# nLab Gabriel-Popescu theorem

Girauds subcategories and Gabriel-Popescu theorem

# Giraud’s subcategories and Gabriel-Popescu theorem

The Gabriel-Popescu theorem, also quoted as Popesco-Gabriel theorem because of that spelling in the original published paper (cf. the spelling in that order, which is sometimes 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).

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

Last revised on December 8, 2019 at 10:03:07. See the history of this page for a list of all contributions to it.