Mitchell's embedding theorem

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

After that we prove that <math>\mathcal{L}</math> 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 <math>\mathcal{L}</math> is an AB5 category with a generator <math>\bigoplus_{A\in\mathcal{A}} h^A</math>. In other words it is a Grothendieck category and therefore has an injective cogenerator <math>I</math>.

The endomorphism ring <math>R := \operatorname{Hom}_{\mathcal{L}} (I,I)</math> is the ring we need for the category of R-modules.

By <math>G(B) = \operatorname{Hom}_{\mathcal{L}} (B,I)</math> we get another contravariant, exact and fully faithful embedding <math>G:\mathcal{L}\to R\operatorname{-Mod}.</math> The composition <math>GH:\mathcal{A}\to R\operatorname{-Mod}</math> 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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