I know that on $\mathbb{P}^3$ (it can be in $\mathbb{P}^n$, but I just want to focus on $\mathbb{P}^3$), if one has an hiperplane $H \subset \mathbb{P}^3$, the equation of the hiperplane provides an exact sequence :
$$0 \to \mathcal{O}_{\mathbb{P}^3}(-1) \to \mathcal{O}_{\mathbb{P}^3} \to \mathcal{O}_{H} (*)$$
from there, we can twist the sequence by a torsion free sheaf $F$ on $\mathbb{P}^3$, one has :
$F(-1) \to F \to F|_H \to 0$,
my question is, how to calculate the tor sheaves that appear when we tensor the sequence (*) by $F$ in order to obtain the second sequence, furthemore, it is possible to obtain a exact sequence of group cohomology like the following?
$$H^0(\mathbb{P}^3,F|_H) \to H^1(\mathbb{P}^3,F(-1)) \to H^1(\mathbb{P}^3,F) $$
Any reference in Tor groups of torsion free sheaves are welcome.
Thank you.