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

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

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