Wednesday, September 15, 2010

A Minor Problem with the Étale groupoids of Bost-Connes

EDIT: I was confused and delusional when writing this blog; the problem discussed below does not actually exist. I didn't know how to take the inverse in the groupoids mentioned. It's actually quite simple and will be discussed in a later post. This entry can be safely ignored.

I had hoped to move beyond my study of the groupoids mentioned in the construction of the Bost Connes system on Monday. Yet it wasn't til Tuesday that I had fully understood what an Étale groupoid was. Though I had thought I had understood the aforementioned groupoids and the proofs of their isomorphisms, I realized in the middle of my writeup last night that I was still confused about things. I worked out some details on paper, but there's one thing left that I'm hung up on. It has to do with the inversion of the elements in the groupoid. Here's the detail isolated all by itself:

Given a $\rho \in \hat{\mathbb{Z}}$ and a positive rational $r$ such that $r \rho \in \hat{\mathbb{Z}}$. I need that $\frac{1}{r} \rho \in \hat{\mathbb{Z}}$ as well. But I don't see how this could possible be true, at least not when I'm thinking of $\hat{\mathbb{Z}}$ as an inverse limit. However....

As I proved earlier, $\hat{\mathbb{Z}}$ is isomorphic to $End(\mathbb{Q}/\mathbb{Z})$, so we can think of $\rho$ as being a homomorphism $\rho : \mathbb{Q}/\mathbb{Z} \to \mathbb{Q}/\mathbb{Z}$. Now if $ r \rho$ is still such a homomorphism ($(r \rho )(x) =r \rho(x)$) then I can't see any reason why $\frac{1}{r} \rho$ wouldn't be. It would preserve addition and the identity on $\mathbb{Q}/\mathbb{Z}$ just fine (I think).

So that brings me to another there any positive rational $r$ and homomorphism $\rho \in End(\mathbb{Q}/\mathbb{Z})$ such that $r \rho$ is not a homomorphism? If not, then why does Connes state that requirement?

Some thoughts immediately after writing this post
A homomorphism $\rho \in End(\mathbb{Q}/\mathbb{Z})$ must preserve torsion, id est, there's a smallest $n$ such that an $x \in \mathbb{Q}/\mathbb{Z}$ will have $n x = 0$. $\rho$ must preserve $n \rho(x) = 0$ if it's going to be a homomorphism. Multiplication by a natural number $k$ is an a homomorphism on $\mathbb{Q}/\mathbb{Z}$, but multiplication by $\frac{1}{k}$ is not. Hence I have my $r=k$ where $r \rho \in End(\mathbb{Q}/\mathbb{Z})$ but $\frac{1}{r} \rho$ is not. And thus I still have a problem inverting elements in the groupoids mentioned. I'll have to spell out this problem more explicitly in another post.

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