Effective Cartier divisor

An effective Cartier divisor is cut out locally by one non-zero-divisor; it is a codimension-one closed subscheme. On an integral scheme a nonzero global section of a line bundle defines such a divisor, possibly empty. On a nonreduced scheme a nonzero section need not be regular, so its zero locus need not be a Cartier divisor.

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 $E$.