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Picard group
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(Definition)
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The Picard group of a variety, scheme, or more generally locally ringed space  is the group of locally free  modules of rank  with tensor product over  as the operation, usually denoted by
 . Alternatively, the Picard group is the group of isomorphism classes of invertible sheaves on  , under tensor products.
It is not difficult to see that
 is isomorphic to
 , the first sheaf cohomology group of the multiplicative sheaf  which consists of the units of  .
Finally, let
 be the group of Cartier divisors on  modulo linear equivalence. If  is an integral scheme then the groups
 and
 are isomorphic. Furthermote, if we let
 be the class group of Weil divisors (divisors modulo principal divisors) and  is a noetherian, integral and separated locally factorial scheme, then there is a
natural isomorphism
 . Thus, the Picard group is sometimes called the divisor class group of  .
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"Picard group" is owned by alozano. [ full author list (2) | owner history (1) ]
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(view preamble)
| Other names: |
divisor class group |
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Cross-references: natural isomorphism, factorial, separated, Noetherian, principal divisors, Weil divisors, class group, integral, equivalence, Cartier divisors, units, multiplicative, sheaf cohomology, isomorphic, sheaves, invertible, classes, isomorphism, operation, tensor product, rank, modules, locally free, group, locally ringed space, scheme, variety
There are 3 references to this entry.
This is version 3 of Picard group, born on 2002-07-27, modified 2007-02-15.
Object id is 3216, canonical name is PicardGroup3.
Accessed 3317 times total.
Classification:
| AMS MSC: | 14-00 (Algebraic geometry :: General reference works ) |
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Pending Errata and Addenda
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