Picard group is a recurring notion that I've been seeing very often recently recently. For projective varieties, Picard group is the divisor class group. For Dedekind domains, it is the ideal class group. For rings, Picard group can be defined on its algebraic line bundles. In the Wikepedia page, it says it can be defined for any ringed space. Or even more generally, it can be defined for any symmetric monoidal category. But I have yet to find a good exposition to link these together.
So my questions are:
How does one see that the Picard groups in the cases of divisor class groups, ideal class groups and line bundles are related?
Follow up to (1.), what are some examples (using the relation of Picard groups in different settings) that allow us to understand a number theoretic problem geometrically, or a geometry problem arithmetically?
Are there results or properties of Picard groups in the general settings (ringed spaces or symmetric monoidal categories) that turn out to be significant when we examine it in the more specific settings?