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Concept
Picard group
Summary
In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.
Alternatively, the Picard group can be defined as the sheaf cohomology group
For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group.
The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces.
The Picard group of the spectrum of a Dedekind domain is its ideal class group.
The invertible sheaves on projective space Pn(k) for k a field, are the twisting sheaves so the Picard group of Pn(k) is isomorphic to Z.
The Picard group of the affine line with two origins over k is isomorphic to Z.
The Picard group of the -dimensional complex affine space: , indeed the exponential sequence yields the following long exact sequence in cohomology
and since we have because is contractible, then and we can apply the Dolbeault isomorphism to calculate by the Dolbeault-Grothendieck lemma.
The construction of a scheme structure on (representable functor version of) the Picard group, the Picard scheme, is an important step in algebraic geometry, in particular in the duality theory of abelian varieties. It was constructed by , and also described by and .
In the cases of most importance to classical algebraic geometry, for a non-singular complete variety V over a field of characteristic zero, the connected component of the identity in the Picard scheme is an abelian variety called the Picard variety and denoted Pic0(V). The dual of the Picard variety is the Albanese variety, and in the particular case where V is a curve, the Picard variety is naturally isomorphic to the Jacobian variety of V.
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