nLab
Picard group

Context

Monoidal categories

Group Theory

Contents

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)Pic(X) of some kind of space and by default means the invertible objects in some monoidal category of something like vector bundles over XX. Specifically for XX 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 𝒪 X\mathcal{O}_X-modules of rank 11 (i.e. the line bundles).

Specifically in complex geometry these objects on a complex manifold XX are holomorphic vector bundles and hence in this case the Picard group of a XX 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 𝔾 m\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,)(C, \otimes), the Picard group of (C,)(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)Pic(X), then \mathcal{L}\otimes \mathcal{M} is still locally free of rank 11 as can be seen by taking intersections of the trivializing covers. So Pic(X)Pic(X) is closed under tensor product.

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

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

In algebraic geometry

Suppose that XX is an integral scheme over a field. The correspondence between Cartier divisors and invertible sheaves? is given by D𝒪 X(D)D\mapsto \mathcal{O}_X(D). If DD is represented by {(U i,f i)}\{(U_i, f_i)\}, then 𝒪 X(D)\mathcal{O}_X(D) is 𝒪 X\mathcal{O}_X-submodule of 𝒦\mathcal{K}, the sheaf of quotients, generated by f i 1f_i^{-1} on U iU_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 XX is separated and locally factorial, we get an isomorphism between the class divisor group and Pic(X)Pic(X).

Another form the Picard group takes is from the isomorphism Pic(X)H 1(X,𝒪 X *)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 (ϕ i)(\phi_i) trivialize \mathcal{L} over the cover (U i)(U_i). Then ϕ i 1ϕ j\phi_i^{-1}\circ \phi_j is an automorphism of 𝒪 U iU j\mathcal{O}_{U_i\cap U_j}, i.e. a section of 𝒪 X *(U iU j)\mathcal{O}_X^*(U_i\cap U_j). One can check this defines a Čech cocycle Hˇ 1(𝒰,𝒪 X *)\check{H}^1(\mathcal{U}, \mathcal{O}_X^*) which is isomorphic to the abelian sheaf cohomology H 1(X,𝒪 X *)H^1(X, \mathcal{O}_X^*).

Examples

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-Cau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
n=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian

References

General

In algebraic geometry

  • Robin Hartshorne, Algebraic Geometry

Revised on May 21, 2014 13:15:53 by Urs Schreiber (82.136.246.44)