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
H 1 ( X , O X ∗ ) . {\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*}).\,}
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.
Examples 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 O ( m ) , {\displaystyle {\mathcal {O}}(m),\,} 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 n {\displaystyle n} -dimensional complex affine space: Pic ( C n ) = 0 {\displaystyle \operatorname {Pic} (\mathbb {C} ^{n})=0} , indeed the exponential sequence yields the following long exact sequence in cohomology
⋯ → H 1 ( C n , Z _ ) → H 1 ( C n , O C n ) → H 1 ( C n , O C n ⋆ ) → H 2 ( C n , Z _ ) → ⋯ {\displaystyle \dots \to H^{1}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\to H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}})\to H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}}^{\star })\to H^{2}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\to \cdots }
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