# Picard group

A group of classes of invertible sheaves (or line bundles). More precisely, let $(X,\mathcal{O}_{X})$ be a ringed space. A sheaf $\mathcal{L}$ of $\mathcal{O}_{X}$-modules is called invertible if and only if it is locally isomorphic to the structure sheaf $\mathcal{O}_{X}$. The set of classes of isomorphic invertible sheaves on $X$ is denoted by $\operatorname{Pic}(X)$. The tensor product $\mathcal{L} \otimes_{\mathcal{O}_{X}} \mathcal{L}'$ defines an operation on the set $\operatorname{Pic}(X)$, making it an Abelian group called the Picard group of $X$. The group $\operatorname{Pic}(X)$ is naturally isomorphic to the cohomology group ${H^{1}}(X,\mathcal{O}_{X}^{*})$, where $\mathcal{O}_{X}^{*}$ is the sheaf of invertible elements in $\mathcal{O}_{X}$.

For a commutative ring $A$, the Picard group $\operatorname{Pic}(A)$ is the group of classes of invertible $A$-modules; $\operatorname{Pic}(A) \cong \operatorname{Pic}(\operatorname{Spec}(A))$. For a Krull ring, the group $\operatorname{Pic}(A)$ is closely related to the divisor class group for this ring.

The Picard group of a complete normal algebraic variety $X$ has a natural algebraic structure (see Picard scheme). The reduced connected component of the zero of $\operatorname{Pic}(X)$ is denoted by ${\operatorname{Pic}^{0}}(X)$ and is called the Picard variety for $X$; it is an algebraic group (an Abelian variety if $X$ is a complete non-singular variety). The quotient group $\operatorname{Pic}(X) / {\operatorname{Pic}^{0}}(X)$ is called the Néron–Severi group, and it has a finite number of generators; its rank is called the Picard number. In the complex case, where $X$ is a smooth projective variety over $\mathbb{C}$, the group ${\operatorname{Pic}^{0}}(X)$ is isomorphic to the quotient group of the space ${H^{0}}(X,\Omega_{X})$ of holomorphic $1$-forms on $X$ by the lattice ${H^{1}}(X,\mathbb{Z})$.

#### References

 [1] D. Mumford, “Lectures on curves on an algebraic surface”, Princeton Univ. Press (1966). MR0209285 Zbl 0187.42701

#### References

 [a1] R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. MR0463157 Zbl 0367.14001
How to Cite This Entry:
Picard group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Picard_group&oldid=40000
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article