Let $s_1$ and $s_2$ be disjoint codimension 1 subvarieties in a variety $X$. Let $i_1 : s_1 \to X$ and $i_2 : s_2 \to X$ be the respective immersions.
My question is whether there is an isomorphism, $$ \mathcal{O}_X(s_1+s_2)\otimes_{\mathcal{O}_X}i_{1*}\mathcal{O}_{s_1} \cong \mathcal{O}_X(s_1)\otimes_{\mathcal{O}_X}i_{1*}\mathcal{O}_{s_1} \,. $$ I would expect this to be true, since $i_{1*}\mathcal{O}_{s_1}$ only has support on $s_1$. But it seems to me that these two sheaves assign different objects to any open set intersecting both $s_1$ and $s_2$.