Dualities of locally compact modules over the rationals

AbstractThe concept of continuity of a duality (i.e., involutive contravariant endofunctor) of the category LR of locally compact modules over a discrete commutative ring R, was introduced by Prodanov. Orsatti and the first-named author proved that the category LR admits discontinuous dualities when R is a large field of characteristic zero. We prove that all dualities of LR 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 LQ 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 LQ is related to the fact that the adele ring AQ of the rationals has no ring automorphisms beyond the identity