Topic summary
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, u : X → Y is a finite typemorphism of schemes, and F is a coherentOX-module, then there is a non-empty open subset U of Y such that the restriction of F to u(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, u : X → S is a finite type morphism, and F is a coherent OX-module. Then there exists a partition of S into locally closed subsets S1, ..., Sn with the following property: Give each Si its reduced scheme structure, denote by Xi the fiber productX ×SSi, and denote by Fi the restriction F ⊗OSOSi; then each Fi is flat.