Gabriel-Popescu theorem

The **Gabriel-Popescu theorem**, also quoted as Popesco-Gabriel theorem (cf. the spelling in that order, which is considered wrong when in the order of the authors!) asserts that every Grothendieck category can be represented as a reflective subcategory of a category ${}_R Mod$ of modules over a unital ring, where the localization functor is exact (one calls such subcategories of module categories Giraud subcategories).

- Pierre Gabriel, Nicolae Popescu,
*Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes*, Les Comptes rendus de l’Académie des sciences Paris**258**: 4188–4190 MR 0166241 - wikipedia Gabriel-Popesco theorem
- Nicolae Popescu, Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. MR0340375

There are various generalizations, e.g.

- Wendy Lowen,
*A generalization of the Gabriel-Popescu theorem*, Journal of Pure and Applied Algebra 190 (1) (2004): 197–211, doi MR2043328

N. Kuhn had related results in the study of Steenrod algebra, cf. also

- Nicholas J. Kuhn,
*Generic representations of the finite general linear groups and the Steenrod algebra. I*, American Journal of Mathematics 116 (2): 327–360 (1994) doi - Jacob Lurie,
*A theorem of Gabriel-Kuhn-Popesco*, MIT math 917 notes, lecture 8

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