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Picard group

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Monoidal categories

Group Theory

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Definition

Given a 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 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)Pic(X) of a ringed space XX is the Picard group of the monoidal category of invertible sheaves?, i.e. the locally free sheaves of 𝒪 X\mathcal{O}_X-modules of rank 11 (i.e. the line bundles).

Pic(X) is 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.

Alternate Forms

Suppose that XX is an integral scheme over a field. The correspondence between Cartier divisor?s 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^*).

References

  • Robin Hartshorne, Algebraic Geometry

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