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, {{nowrap|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.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, {{nowrap|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 {{nowrap|X ×_S S_i}}, and denote by F_i the restriction {{nowrap|F ⊗_OS O_Si}}; then each F_i is flat.