Weil divisors on schemes

Let $X$ be a noetherian integral separated scheme such that every local ring ${𝒪}_x$ 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 $\mathrm{Div}\mathit{}\mathrm{(}X\mathrm{)}$ generated by the prime divisors on $X$. A Weil divisor is an element of $\mathrm{Div}\mathit{}\mathrm{(}X\mathrm{)}$. Thus, a Weil divisor $\mathrm{W}$ can be written as:

$$𝒲=\sum n_{Y}Y$$

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 $\mathrm{deg}\mathit{}\mathrm{(}\mathrm{W}\mathrm{)}\mathrm{=}\mathrm{\sum }{n}_Y$. Finally, a divisor is said to be effective if $n_Y\mathrm{\ge }\mathrm{0}$ 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
Canonical name	WeilDivisorsOnSchemes
Date of creation	2013-03-22 15:34:08
Last modified on	2013-03-22 15:34:08
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Definition
Classification	msc 14C20
Related topic	BibliographyForAlgebraicGeometry
Defines	prime divisor
Defines	effective divisor
Defines	regular in codimension one