nLab Gabriel-Oberst duality

The category Ab of discrete abelian groups is (as a case of Pontrjagin duality for locally compact Hausdorff groups) dual category to the category of compact Hausdorff abelian groups.

Gabriel-Roos-Oberst duality refers to several generalizations of this duality for more general Grothendieck categories in terms of linearly compact topological rings and modules.

In Gabriel’s thesis

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France 90 (1962), 323-448 (numdam)

the dual of any locally finite Grothendieck category has been found. Roos has generalized this to locally noetherian Grothendieck categories in

• J. E. Roos, Locally noetherian categories and generalized strictly linearly compact rings. Applications., in: Cat. theory homology theory and their appl., “Battelle Institute Conference 1968”, vol. 2, Springer 1969, pp. 197-227

The case of (dual of) a general Grothendieck category is found in

• U. Oberst, Duality theory for Grothendieck categories, Bull. Amer. Math. Soc. 75, (1969) 1401–1408 pdf; Duality theory for Grothendieck categories and linearly compact rings, J. Alg. 15 (1970) 473–542 journal doi

A textbook exposition is in the chapter 6, Duality of

• N. Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375