Holy Crap, Do You Know What A Compact Ring Is?

Posted by Tom Leinster

You know how sometimes someone tells you a theorem, and it’s obviously false, and you reach for one of the many easy counterexamples only to realize that it’s not a counterexample after all, then you reach for another one and another one and find that they fail too, and you begin to concede the possibility that the theorem might not actually be false after all, and you feel your world start to shift on its axis, and you think to yourself: “Why did no one tell me this before?”

That’s what happened to me today, when my PhD student Barry Devlin — who’s currently writing what promises to be a rather nice thesis on codensity monads and topological algebras — showed me this theorem:

Every compact Hausdorff ring is totally disconnected.

I don’t know who it’s due to; Barry found it in the book Profinite Groups by Ribes and Zalesskii. And in fact, there’s also a result for rings analogous to a well-known one for groups: a ring is compact, Hausdorff and totally disconnected if and only if it can be expressed as a limit of finite discrete rings. Every compact Hausdorff ring is therefore “profinite”, that is, expressible as a limit of finite rings. So the situation for compact rings is completely unlike the situation for compact groups. There are loads of compact groups (the circle, the torus, SO ( n ) SO(n) , U ( n ) U(n) , E 8 E_8 , …) and there’s a very substantial theory of them, from Haar measure through Lie theory and onwards. But compact rings are relatively few: it’s just the profinite ones. I only laid eyes on the proof for five seconds, which was just long enough to see that it used Pontryagin duality. But how should I think about this theorem? How can I alter my worldview in such a way that it seems natural or even obvious?

Posted at August 20, 2014 11:54 PM UTC