category with duals (list of them)
dualizable object (what they have)
Given a monoidal category , the Picard group of 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 question is often assumed by default to be a category of vector bundles or quasicoherent sheaves over some space. For instance The Picard group of a ringed space is the Picard group of the monoidal category of invertible sheaves?, i.e. the locally free sheaves of -modules of rank (i.e. the line bundles).
First, if and are elements of , then is still locally free of rank as can be seen by taking intersections of the trivializing covers. So is closed under tensor product.
There is an identity element, since . The tensor product is associative.
Lastly, given any invertible sheaf we check that is its inverse. Consider .
Suppose that is an integral scheme over a field. The correspondence between Cartier divisor?s and invertible sheaves is given by . If is represented by , then is -submodule of , the sheaf of quotients, generated by on . 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 is separated and locally factorial, we get an isomorphism between the class divisor group and .
Another form the Picard group takes is from the isomorphism . The isomorphism is most easily seen by looking at the transition functions for a trivializing cover of . Suppose trivialize over the cover . Then is an automorphism of , i.e. a section of . One can check this defines a Čech cocycle which is isomorphic to the abelian sheaf cohomology .