nLab Picard group

Idea

Fully generally, a Picard group is an abelian group defined for a symmetric monoidal category as the group of isomorphism classes of objects which are invertible with respect to the tensor product.

Traditionally though one speaks in the context of geometry of the Picard group $Pic(X)$ of some kind of space and by default means the invertible objects in some monoidal category of something like vector bundles over $X$. Specifically for $X$ a ringed topos (in particular a ringed space), then the monoidal category to be understood is that of locally free module sheaves over the structure sheaf and hence the Picard group in this case is that of locally free sheaves of $\mathcal{O}_X$-modules of rank $1$ (i.e. the line bundles).

Specifically in complex geometry these objects on a complex manifold $X$ are holomorphic vector bundles and hence in this case the Picard group of a $X$ is that of isomorphism classes of holomorphic line bundles. This case has an obvious genralization to schemes in algebraic geometry, and in much of the literature a Picard group is meant to be a Picard group of $\mathbb{G}_m$-torsors over a given scheme. In this (and other) geometric situations, the Picard group naturally inherits geometric structure itself and equipped with that it is then called the Picard scheme (with formal completion the formal Picard group), see there for more.

Not decategorifying by passing to isomorphism classes instead yields the concept of Picard 2-group and geometrically that of Picard stack, see there for more.

Definition

General abstract

Definition

Given a (symmetric monoidal category) monoidal category $(C, \otimes)$, the Picard group of $(C,\otimes)$ is the group of isomorphism classes of invertible objects, those that have an inverse under the tensor product – the line objects. Equivalently, this is the decategorification or 0-truncation of the Picard 2-group, the maximal 2-group inside a monoidal category.

Remark

The Picard group is indeed a group: First, if $\mathcal{L}$ and $\mathcal{M}$ are elements of $Pic(X)$, then $\mathcal{L}\otimes \mathcal{M}$ is still locally free of rank $1$ as can be seen by taking intersections of the trivializing covers. So $Pic(X)$ is closed under tensor product.

There is an identity element, since $\mathcal{O}_X\otimes \mathcal{L}\simeq \mathcal{L}$. The tensor product is associative.

Lastly, given any invertible sheaf $\mathcal{L}$ we check that $\mathcal{L}^\wedge=\mathcal{Hom}(\mathcal{L}, \mathcal{O}_X)$ is its inverse. Consider $\mathcal{L}^\wedge \otimes \mathcal{L}\simeq \mathcal{Hom}(\mathcal{L}, \mathcal{L})\simeq \mathcal{O}_X$.

In algebraic geometry

Suppose that $X$ is an integral scheme over a field. The correspondence between Cartier divisors and invertible sheaves? is given by $D\mapsto \mathcal{O}_X(D)$. If $D$ is represented by $\{(U_i, f_i)\}$, then $\mathcal{O}_X(D)$ is $\mathcal{O}_X$-submodule of $\mathcal{K}$, the sheaf of quotients, generated by $f_i^{-1}$ on $U_i$. Under our assumptions, this map is an isomorphism between the Cartier class divisor group and Picard group, but for a general scheme it is only injective. Under the additional assumptions that $X$ is separated and locally factorial, we get an isomorphism between the class divisor group and $Pic(X)$.

Another form the Picard group takes is from the isomorphism $Pic(X)\simeq H^1(X, \mathcal{O}_X^*)$. The isomorphism is most easily seen by looking at the transition functions for a trivializing cover of $\mathcal{L}$. Suppose $(\phi_i)$ trivialize $\mathcal{L}$ over the cover $(U_i)$. Then $\phi_i^{-1}\circ \phi_j$ is an automorphism of $\mathcal{O}_{U_i\cap U_j}$, i.e. a section of $\mathcal{O}_X^*(U_i\cap U_j)$. One can check this defines a ?ech cocycle $\check{H}^1(\mathcal{U}, \mathcal{O}_X^*)$ which is isomorphic to the abelian sheaf cohomology $H^1(X, \mathcal{O}_X^*)$.

Examples

• over a Riemann surface: see at Riemann surface – Picard group of holomorphic line bundles.

Properties

• Let $R$ be the ring of integers of an algebraic number field. Then $R$ is a unique factorization domain if and only if the Picard group of $R$ is trivial.

Related concepts

• group of units, Picard group, Picard stack

  • Brauer group, Azumaya algebra

• Picard 2-group, Picard ∞-group, Picard ∞-stack

• Picard scheme

• Tate?Shafarevich group?

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian

References

General

• John Baez, Hoàng Xuân Sính’s thesis: categorifying group theory (pdf)

In algebraic geometry

• Robin Hartshorne, Algebraic Geometry