High Quality Content by WIKIPEDIA articles! In mathematics, the Picard group of a ringed space X, denoted by operatorname{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, mathcal{O}_X^{*})., For integral schemes the Picard group can be shown to be isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group.

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