When is the pushforward / direct image of a reflexive sheaf locally free?
Solution 1:
A standard example of this kind of theorem is: If $X$ and $Y$ are surfaces, $Y$ is nonsingular, $X$ is normal, $f$ is finite, and $S$ is reflexive, then $f_*S$ is locally free. The reason is that reflexive over normal implies depth $\geq 2$ at closed points. Then $f_*S$ has depth $\geq 2$ at closed points too (by finiteness of $f$). Finally, if you have a maximal Cohen-Macaulay module over a regular local ring, then it is free (see Lemma Tag 00NT).