Picard group

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

${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*}).\,}$

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.