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 [math]\displaystyle{ \mathcal{L} \subset \operatorname{Fun}(\mathcal{A}, Ab) }[/math] be the category of left exact functors from the abelian category [math]\displaystyle{ \mathcal{A} }[/math] to the category of abelian groups [math]\displaystyle{ Ab }[/math]. First we construct a contravariant embedding [math]\displaystyle{ H:\mathcal{A}\to\mathcal{L} }[/math] by [math]\displaystyle{ H(A) = h^A }[/math] for all [math]\displaystyle{ A\in\mathcal{A} }[/math], where [math]\displaystyle{ h^A }[/math] is the covariant hom-functor, [math]\displaystyle{ h^A(X)=\operatorname{Hom}_\mathcal{A}(A,X) }[/math]. The Yoneda Lemma states that [math]\displaystyle{ H }[/math] is fully faithful and we also get the left exactness of [math]\displaystyle{ H }[/math] very easily because [math]\displaystyle{ h^A }[/math] is already left exact. The proof of the right exactness of [math]\displaystyle{ H }[/math] is harder and can be read in Swan, Lecture Notes in Mathematics 76.
After that we prove that [math]\displaystyle{ \mathcal{L} }[/math] 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 [math]\displaystyle{ \mathcal{L} }[/math] is an AB5 category with a generator [math]\displaystyle{ \bigoplus_{A\in\mathcal{A}} h^A }[/math]. In other words it is a Grothendieck category and therefore has an injective cogenerator [math]\displaystyle{ I }[/math].
The endomorphism ring [math]\displaystyle{ R := \operatorname{Hom}_{\mathcal{L}} (I,I) }[/math] is the ring we need for the category of R-modules.
By [math]\displaystyle{ G(B) = \operatorname{Hom}_{\mathcal{L}} (B,I) }[/math] we get another contravariant, exact and fully faithful embedding [math]\displaystyle{ G:\mathcal{L}\to R\operatorname{-Mod}. }[/math] The composition [math]\displaystyle{ GH:\mathcal{A}\to R\operatorname{-Mod} }[/math] 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.
- Peter Freyd (1964). Abelian Categories: An Introduction to the Theory of Functors. Harper and Row. https://archive.org/details/abeliancategorie00frey. reprinted with a forward as "Abelian Categories". Reprints in Theory and Applications of Categories 3: 23-164. 2003. http://www.emis.de/journals/TAC/reprints/articles/3/tr3abs.html.
- Mitchell, Barry (July 1964). "The Full Imbedding Theorem". American Journal of Mathematics (The Johns Hopkins University Press) 86 (3): 619-637. doi:10.2307/2373027.
- Charles A. Weibel (1993). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics.
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