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).

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

, 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.