nLab Freyd-Mitchell embedding theorem

Idea

The Freyd–Mitchell embedding theorem says that every abelian category $\mathcal{A}$ is a full subcategory of a category of modules over some ring $R$, such that the embedding functor $\mathcal{A} \hookrightarrow R Mod$ 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 $R Mod$ has all small limits and colimits. But for instance, for $R$ Noetherian, the category of finitely generated $R$-modules is an abelian category but lacks these properties.

However, we have:

Theorem

(Mitchell 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 (The terminology there is a bit outdated, in that it calls an abelian category “fully abelian” if it admits a full and faithful exact functor to a category of $R$-modules.)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 $C \simeq R Mod$ for some ring $R$.

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

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

$$B \otimes_R -\colon R Mod \to S Mod$$

is right exact and preserves small coproducts. Conversely, if $F\colon Mod_R \to Mod_S$ is right exact and that preserves small coproducts, it is naturally isomorphic to $B \otimes_R -$ where $B$ is the $S$-$R$-bimodule $F R$.

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.

Related concepts

• stable model category – as A-infinity algebroid module categories

• reconstruction theorem

• derived category

References

Original articles:

• Barry Mitchell, The Full Imbedding Theorem, American Journal of Mathematics 86 3 (1964) 619-637 [doi:10.2307/2373027]

Textbook account:

• Peter Freyd, Abelian Categories, Harper and Row (1964), Reprints in Theory and Applications of Categories 3 (2003) 23-164 [tac:tr3]

Details on the proof and its variants are also in

• Charles Weibel, section 1.6 of: 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