Let $H$ be a hypeplane in $\mathbb{P}^3$ containing a point $p$ and $I_p$ be the ideal sheaf corresponding to $p$. Consider the natural exact sequence :

$0 \to \mathcal{O} \to \mathcal{O}(H) \to \mathcal{O}(H) \mid_H \to 0$.

Is it true that the tensoring the exact sequence by $I_p$ remains exact ? I guess not, because if it is exact then we get the following exact sequence:

$0 \to I_p \to I_p(1) \to I_p(1) \mid_H \to 0$.

Note that $h^0(I_p) = h^1(I_p) = 0$. Thus considering the long exact sequence of cohomology of the above sequence, gives that $H^0(I_p(1)) \cong H^0(I_p(1)\mid_H)$, which is a contradiction as their dimensions are $3$ and $2$ respectively. Please correct me if i am wrong.