Mitchell's embedding theorem

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 (associative, with 1, not necessarily commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all right 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. Sch 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 and sums of morphisms being computed 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 very easily because $h_{A}$ is already left exact. The proof of the right exactness is harder and can be read in Swan, Lecture notes on mathematics 76.

After that we prove that ${\mathcal {L}}$ is abelian by using localization theory (also Swan). ${\mathcal {L}}$ also has enough injective objects and a cogenerator. This follows easily from $\operatorname {Fun} ({\mathcal {A}},Ab)$ having these properties.

By taking the dual category of ${\mathcal {L}}$ which we call ${\mathcal {L}}^{op}$ we get an exact and fully faithful embedding from our category ${\mathcal {A}}$ to an abelian category which has enough projective objects and a generator.

We can then construct a projective generator $P$ in ${\mathcal {L}}^{op}$ whose endomorphism ring $R:=\operatorname {Hom} _{{\mathcal {L}}^{op}}(P,P)$ is the ring we need for the category of R-modules.

By $T(B)=\operatorname {Hom} _{{\mathcal {L}}^{op}}(P,B)$ we get an exact and fully faithful embedding $T:{\mathcal {L}}^{op}\to R\operatorname {-Mod}$.

References

• 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.