Mitchell's embedding theorem

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Mitchell's embedding theorem, also known as the Freyd–Mitchell 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 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: AR-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[edit]

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 \mathcal{L} has an injective cogenerator

I=\prod_{A\in\mathcal{A}} h_A.

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[edit]

  • R. G. Swan (1968). Lecture Notes in Mathematics 76. Springer. 
  • Peter Freyd (1964). Abelian categories. Harper and Row. 
  • Barry Mitchell (1964). The full imbedding theorem. The Johns Hopkins University Press. 
  • Charles A. Weibel (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics.