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单词 WeilDivisorsOnSchemes
释义

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).

Definition.

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:

𝒲=nYY

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.

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更新时间:2025/5/4 19:21:12