Let $f: X \to Y$ be a finite surjective morphism of quasi-projective varieties. Let $X$ be Cohen-Macaulay and let $Y$ be smooth. Now let $\mathcal{F}$ be a coherent sheaf on $X$. Then $f_* \mathcal{F}$ is a coherent sheaf on $Y$.
Question: If $\mathcal{F}$ is a Cohen-Macaulay sheaf with $\textrm{supp}(\mathcal{F})=X$, is it true that $f_* \mathcal{F}$ is locally free?
I really think this should be true and I tried to prove it like Exercise 2.2.26 in "Cohen-Macaulay rings" by Bruns and Herzog. But then I was confused since the local ring $\mathcal{O}_{X,x}$ does not need to be a finite $\mathcal{O}_{Y,f(x)}$-module, right? Is it possible to avoid this?
As $f$ is finite, and as its target is smooth and its source is CM, $f$ is flat. (Actually the result is the following : if $f$ is finite and if the target is smooth, then the source is CM if and only if $f$ is flat.) Now, finite and flat implies locally free.