I am working on a certain problem about commutative rings which has an obstruction involving the Picard group and the (algebraic) Brauer group of the ring. The obstruction is trivial at least when the Picard group is trivial or when the Brauer groups is finite.
I would like to find some suggestions of where to look for examples/counter-examples to this obstruction, so the question is:
What would you say are the simplest examples of (commutative) rings with a non-trivial Picard group and infinite Brauer group?
Here I mean "simplest" not in the sense that the rings are the simplest to describe, but rather where the Picard/Brauer groups are the simplest to compute and work with.
Of course in the question you can replace "commutative rings" with "affine schemes", but if you have some nice examples of non-affine schemes that you think are more interesting than any affine ones, I would also be interested (the problem I'm working on has clear extensions to general schemes, I just would like to start with the affine case).