Is it true that one has an exact sequence of the following form ? $0 \to \mathcal{O}_Z \to I_{Z, \mathbb{P}^3}(1)\otimes \mathcal{O}_H \to I_{Z, H}(1) \to 0$, where $Z$ is a finite set of points in $\mathbb{P}^3$ contained in a hyperplane $H$ and $I_{Z, \mathbb{P}^3}$ and $I_{Z, H}$ denote the ideal sheaf of $Z$ in $\mathbb{P}^3$ and $H$ respectively

Is it true that one has an exact sequence of the following form: $$0 \to \mathcal{O}_Z \to I_{Z, \mathbb{P}^3}(1)\otimes \mathcal{O}_H \to I_{Z, H}(1) \to 0,$$ where $Z$ is a finite set of points in $\mathbb{P}^3$ contained in a hyperplane $H$ and $I_{Z, \mathbb{P}^3}$ and $I_{Z, H}$ denote the ideal sheaf of $Z$ in $\mathbb{P}^3$ and $H$ respectively?