Weil divisors on schemes

Let X be a noetherian integral separated scheme such that every local ring 𝒪x 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).


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


where the sum is over all the prime divisors Y of X, the nY are integers and only finitely many of them are non-zero. A degree of a divisorMathworldPlanetmathPlanetmath is defined to be deg(W)=nY. Finally, a divisor is said to be effective if nY0 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