Lemme 6.9.2〕 Generic freeness can be extended to the graded situation: If ''B'' is graded by the natural numbers, ''A'' acts in degree zero, and ''M'' is a graded ''B''-module, then ''f'' may be chosen such that each graded component of ''M''''f'' is free.〔Eisenbud, Theorem 14.4〕 Generic freeness is proved using Grothendieck's technique of dévissage. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「generic flatness」の詳細全文を読む スポンサード リンク
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