×

Dualities over compact commutative rings. (English) Zbl 0545.22004

Let S be a compact topological ring with 1 and \({\mathcal L}_ S\) the category of locally compact topological S-modules and continuous S-module morphisms. If \({\mathcal M}\) and \({\mathcal N}\) are full subcategories closed under the passage to isomorphic copies and the formation of closed submodules and of quotients, and if \({\mathcal M}\) contains all compact modules and \({\mathcal N}\) all discrete ones, then a duality implemented by two contravariant functors F:\({\mathcal M}\to {\mathcal N}\) and G:\({\mathcal N}\to {\mathcal M}\) is said to be a duality over S if \(F(sf)=sF(f)\) for all \(s\in S\) and all \({\mathcal M}\)-morphisms f and if a similar condition holds for G. In the paper it is shown that up to a natural equivalence, every duality over S between \({\mathcal M}\) and \({\mathcal N}\) is Pontryagin duality. For this purpose, certain properties of compact rings are used and the methods introduced by D. W. Roeder are applied [Trans. Am. Math. Soc. 154, 151-175 (1971; Zbl 0216.343)]. Not all existing literature on locally compact topological rings and modules is covered [cf. e. g. O. Goldman and C.-H. Sah, J. Algebra 4, 71-95 (1966; Zbl 0163.036), and ibid 11, 363-454 (1969; Zbl 0209.339), F. Eckstein and G. Michler, Arch. Math. 23, 10-18 (1972; Zbl 0245.16028)].
Reviewer: K.H.Hofmann

MSC:

22D35 Duality theorems for locally compact groups
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
PDFBibTeX XMLCite