Gabriel–Popescu theorem
In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain abelian categories, introduced by. It characterizes certain abelian categories as quotients of module categories.
There are several generalizations and variations of the Gabriel–Popescu theorem, given by ,, .Theorem
Let A be a Grothendieck category, 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 defined by S = Hom. 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. 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.
