Dualities of locally compact modules over the rationals

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