Mitchell's embedding theorem explained

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

l{L} \subset \operatorname{Fun}( l{A}, Ab) be the category of left exact functors from the abelian category l{A} to the category of abelian groups Ab . First we construct a contravariant embedding H: l{A}\to l{L} by H(A) = h ^{A} for all A\in l{A} , where h ^{A} is the covariant hom-functor, h ^{A(X) = \operatorname{Hom}} _{l{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

l{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

l{L} is an AB5 category with a generator o plus _{A\inl{A} } h^A.In other words it is a Grothendieck category and therefore has an injective cogenerator I .R : = \operatorname{Hom} _{l{L} } (I,I) is the ring we need for the category of R-modules.

By

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

• Book: R. G. Swan . Algebraic K-theory, Lecture Notes in Mathematics 76 . 1968 . Springer.
• Book: Peter Freyd . Abelian Categories: An Introduction to the Theory of Functors . registration . 1964 . Harper and Row. reprinted with a forward as Abelian Categories . Reprints in Theory and Applications of Categories . 2003 . 3 . 23-164 .
• Mitchell . Barry . The Full Imbedding Theorem . American Journal of Mathematics . July 1964 . 86 . 3 . 619-637 . 10.2307/2373027 . The Johns Hopkins University Press.
• Book: Charles A. Weibel . An introduction to homological algebra . 1993 . Cambridge Studies in Advanced Mathematics.