(Miracle flatness)
Let \(f : X \to Y\) be a local homomorphism of Noetherian local rings, or a morphism of schemes locally of finite type, with \(X\) Cohen–Macaulay and \(Y\) regular. If \[ \dim_x X_{f(x)} = \dim_x X - \dim_{f(x)} Y \] at every point, that is, the fibres have the expected dimension, then \(f\) is flat.
This is the theorem that makes flatness checkable, since the hypotheses are about dimension and the conclusion is about modules. In the form one uses it: a morphism between smooth varieties with equidimensional fibres of the expected dimension is flat.
It also explains the blowup counterexample rather than merely contradicting it: \(\Bl_0 \AA^2 \to \AA^2\) has a fibre of dimension \(1\) where the expected dimension is \(0\), so the hypothesis fails at exactly the point where flatness does.
The displayed dimension formula is itself a consequence of flatness in general, so the theorem says that for a Cohen–Macaulay source over a regular base the necessary condition is sufficient. The Cohen–Macaulay hypothesis rules out embedded depth defects that would otherwise allow the expected dimension formula without flatness.
Dependencies
D-MORFLATFlat morphismsuses