Dualities of locally compact modules over the rationals

✍ Scribed by Dikran Dikranjan; Chiara Milan

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 316 KB
• Volume: 256
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00115-1

✦ Synopsis

The concept of continuity of a duality (i.e., involutive contravariant endofunctor) of the category L R of locally compact modules over a discrete commutative ring R, was introduced by Prodanov. Orsatti and the first-named author proved that the category L R admits discontinuous dualities when R is a large field of characteristic zero. We prove that all dualities of L R are continuous when R = Q is the discrete field of rationals numbers, while this fails to be true for the discrete fields R and C of the real and of the complex numbers, respectively. More generally, we describe the finitely closed subcategories L of L Q such that all dualities of L are continuous. All dualities of such a category L turn out to be naturally equivalent to the Pontryagin duality. This property extends to R and C. The continuity of all dualities of L Q is related to the fact that the adele ring A Q of the rationals has no ring automorphisms beyond the identity.