Generic flatness
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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. 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. 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.
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. 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. 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.
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