Finite flat rank over an irreducible scheme (${\dim }_{\kappa (y)}{H}^0(X_y,{\mathcal{O}}_{X_y})=r$)

For a finite morphism that is flat, its direct-image algebra is finite free at every local stalk. On an irreducible scheme, every point specializes from the same generic point. Localizing each stalk further at that generic point shows that all these free ranks equal the generic rank. Hence all fibre algebras have the same dimension, even without assuming finite presentation in advance.