Mitchell's Embedding Theorem - Sketch of The Proof

Sketch of The Proof

First we construct an embedding from an abelian category to the category of left exact functors from the abelian category to the category of abelian groups through the functor by for all, where is the covariant hom-functor. The Yoneda Lemma states that is fully faithful and we also get the left exactness very easily because is already left exact. The proof of the right exactness is harder and can be read in Swan, Lecture notes on mathematics 76.

After that we prove that is abelian by using localization theory (also Swan). also has enough injective objects and a generator. This follows easily from having these properties.

By taking the dual category of which we call we get an exact and fully faithful embedding from our category to an abelian category which has enough projective objects and a cogenerator.

We can then construct a projective cogenerator which leads us via to the ring we need for the category of R-modules.

By we get an exact and fully faithful embedding from to the category of R-modules.