Mitchell's embedding theorem

{{short description|Abelian categories, while abstractly defined, are in fact concrete categories of modules}}

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

{{cite book

| author = R. G. Swan

| title = Algebraic K-theory, Lecture Notes in Mathematics 76

| year = 1968

| publisher = Springer

|isbn = 978-3-540-04245-7

|doi = 10.1007/BFb0080281}}

{{cite book

| author = Peter Freyd

| title = Abelian Categories: An Introduction to the Theory of Functors

| url = https://archive.org/details/abeliancategorie00frey

| url-access = registration

| year = 1964

| publisher = Harper and Row

}} reprinted with a forward as {{cite journal |title=Abelian Categories |journal=Reprints in Theory and Applications of Categories |date=2003 |volume=3 |pages=23-164 |url=http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html}}

{{cite journal

|last1 = Mitchell

|first1 = Barry

|title = The Full Imbedding Theorem

|journal = American Journal of Mathematics

|date = July 1964

|volume = 86

|issue = 3

|pages = 619–637

|doi = 10.2307/2373027

|jstor = 2373027

|publisher = The Johns Hopkins University Press}}

{{cite book

| author = Charles A. Weibel

| title = An introduction to homological algebra

| year = 1993

| publisher = Cambridge Studies in Advanced Mathematics

|isbn=9781139644136

|doi=10.1017/CBO9781139644136

}}

Category:Module theory

Category:Additive categories

Category:Theorems in algebra