Divisor restriction exact sequence ($0\to {\mathcal{O}}_X(D-E)\to {\mathcal{O}}_X(D)\to {\mathcal{O}}_E(D)\to 0$)

For an effective Cartier divisor $E$ and any Cartier divisor $D$, multiplication by a defining section of $E$ gives the displayed short exact sequence of sheaves. Its long exact sequence in sheaf cohomology compares sections and cohomology on $X$ with their restrictions to $E$.