# nLab Picard group

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

group theory

# Contents

## Definition

Given a monoidal category $\left(C,\otimes \right)$, the Picard group of $\left(C,\otimes \right)$ is the group of isomorphism classes of objects 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 queszion is often assumed by default to be a category of vector bundles or quasicoherent sheaves over some space. For instance The Picard group $\mathrm{Pic}\left(X\right)$ of a ringed space $X$ is the Picard group of the monoidal category of invertible sheaves?, i.e. the locally free sheaves of ${𝒪}_{X}$-modules of rank $1$ (i.e. the line bundles).

## Pic(X) is a Group

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

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

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

## Alternate Forms

Suppose that $X$ is an integral scheme over a field. The correspondence between Cartier divisor?s and invertible sheaves is given by $D↦{𝒪}_{X}\left(D\right)$. If $D$ is represented by $\left\{\left({U}_{i},{f}_{i}\right)\right\}$, then ${𝒪}_{X}\left(D\right)$ is ${𝒪}_{X}$-submodule of $𝒦$, 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 $\mathrm{Pic}\left(X\right)$.

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

## References

• Robin Hartshorne, Algebraic Geometry

Revised on January 16, 2013 20:44:54 by Urs Schreiber (203.116.137.162)