Gabriel–Popescu theorem

In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Pierre Gabriel and Nicolae Popescu (1964). It characterizes certain abelian categories (the Grothendieck categories) as quotients of module categories.

There are several generalizations and variations of the Gabriel–Popescu theorem, given by Kuhn (1994) (for an AB5 category with a set of generators), Lowen (2004), Porta (2010) (for triangulated categories).

Theorem

Let A be a Grothendieck category (an AB5 category with a generator), G a generator of A and R be the ring of endomorphisms of G; also, let S be the functor from A to Mod-R (the category of right R-modules) defined by S(X) = Hom(G,X). Then the Gabriel–Popescu theorem states that S is full and faithful and has an exact left adjoint.

This implies that A is equivalent to the Serre quotient category of Mod-R by a certain localizing subcategory C. (A localizing subcategory of Mod-R is a full subcategory C of Mod-R, closed under arbitrary direct sums, such that for any short exact sequence of modules ${\displaystyle 0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0}$, we have M₂ in C if and only if M₁ and M₃ are in C. The Serre quotient of Mod-R by any localizing subcategory is a Grothendieck category.) We may take C to be the kernel of the left adjoint of the functor S.

Note that the embedding S of A into Mod-R is left-exact but not necessarily right-exact: cokernels of morphisms in A do not in general correspond to the cokernels of the corresponding morphisms in Mod-R.

References

• Castaño Iglesias, Florencio; Enache, P.; Năstăsescu, Constantin; Torrecillas, Blas (2004), "Un analogue du théorème de Gabriel-Popescu et applications", Bulletin des Sciences Mathématiques, 128 (4): 323–332, doi:10.1016/j.bulsci.2003.12.004, ISSN 0007-4497, MR 2052174
• Gabriel, Pierre; Popesco, Nicolae (1964), "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes", Les Comptes rendus de l'Académie des sciences, 258: 4188–4190, MR 0166241 [Remark: "Popesco" is very likely a misspelling in that paper.]
• Kuhn, Nicholas J. (1994), "Generic representations of the finite general linear groups and the Steenrod algebra. I", American Journal of Mathematics, 116 (2): 327–360, doi:10.2307/2374932, ISSN 0002-9327, MR 1269607
• Lowen, Wendy (2004), "A generalization of the Gabriel-Popescu theorem", Journal of Pure and Applied Algebra, 190 (1): 197–211, doi:10.1016/j.jpaa.2003.11.016, ISSN 0022-4049, MR 2043328
• Mitchell, Barry (1981), "A quick proof of the Gabriel-Popesco theorem", Journal of Pure and Applied Algebra, 20 (3): 313–315, doi:10.1016/0022-4049(81)90065-7, ISSN 0022-4049, MR 0604322
• Porta, Marco (2010), "The Popescu-Gabriel theorem for triangulated categories", Advances in Mathematics, 225 (3): 1669–1715, arXiv:0706.4458, doi:10.1016/j.aim.2010.04.002, ISSN 0001-8708, MR 2673743

External links

• Lurie (2008), A Theorem of Gabriel-Kuhn-Popesco (PDF)