Let $S$ be a smooth variety. Assume we have an exact sequence of coherent sheaves
$$0 \to F \to E \to T \to 0$$
such that $E$ is locally free and $T$ is torsion. Is it true that $F$ is also locally free?
I know that this is trivially to be true for curves, as torsion-free is equivalent to free for modules over PIDs. However, I have no idea for higher-dimensional cases.
Any references or counter-examples on this would be very helpful. Thanks!
No, take $S=\operatorname{Spec}R$ where $R=k[x,y]$. Since $S$ is affine, we might as well work with modules. Take then $E=R=k[x,y]$, $T=R/(x,y)$ and $F=(x,y)$. I claim that $F$ is not locally free. To see this, we will show that it is not projective and this can be done by computing $\operatorname{Ext}^1(F,T)$. We have the free resolution of $F$ $$ 0\to R\xrightarrow{\begin{pmatrix}-y\\x\end{pmatrix}} R\oplus R\xrightarrow{\begin{pmatrix}x&y\end{pmatrix}} F\to 0$$ This gives $\operatorname{Ext}^1(F,T)=H^1(0\to\operatorname{Hom}(R\oplus R,T)\to\operatorname{Hom}(R,T)\to 0)=\operatorname{coker}(T\oplus T\xrightarrow{(-y\;x)} T)=T$ since $x=y=0$ in $T$.