Stoyanov, Luchezar Dualities over compact commutative rings. (English) Zbl 0545.22004 Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat. 7, No. 1, 155-176 (1983). 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 Cited in 3 Documents MSC: 22D35 Duality theorems for locally compact groups 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) Keywords:compact ring; Pontryagin duality Citations:Zbl 0216.343; Zbl 0163.036; Zbl 0209.339; Zbl 0245.16028 PDFBibTeX XMLCite \textit{L. Stoyanov}, Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat. 7, No. 1, 155--176 (1983; Zbl 0545.22004)