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Freyd-Mitchell embedding theorem

Context

Category theory

Additive and abelian categories

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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

Details

Remark

It is easy to see that not every abelian category is equivalent to RRMod for some ring RR. The reason is that RModR Mod has all small limits and colimits. But for instance the category of finitely generated RR-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 RRMod for some ring RR.

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 (The terminology there is a bit outdated, in that it calls an abelian category “fully abelian” if it admits a full and faithful exact functor to a category of RR-modules.)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 RR-modules:

Theorem

Let CC be an abelian category. If CC has all small coproducts and has a compact projective generator, then CRModC \simeq R Mod for some ring RR.

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

This theorem, minus the explicit description of RR, 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 RR is a compact projective generator of RModR Mod.

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

Watts’ Theorem

If BB is an an SS-RR-bimodule, the tensor product functor

B R:RModSMod B \otimes_R -\colon R Mod \to S Mod

is right exact and preserves small coproducts. Conversely, if F:Mod RMod SF\colon Mod_R \to Mod_S is right exact and that preserves small coproducts, it is naturally isomorphic to B RB \otimes_R - where BB is the SS-RR-bimodule FRF 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 nn-Cafe.

References

A standard textbook is

Details on the proof and its variants are also in

section 1.6 of

and

  • 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

Revised on August 25, 2012 23:08:16 by Urs Schreiber (82.113.106.150)