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Picard group

Definition

Given a monoidal category $(C,\otimes )$, the Picard group of $(C,\otimes )$ 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 $\mathrm{Pic}(X)$ 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 $\mathrm{Pic}(X)$, then $ℒ\otimes ℳ$ is still locally free of rank $1$ as can be seen by taking intersections of the trivializing covers. So $\mathrm{Pic}(X)$ 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 ${ℒ}^{\wedge }=\mathrm{\mathscr{H}ℴ𝓂}(ℒ,{𝒪}_X)$ is its inverse. Consider ${ℒ}^{\wedge }\otimes ℒ\simeq \mathrm{\mathscr{H}ℴ𝓂}(ℒ,ℒ)\simeq {𝒪}_X$.

Alternate Forms

Suppose that $X$ is an integral scheme over a field. The correspondence between Cartier divisor?s and invertible sheaves is given by $D\mapsto {𝒪}_X(D)$. If $D$ is represented by $\{(U_i,f_i)\}$, then ${𝒪}_X(D)$ 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 $\mathrm{Pic}(X)$.

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

Related concepts

• group of units, Picard group, Brauer group, Azumaya algebra

• Picard 2-group, Picard ∞-group

• Picard scheme

References

• Robin Hartshorne, Algebraic Geometry