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?



How does torsion behave under the direct image functor?
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 $T$.


