Let $X$ be a smooth projective variety and $Z$ a closed subvariety of co-dimension $k$. The first $k-1$ chern classes of the ideal sheaf of $Z$ vanishes and the $k$-th chern class is given by something like $n[Z]$, where $n$ is an integer. My question is that, is there always a torsion-free coherent $\mathcal{O}_X$-module on $X$ such that the first $k-1$ chern classes of it vanishes just like the ideal sheaf of $Z$, but the $k$-th chern class is given by $m[Z]$ such that $m$ has the opposite sign to $n$?
My thoughts were that, this should be straightforward. I was thinking probably it is even possible for $m=-n$. We just want a torsion-free sheaf such that its chern classes are same as the torsion sheaf $i_*\mathcal{O}_Z$. Probably there is an extension of a trivial vector bundle and $i_*\mathcal{O}_Z$ which is torsion-free.
Extra thoughts: We can take the ideal sheaf $I_Z$, for some non-negative $k$ we have a surjection $\oplus \mathcal{O}(-k)\rightarrow I_Z$. The kernel is torsion-free, let's call it $E$. $E$ is very close to what I want, but the only problem is the extra chern classes produced by $\mathcal{O}(-k)$. So if we could construct a torsion free sheaf such that its total chern class is inverse of $\mathcal{O}(-k)$ we are done. (Then we will just need to direct sum them with $E$). Now is there a torsion-free sheaf that its total chern class is inverse of $\mathcal{O}(-k)$? (It is possible to reduce the problem to the case $k=1$). Does $\mathcal{O}(-1)$ inject to a trivial vector bundle such that quotient is torsion-free?
Continuation of thoughts: Now the problem is reduced to finding a tf sheaf with inverse chern classes of $\mathcal{O}(-1)$. Furthermore we can reduce the problem to the case of projective space. Because there is an embedding into $\mathbb{P}^n$ such that the pull back of tautological bundle is $\mathcal{O}(-1)$, so it suffices to inject $\mathcal{O}(-1)$ into a trivial bundle with tf quotient on $\mathbb{P}^n$. But $\mathcal{O}(-1)$ injects into $\mathcal{O}^{n+1}$ and the quotient is a vector bundle. So I guess this finishes the proof.
