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 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, u : X → S is a finite type morphism, and F is a coherent O_X module. Then there exists a partition of S into locally closed subsets S₁, ..., S_n with the following property: Give each S_i its reduced scheme structure, denote by X_i the fiber product X ×_S S_i, and denote by F_i the restriction F ⊗_OS O_Si; then each F_i is flat.