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[parent] 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:
$\displaystyle \mathcal{W}=\sum n_Y Y$
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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See Also: bibliography for algebraic geometry

Also defines:  prime divisor, effective divisor, regular in codimension one

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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
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This is version 1 of Weil divisors on schemes, born on 2005-11-09.
Object id is 7473, canonical name is WeilDivisorsOnSchemes.
Accessed 2698 times total.

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