Editing 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 category|abelian categories]]; it essentially states that these categories, while rather abstractly defined, are in fact [[concrete category|concrete categories]] of [[module (mathematics)|modules]]. This allows one to use element-wise [[diagram chasing]] proofs in these categories. The theorem is named after [[Barry Mitchell (mathematician)|Barry Mitchell]] and [[Peter Freyd]].

==Details==
The precise statement is as follows: if '''A''' is a small abelian category, then there exists a [[ring (mathematics)|ring]] ''R'' (with&nbsp;1, not necessarily commutative) and a [[full functor|full]], [[faithful functor|faithful]] and [[exact functor]] ''F'': '''A''' → ''R''-Mod (where the latter denotes the category of all [[module (mathematics)|left ''R''-modules]]).

The functor ''F'' yields an [[equivalence of categories|equivalence]] between '''A''' and a [[full subcategory]] of ''R''-Mod in such a way that [[kernel (category theory)|kernels]] and [[cokernel]]s computed in '''A''' correspond to the ordinary kernels and cokernels computed in ''R''-Mod. Such an equivalence is necessarily [[additive functor|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 sequence]]s and sums of morphisms being determined as in the case of modules. However, [[projective object|projective]] and [[injective object|injective]] objects in '''A''' do not necessarily correspond to projective and injective ''R''-modules.

== Sketch of the proof ==
Let <math>\mathcal{L} \subset \operatorname{Fun}(\mathcal{A}, Ab)</math> be the category of [[left exact functor]]s from the abelian category <math>\mathcal{A}</math> to the [[category of abelian groups]] <math>Ab</math>. First we construct a [[Covariance and contravariance of functors|contravariant]] embedding <math>H:\mathcal{A}\to\mathcal{L}</math> by <math>H(A) = h^A</math> for all <math>A\in\mathcal{A}</math>, where <math>h^A</math> is the covariant hom-functor, <math>h^A(X)=\operatorname{Hom}_\mathcal{A}(A,X)</math>. The [[Yoneda Lemma]] states that <math>H</math> is fully faithful and we also get the left exactness of <math>H</math> very easily because <math>h^A</math> is already left exact. The proof of the right exactness of <math>H</math> is harder and can be read in Swan, ''Lecture Notes in Mathematics 76''.

After that we prove that <math>\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>\mathcal{L}</math> is an [[AB5 category]] with a [[Generator (category theory)|generator]] 
<math>\bigoplus_{A\in\mathcal{A}} h^A</math>.
In other words it is a [[Grothendieck category]] and therefore has an injective cogenerator <math>I</math>.

The [[endomorphism ring]] <math>R := \operatorname{Hom}_{\mathcal{L}} (I,I)</math> is the ring we need for the category of ''R''-modules.

By <math>G(B) = \operatorname{Hom}_{\mathcal{L}} (B,I)</math> we get another contravariant, exact and fully faithful embedding <math>G:\mathcal{L}\to R\operatorname{-Mod}.</math> The composition <math>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 category|exact categories]] is almost identical.

== References ==
{{refbegin}}
*{{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
}}
{{refend}}

[[Category:Module theory]]
[[Category:Additive categories]]
[[Category:Theorems in algebra]]