(Miracle flatness)
Let \(f \colon X \to Y\) be a finite morphism of varieties over a field with \(X\) Cohen–Macaulay, \(Y\) smooth, and \(\dim X = \dim Y\) at every point of \(X\). Then \(f\) is flat. In particular a finite surjective morphism between smooth varieties of the same dimension is flat. [@Har10a, Exercise III.10.9]
The hypotheses cannot be dropped. The inclusion of the origin \(\Spec k \to \AA^1_k\) is finite and both schemes are smooth, but it is not flat, since \(k = k[t]/(t)\) is a torsion \(k[t]\)-module; here the dimensions differ. The normalization \(\AA^1 \to V(y^2 - x^3)\), \(t \mapsto (t^2, t^3)\), is finite between varieties of the same dimension with smooth source, but the target is not smooth, and the map is not flat: it is birational of degree one and not an isomorphism.
Dependencies
D-SCHCMGORCohen--Macaulay, Gorenstein and local complete intersection schemesusesD-MORFINFinite morphisms, and finite against finite typeuses