# Weil divisors on schemes

Let $X$ be a noetherian integral separated scheme such that every local ring $\mathcal{O}_{x}$ of $X$ of dimension one is regular (such a scheme $X$ is said to be regular in codimension one, or non-singular in codimension one).

###### Definition.

A prime divisor on $X$ is a closed integral subscheme $Y$ of codimension one. We define an abelian group $\operatorname{Div}(X)$ generated by the prime divisors on $X$. A Weil divisor is an element of $\operatorname{Div}(X)$. Thus, a Weil divisor $\mathcal{W}$ can be written as:

 $\mathcal{W}=\sum n_{Y}Y$

where the sum is over all the prime divisors $Y$ of $X$, the $n_{Y}$ are integers and only finitely many of them are non-zero. A degree of a divisor is defined to be $\deg(\mathcal{W})=\sum n_{Y}$. Finally, a divisor is said to be effective if $n_{Y}\geq 0$ for all the prime divisors $Y$.

For more information, see Hartshorne’s book listed in the bibliography for algebraic geometry.

Title Weil divisors on schemes WeilDivisorsOnSchemes 2013-03-22 15:34:08 2013-03-22 15:34:08 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 14C20 BibliographyForAlgebraicGeometry prime divisor effective divisor regular in codimension one