PlanetMath: Weil divisors on schemes

(more info)

Math for the people, by the people.

donor list - find out how

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (236)
Orphanage
Unclass'd (1)
Unproven (540)
Corrections (51)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Copyright
About

Entry average rating: No information on entry rating

Weil divisors on schemes

(Definition)

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 1 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.

"Weil divisors on schemes" is owned by alozano.

(view preamble | get metadata)

Also defines:

prime divisor, effective divisor, regular in codimension one

This object's parent.

Attachments:: algebraic equivalence of divisors (Definition) by alozano

Log in to rate this entry.
(view current ratings)

Cross-references: bibliography for algebraic geometry, information, effective, degree, integers, sum, Weil divisor, generated by, abelian group, closed, codimension, non-singular, scheme, regular, dimension, local ring, separated scheme, integral, Noetherian
There are 3 references to this entry.

This is version 1 of Weil divisors on schemes, born on 2005-11-09.
Object id is 7473, canonical name is WeilDivisorsOnSchemes.
Accessed 3985 times total.

Classification:

AMS MSC:

14C20 (Algebraic geometry :: Cycles and subschemes :: Divisors, linear systems, invertible sheaves)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact