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.
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 .
|Date of creation||2013-03-22 12:52:30|
|Last modified on||2013-03-22 12:52:30|
|Last modified by||alozano (2414)|
|Synonym||divisor class group|