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Cartier divisor
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(Definition)
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On a scheme  , a Cartier divisor is a global section of the sheaf
 , where
 is the multiplicative sheaf of meromorphic functions, and  the multiplicative sheaf of invertible regular functions (the units of the structure sheaf).
More explicitly, a Cartier divisor is a choice of open cover  of  , and meromorphic functions
 , such that
 , along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if  is replaced by  with  .
Intuitively, the only information carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a Weil divisor, and vice versa. On “nice” (for example, nonsingular over an algebraically closed field) schemes, it does.
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"Cartier divisor" is owned by mathcam. [ full author list (3) | owner history (2) ]
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(view preamble)
Cross-references: field, algebraically closed, nonsingular, Weil divisor, order, vanishes, open sets, refinement, functions, meromorphic, open cover, structure sheaf, units, regular functions, invertible, sheaf of meromorphic functions, multiplicative, sheaf, global section, scheme
There are 3 references to this entry.
This is version 3 of Cartier divisor, born on 2003-08-19, modified 2004-10-25.
Object id is 4617, canonical name is CartierDivisor.
Accessed 3453 times total.
Classification:
| AMS MSC: | 14A99 (Algebraic geometry :: Foundations :: Miscellaneous) |
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Pending Errata and Addenda
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