Mitchell's embedding theorem - Alchetron, the free social encyclopedia

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 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 ⊂ Fun ⁡ ( A , A b ) be the category of left exact functors from the abelian category A to the category of abelian groups A b . First we construct a contravariant embedding H : A → L by H ( A ) = h A for all A ∈ A , where h A is the covariant hom-functor, h A ( X ) = Hom 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 is an abelian category by using localization theory (also Swan). This is the hard part of the proof.

It is easy to check that L has an injective cogenerator

I = ∏ A ∈ A h A .

The endomorphism ring R := Hom L ⁡ ( I , I ) is the ring we need for the category of R-modules.

By G ( B ) = Hom L ⁡ ( B , I ) we get another contravariant, exact and fully faithful embedding G : L → R - M o d . The composition G H : A → R - M o d 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

Mitchell's embedding theorem Wikipedia

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