# nLab Picard group

### Context

#### Monoidal categories

monoidal categories

group theory

# Contents

## Definition

Given a 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 of the Picard 2-group, the maximal 2-group inside a monoidal category.

In geometry, the monoidal category in question is often assumed by default to be a category of vector bundles or quasicoherent sheaves over some space. For instance The Picard group $Pic(X)$ of a ringed space $X$ is the Picard group of the monoidal category of invertible sheaves?, i.e. the locally free sheaves of $\mathcal{O}_X$-modules of rank $1$ (i.e. the line bundles).

## Pic(X) is 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 $\mathcal{L}^\wedge \otimes \mathcal{L}\simeq \mathcal{Hom}(\mathcal{L}, \mathcal{L})\simeq \mathcal{O}_X$.

## Alternate Forms

Suppose that $X$ is an integral scheme over a field. The correspondence between Cartier divisor?s 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 Čech cocycle $\check{H}^1(\mathcal{U}, \mathcal{O}_X^*)$ which is isomorphic to the abelian sheaf cohomology $H^1(X, \mathcal{O}_X^*)$.

## References

• Robin Hartshorne, Algebraic Geometry

Revised on February 15, 2014 10:54:13 by Urs Schreiber (89.204.139.93)