Gabriel–Popescu theorem

Theorem

Let A be a Grothendieck category (an AB5 category with a generator), U a generator of A and R be the ring of endomorphisms of U; also, let S be the functor from A to Mod-R (the category of right R-modules) defined by S(X) = Hom(U,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 → M 1 → M 2 → M 3 → 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.