Let $C,D$ curves (so $1$-dimensional proper $k$-schemes). Assume futhermore that they are also regular and $f:C \to D$ is a finite morphism.
It is known that in this case the pushforward functor $f_*: (QCoh-O_C-Mod) \to (QCoh-O_D-Mod)$ provides a category equivalence between quasi coherent modules of $O_C$ and $O_D$-modules with extra $f_*(O_C)$-structure. .
My question is if this functor also preserves local freeness for coherent modules: Namely if $F$ is a coherent locally free $O_C$-module does this also hold for $f_*F$?
The problem is local since $f$ is a affine morphism so by regularity we can assume that $C=Spec(A),D=Spec(R)$ with $A,R$ Dedekind-rings and $F$ is a free $A$-module $A^n$.
By classification of finitely generated modules over Dedekind rings every f.g. $A$-module $M$ has the shape $F= A^k \oplus T$ with free component $A^k$ and $T$ torsion.
My idea is firstly to observe that $f_*$ preserves (finite) direct sums. I guess that this follows from that as category equivalence of modules is preserves exact sequences?)
And then need to show that:
-$f_*A$ is a free $R$-module
-$f_*T$ is also torsion wrt $R$
Here I'm stuck. Why these two statements hold?
Futhermore does this category equivalence allow a "reverse" argument: namely if $f_*F$ is free then also $F$?