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.
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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 $f_*F$?
My ideas:
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 $F$?
Best Answer
A module (finitely generated or not) over a Dedekind domain $R$ is flat if and only it is torsion-free.
So in your case of a finite morphism $R\to A$ the $R$-module $A$ is flat and finitely presented, hence projective.
We conclude that $A$, and all $A^n$, are locally free over $R$.
Geometrically this means that given a finite morphisms $f:C\to D$ between regular curves, the direct image $ f_*\mathcal F$ on $D$ of a locally free sheaf $\mathcal F$ on $C$ is a locally free sheaf on $D$.
This is always false if the target curve $D$ is not regular:
Indeed, in that case if $f:C\to D$ is the normalization, then the sheaf $f_*\mathcal O_C$ is never locally free on $D$.