Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can conclude the following $$\wedge^r\left[(f^*\mathcal F)/\operatorname{Tor}\right] \cong \left[\wedge^r\left(f^*\mathcal F\right)\right]/\operatorname{Tor}.$$
It seems for me that this is just a question in commutative algebra, but I do not find any reference for this. Do anyone have some ideas or comments?
Thank you.