The groupalgebra of [tex]\mathbb{Q}/\mathbb{Z}[/tex] is ‘somewhat’ nicer in that it is at least countably generated.

So even if there is no affine variety associated to the group-algebra I’ll ask them what they think the ‘points’ of any geometrical object associated to it should be. Again, hopefully, they’ll respond something like : the one-dimensional representations of the group.

Fine! so what are they? this may take some time to explain to them but eventually we’ll agree that this is something like [tex]\hat{\mathbb{Z}}[/tex] which then may take us eventually to [tex]\prodp \mathbb{Z}_p[/tex] and then if there is still plenty of time and interest left to the adeles…

A small remark, the books by Bratteli are available at his website as (big) pdf files:

http://folk.uio.no/bratteli/bratrob/VOL-1S~1.PDF
http://folk.uio.no/bratteli/bratrob/VOL-2.pdf

Also, a naive question I’d like to know your answer to. Imagine that you are talking about this to people who just finished a first course in algebraic geometry, and once you start talking about the groupring [tex]\mathbb{Q}[\mathbb{Q}/\mathbb{Z}] [/tex] someone comes with the following:

“Ok, we know that the real circle can be defined as the quotient [tex]\mathbb{R}/\mathbb{Z} [/tex], and that its coordinate ring is the Laurent polynomial ring [tex] \mathbb{R}[x,x^{-1}] [/tex]. Why don’t we consider that quotient to be something like a rational circle and study it using the ring [tex] \mathbb{Q}[x,x^{-1}] [/tex] instead of making all that mess with adeles and the so?”

(In case I wasn’t clear, I am more intrested in the “pedagogical” part on how to answer these questions that in the math of it)