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 OX-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.
This can be applied to deduce a variant of generic flatness which is true when the base is not integral.
[2] 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 product X ×S Si, and denote by Fi the restriction F ⊗OS OSi; then each Fi is flat.
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 Mf is a free Af-module.