category with duals (list of them)
dualizable object (what they have)
Traditionally though one speaks in the context of geometry of the Picard group of some kind of space and by default means the invertible objects in some monoidal category of something like vector bundles over . Specifically for a ringed topos (in particular a ringed space), then the monoidal category to be understood is that of locally free module sheaves over the structure sheaf and hence the Picard group in this case is that of locally free sheaves of -modules of rank (i.e. the line bundles).
Specifically in complex geometry these objects on a complex manifold are holomorphic vector bundles and hence in this case the Picard group of a is that of isomorphism classes of holomorphic line bundles. This case has an obvious genralization to schemes in algebraic geometry, and in much of the literature a Picard group is meant to be a Picard group of -torsors over a given scheme. In this (and other) geometric situations, the Picard group naturally inherits geometric structure itself and equipped with that it is then called the Picard scheme (with formal completion the formal Picard group), see there for more.
Given a (symmetric monoidal category) monoidal category , the Picard group of 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 or 0-truncation of the Picard 2-group, the maximal 2-group inside a monoidal category.
The Picard group is indeed a group: 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 divisors 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 .
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|