Assume that $\psi: X\rightarrow Y$ is a morphism between two irreducible normal schemes. And $\psi$ maps the generic point of $X$ to the generic of $Y$.
Then can we prove that $\psi_* \mathscr{O}_X$ is a locally free $\mathscr{O}_Y$-module with rank $[k(X):k(Y)]$.
Please try $k[x^2,xy,y^2]\subset k[x,y]$.