Re: dualities

["Ronnie Brown" <Ronnie@LL319dg.fsnet.co.uk>]

Peter Freyd writes on Pontrjagin duality. I would like to mention the
generalisation given in

 (with P.J. HIGGINS and S.A. MORRIS), ``Countable products of lines  and
circles: their closed subgroups, quotients and duality properties'',  {\em
Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32.

One point made is that a duality is not necessarily inherited by closed
subgroups and Hausdorff quotients. If it is, it is called a strong duality.
Then strong duality is inherited by closed subgroups and Hausdorff
quotients!

I have taught the classification of closed subgroups of R^n in an analysis
course. It is a nice result, and the sums you can set use duality in a nice
way - treatment borrowed from Bourbaki.

Ronnie Brown

> Next is Pontryagin's: the category of locally compact groups.  The
> original Pontryagin duality easily generalizes: the category of
> locally compact modules over a given commutative ring is self-dual.
> (In the non-commutative case one also obtains a duality but not a
> self-duality -- unless, of course, the ring is self-dual.) A corollary
> is that the category of discrete left  R-modules is dual to the
> category of compact right  R-modules