PlanetMath: Weil divisors on schemes

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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.

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Also defines:

prime divisor, effective divisor, regular in codimension one

This object's parent.

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

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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 3935 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

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