Generic flatness

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.
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 a non-zero element "f" of "A" such that "M""f" is a free "A""f"-module. 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.
Generic freeness is proved using Grothendieck's technique of dévissage. See for a proof of a version of generic freeness.