suppose $D$ is an effective smooth irreducible divisor on a smooth variety $X$. suppose $W$ is a closed subvariety of $X$ not contained in $D$. Suppose $L$ is a line bundle on $X$ and consider the exact sequence: $$ 0\rightarrow L\otimes O(-D) \rightarrow L \rightarrow i_*(L_{|D})\rightarrow 0. $$ Here $i:D\hookrightarrow X$. Now restrict this exact sequence on $W$ to get the exact sequence: $$ Tor^1(i_*(L_{|D}), O_W)\rightarrow L\otimes O(-D)\otimes O_W\rightarrow L\otimes O_W \rightarrow i_*(L_{|D}) \otimes O_W \rightarrow 0. $$ What is the support of the sheaves $Tor^1$ and of $i_*(L_D)\otimes O_W$.

Can we assume that outside a codimension two subset of $W$, these two sheaves are zero ?