Mitchell's embedding theorem

Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows one to use element-wise diagram chasing proofs in these categories. The theorem is named after Barry Mitchell and Peter Freyd.

Details

The precise statement is as follows: if A is a small abelian category, then there exists a ring ''R'' (with 1, not necessarily commutative) and a full, faithful and exact functor ''F'': A → ''R''-Mod (where the latter denotes the category of all left ''R''-modules).

The functor ''F'' yields an equivalence between A and a full subcategory of ''R''-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in ''R''-Mod. Such an equivalence is necessarily additive.
The theorem thus essentially says that the objects of A can be thought of as ''R''-modules, and the morphisms as ''R''-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules. However, projective and injective objects in A do not necessarily correspond to projective and injective ''R''-modules.

Sketch of the proof

Let $\mathcal \subset \operatorname(\mathcal, Ab)$ be the category of left exact functors from the abelian category $\mathcal$ to the category of abelian groups $Ab$. First we construct a contravariant embedding $H:\mathcal\to\mathcal$ by $H(A) = h^A$ for all $A\in\mathcal$, where $h^A$ is the covariant hom-functor, $h^A(X)=\operatorname_\mathcal(A,X)$. The Yoneda Lemma states that $H$ is fully faithful and we also get the left exactness of $H$ very easily because $h^A$ is already left exact. The proof of the right exactness of $H$ is harder and can be read in Swan, ''Lecture Notes in Mathematics 76''.

After that we prove that $\mathcal$ is an abelian category by using localization theory (also Swan). This is the hard part of the proof.

It is easy to check that the abelian category $\mathcal$ is an AB5 category with a generator
$\bigoplus_ h^A$.
In other words it is a Grothendieck category and therefore has an injective cogenerator $I$.

The endomorphism ring $R := \operatorname_ (I,I)$ is the ring we need for the category of ''R''-modules.

By $G(B) = \operatorname_ (B,I)$ we get another contravariant, exact and fully faithful embedding $G:\mathcal\to R\operatorname.$ The composition $GH:\mathcal\to R\operatorname$ is the desired covariant exact and fully faithful embedding.

Note that the proof of the Gabriel–Quillen embedding theorem for exact categories is almost identical.

References

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Module theory
Additive categories
Theorems in algebra