From dinakar@its.caltech.edu Thu Oct 7 22:17:24 1999
Return-Path:
Received: from blinky.its.caltech.edu (blinky.its.caltech.edu [131.215.48.132])
by zorn2.math.leidenuniv.nl (8.9.3/8.8.7) with ESMTP id WAA30264
for ; Thu, 7 Oct 1999 22:17:23 +0200
Received: from localhost (dinakar@localhost)
by blinky.its.caltech.edu (8.9.1/8.9.1) with ESMTP id NAA25132;
Thu, 7 Oct 1999 13:15:01 -0700 (PDT)
X-Authentication-Warning: blinky.its.caltech.edu: dinakar owned process doing -bs
Date: Thu, 7 Oct 1999 13:15:01 -0700 (PDT)
From: Dinakar Ramakrishnan
X-Sender: dinakar@blinky
To: Peter Stevenhagen
cc: desmit@math.leidenuniv.nl, drama@ias.edu
Subject: Re: FANF
In-Reply-To: <199910071610.SAA13244@noot.wins.uva.nl>
Message-ID:
MIME-Version: 1.0
Content-Type: TEXT/PLAIN; charset=US-ASCII
Status: RO
Dear Peter and Bart (if I may),
(you may address me as Dinakar; we are not so formal in the States)
Thanks for your message. I am on sabbatical this year at the IAS.
>
> You may be pleased to hear that we have selected your
> book `Fourier analysis on number fields' as a text
> for our local graduate student seminar in number theory.
>
Thanks.
> We are a bit confused by the conventions on topological groups
> that you introduce in the first chapter.
> Topological groups are not assumed to be T_1 (page 5), so for
> all we know the topology might be indiscrete. But how can we
> then prove that the closure of an abelian subgroup (such as {e})
> is always abelian (exercise 1.7)?
You are right. We neglected to tell the reader to assume that the group
is Hausdorff for this exercise.
> Even more troubling, you prove in theorem
1.18 (page
31) that > subgroups of finite index in a profinite group are always open.
> If we take G to be a countably infinite product of fields F2
> (of two elements), then G is a topological ring. Any maximal ideal
> containing the direct sum of all F2's inside the direct product G
> is an additive subgroup of index 2 that is not open in G.
>
Again you are right. I should have proofread carefully as I am aware of
this example and the pitfalls. The statement of
Thm.1-18 should say that a closed subgroup H of a profinite group G is
open iff G/H is finite. The proof as given seems to go through with this
correction.
I have two comments:
(i) As I learnt from Serre a while ago, one can prove the following: If H
is a topologically finitely generated subgroup of a profinite group G with
G/H finite, then H is open.
(ii) I will be very interested if you can come up with a counterexample
which does not use the axiom of choice.
________________________________________________________________________
It is certainly embarrassing to find such errors in one's book. But I am
grateful to be apparaised of them so I can correct them in the next
printing or issue a sheet of errata.
Best.
Dinakar