Mitchell's embedding theorem

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{L} \subset \operatorname{Fun}(\mathcal{A}, Ab) be the category of left exact functors from the abelian category \mathcal{A} to the category of abelian groups Ab. First we construct a contravariant embedding H:\mathcal{A}\to\mathcal{L} by H(A) = h^A for all A\in\mathcal{A}, where h^A is the covariant hom-functor, h^A(X)=\operatorname{Hom}_\mathcal{A}(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{L} 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{L} is an AB5 category with a generator \bigoplus_{A\in\mathcal{A}} h^A. In other words it is a Grothendieck category and therefore has an injective cogenerator I.

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

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

• {{cite book | author-link= Richard Swan
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• {{cite book | author-link= Charles A. Weibel