Mitchell's embedding theorem

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 ${\mathcal {L}}\subset \operatorname {Fun} ({\mathcal {A}},Ab)$ 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

R. G. Swan (1968). Algebraic K-theory, Lecture Notes in Mathematics 76. Springer. doi:10.1007/BFb0080281. ISBN 978-3-540-04245-7.
Peter Freyd (1964). Abelian Categories: An Introduction to the Theory of Functors. Harper and Row. reprinted with a forward as "Abelian Categories". Reprints in Theory and Applications of Categories. 3: 23–164. 2003.
Mitchell, Barry (July 1964). "The Full Imbedding Theorem". American Journal of Mathematics. 86 (3). The Johns Hopkins University Press: 619–637. doi:10.2307/2373027. JSTOR 2373027.
Charles A. Weibel (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. doi:10.1017/CBO9781139644136. ISBN 9781139644136.