Gabriel–Popescu theorem

In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by Template:Harvs. 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 Template:Harvtxt (for an AB5 category with a set of generators), Template:Harvtxt, Template:Harvtxt (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 $0 \to M_{1} \to M_{2} \to M_{3} \to 0$ , 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.