How does torsion behave under the direct image functor?

Question

Assume we have a finite morphism $f: X\rightarrow Y$ of smooth projective varieties of degree $d$ over $k=\mathbb{C}$. Then $f_{*}$ induces an equivalence between the categoy of coherent $O_X$-modules and the category of coherent $O_Y$-modules with a $f_{*}O_X$-module structure.

This equivalence restricts to an equivalence between locally free sheaves of rank $r$ on $X$ and locally free $f_{*}O_X$-modules of rank $r$ on $Y$ (i.e. rank $rd$ as $O_Y$-modules).

But what about torsion free sheaves or torsion sheaves?

Given a torsion free sheaf $E$ on $X$, then $f_{*}E$ is torsion free on $Y$ and has the structure of a $f_{*}O_X$-module. But given a torsion free sheaf $F$ on $Y$, with the structure of a $f_{*}O_X$-module, then there is an $O_X$-module $E$ with $f_{*}E=F$, is $E$ torsion free on $X$? Can torsion on $X$ disapper on $Y$ when applying $f_{*}$?

Or does the functor $f_{*}$ also restrict to an equivalence between the categories of torsion free sheaves?

Furthermore if $T$ is a coherent torsion sheaf on $Y$, there is a coherent $O_X$-module $S$, with $f_{*}S=T$. Must $S$ also be torsion? What can be said about the relationship between $supp(S)$ and $supp(T)$. For example, if $T$ is supported at one point $y\in Y$, can $S$ have support at more than one point, maybe at some points in the fiber $f^{-1}(y)$?

I'm only interested in the case, where $dim(X)=dim(Y)=2$, i.e. the varities are surfaces. Can something more be said in this case? Maybe this is somewhere in the literature?

Answer 1

On an integral scheme of finite type over a field being torsion is the same as having a support that's lower dimensional than the ambient space. Since finite morphisms preserve dimension, being torsion is invariant under push-forward.

As for the support being one (I suppose you mean closed) point, you should make your question more precise. It is not true that any $S$ that satisfies $f_*S=T$ would be supported at only one point. Just take $S=k_P\oplus k_Q$ for two closed points with $f(P)=f(Q)$. Then $T=f_*S$ is supported at $f(P)$ but $S$ is supported on two points. On the other hand if $T$ is supported at (say) the closed point $f(P)$, then under your assumptions you $T$ will be naturally a module over the residue field at $P$, so you can consider $T$ as an $\mathscr O_X$-module supported at $P$ and call it $S$. In other words, you can find an $S$ that's supported at one point whose direct image is $T$.