Generic flatness

Remarks

The statement reduces to the affine case and to the generic freeness lemma. That lemma says the following: let $A$ be a Noetherian domain, $B$ a finitely generated $A$-algebra, and $M$ a finitely generated $B$-module. Then there is a nonzero $a \in A$ such that $M_a$ is a free $A_a$-module. Its proof uses Noether normalization over the fraction field of $A$ and induction on dimension. For example, the blow-up of $\mathbb{A}^2_k$ at the origin is flat over $U = \mathbb{A}^2_k \setminus \set{0}$, where it is an isomorphism. The integrality of $Y$ cannot be dropped. For $Y = \operatorname{Spec} k[\varepsilon]/(\varepsilon^2)$ and the closed immersion $\operatorname{Spec} k \to Y$, the only nonempty open subset of $Y$ is $Y$, and $k$ is not flat over $k[\varepsilon]/(\varepsilon^2)$. The same proof gives a sheaf version: a coherent sheaf on $X$ is flat over $Y$ above some nonempty open subset of $Y$.

Going deeper

For a student

Generic flatness is the starting point of many stratification arguments. One applies it on $Y$, then again on the closed complement with its reduced structure, and so on; Noetherian induction ends the process. The result is a finite partition of $Y$ into locally closed pieces over each of which $f$ is flat. This is how one builds flattening stratifications and proves that Hilbert polynomials of fibres take finitely many values. The theorem also gives a quick proof of Chevalley's theorem on constructible sets.