nLab
Freyd-Mitchell embedding theorem

Idea

The Freyd–Mitchell embedding theorem says that every abelian category is a full subcategory of a category of modules over some ring $R$ and that the embedding is an exact functor.

Details

Remark

It is easy to see that not every abelian category is equivalent to $R$Mod for some ring $R$. The reason is that $R\mathrm{Mod}$ has all small limits and colimits. But for instance the category of finitely generated $R$-modules is an abelian category but lacks these properties.

However, we have

Mitchell’s Embedding Theorem

Every small abelian category admits a full, faithful and exact functor to the category $R$Mod for some ring $R$.

This result can be found as Theorem 7.34 on page 150 of (Freyd). (The terminology there is a bit outdated, in that it calls an abelian category “fully abelian” if it admits a full and faithful exact functor to a category of $R$-modules.) A pedagogical discussion is in section 1.6 of (Weibel). See also (Wikipedia) for the idea of the proof.

Proof

(…)

We can also characterize which abelian categories are equivalent to a category of $R$-modules:

Theorem

Let $C$ be an abelian category. If $C$ has all small coproducts and has a compact projective generator, then $C\simeq R\mathrm{Mod}$ for some ring $R$.

In fact, in this situation we can take $R=C(x,x{)}^{\mathrm{op}}$ where $x$ is any compact projective generator. Conversely, if $C\simeq R\mathrm{Mod}$, then $C$ has all small coproducts and $x=R$ is a compact projective generator.

This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of (Freyd). The first part of this theorem can also be found as Prop. 2.1.7 in (Ginzburg). Conversely, it is easy to see that $R$ is a compact projective generator of $R\mathrm{Mod}$.

Going further, we can try to characterize functors between categories of $R$-modules that come from tensoring with bimodules. Here we have

Watts’ Theorem

If $B$ is an an $S$-$R$-bimodule, the tensor product functor

$$B{\otimes }_R-:R\mathrm{Mod}\to S\mathrm{Mod}$$B \otimes_R -\colon R Mod \to S Mod

is right exact and preserves small coproducts. Conversely, if $F:{\mathrm{Mod}}_R\to {\mathrm{Mod}}_S$ is right exact and that preserves small coproducts, it is naturally isomorphic to $B{\otimes }_R-$ where $B$ is the $S$-$R$-bimodule $FR$.

This theorem was more or less simultaneously proved by Watts and Eilenberg; a generalization is proved in (Nyman-Smith), and references to the original papers can be found there.

Going still further we should be able to obtain a nice theorem describing the image of the embedding of the 2-category of

• rings
• bimodules
• bimodule homomorphisms

into the strict 2-category of

• abelian categories
• right exact functors
• natural transformations.

For more discussion see the $n$-Cafe.

References

A standard textbook is

• Peter Freyd, Abelian Categories_ (pdf)

Details on the proof and its variants are also in

section 1.6 of

• Charles Weibel, An Introduction to Homological Algebra

and

• Victor Ginzburg, Lectures on noncommutative geometry (pdf)

• A. Nyman , S. Paul Smith, A generalization of Watts’s Theorem: Right exact functors on module categories (arXiv:0806.0832)

An introductory survey is for instance also in section 3 of

• Geillan Aly, Abelian Categories and the Freyd-Mitchell Embedding Theorem (pdf)

See also

• Wikipedia, Mitchell’s embedding theorem