### 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, $SO(n)$,
$U(n)$, $E_8$, …) 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?

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

Well, firstly, if G is a compact ring, then the connected component of the identity is also a compact ring, so the basic question is why there are no non-trivial compact connected rings.

The Peter-Weyl theorem implies that every compact group is the inverse limit of Lie groups (or in fact linear groups). In the case of connected abelian groups, they are the inverse limit of tori. So just from the additive structure, a compact connected ring is an inverse limit of tori.

Now, tori are rigid; they don’t have any endomorphisms near the identity. (By Pontryagin duality, this is equivalent to the discreteness of the Pontryagin dual.) But multiplication on a ring by an element close to 1 is an endomorphism. So there are no non-trivial elements close to 1, and so the ring is trivial.

So I think the main ingredients here are (a) compact connected abelian groups are basically tori (in particular, they have much less variability than arbitrary compact groups), and (b) tori are basically rigid.