I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of codimension $1$). We know that we have an exact sequence, $$H^0(\mathcal{O}_U) \to H^0(U-Z,\mathcal{O}_U|_{U-Z}) \xrightarrow{\delta} H^1_Z(\mathcal{O}_U)$$

My question is: Is $\delta$ a $\mathcal{O}_U(U)$-linear morphism? If so, then it seems that $H^1_Z(\mathcal{O}_U) \otimes_{\mathcal{O}_U(U)} \mathcal{O}_Z(Z)$ should be zero. Is this correct? Am I missing something?