Weil divisors on schemes
Let be a noetherian integral separated scheme such that every local ring of of dimension one is regular (such a scheme is said to be regular in codimension one, or non-singular in codimension one).
Definition.
A prime divisor on is a closed integral subscheme of codimension one. We define an abelian group
generated by the prime divisors on . A Weil divisor is an element of . Thus, a Weil divisor can be written as:
where the sum is over all the prime divisors of , the are integers and only finitely many of them are non-zero. A degree of a divisor is defined to be . Finally, a divisor is said to be effective if for all the prime divisors .
For more information, see Hartshorne’s book listed in the bibliography for algebraic geometry.