Clairaut's equation: Difference between revisions

Latest revision as of 16:23, 30 August 2013

Mitchell's embedding theorem, also known as the Freyd–Mitchell 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 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 $ℒ \subset Fun (𝒜, A b)$ be the category of left exact functors from the abelian category $𝒜$ to the category of abelian groups $A b$ . First we construct a contravariant embedding $H : 𝒜 \to ℒ$ by $H (A) = h_{A}$ for all $A \in 𝒜$ , where $h_{A}$ is the covariant hom-functor, $h_{A} (X) = {Hom}_{𝒜} (A, X)$ . The Yoneda Lemma states that $H$ is fully faithful and we also get the left exactness of $H$ very easily because $h_{A}$ is already left exact. The proof of the right exactness of $H$ is harder and can be read in Swan, Lecture Notes in Mathematics 76.

After that we prove that $ℒ$ is an abelian category by using localization theory (also Swan). This is the hard part of the proof.

It is easy to check that $ℒ$ has an injective cogenerator

I = \prod_{A \in 𝒜} h_{A} .

The endomorphism ring $R : = {Hom}_{ℒ} (I, I)$ is the ring we need for the category of R-modules.

By $G (B) = {Hom}_{ℒ} (B, I)$ we get another contravariant, exact and fully faithful embedding $G : ℒ \to R -Mod .$ The composition $G H : 𝒜 \to R -Mod$ 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

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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+'''Mitchell's embedding theorem''', also known as the '''Freyd–Mitchell 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 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|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 <math>\mathcal{L}</math> has an [[injective cogenerator]]
+:<math>I=\prod_{A\in\mathcal{A}} h_A.</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      = Lecture Notes in Mathematics 76
+ | year       = 1968
+ | publisher  = Springer
+}}
+*{{cite book
+ | author     = Peter Freyd
+ | title      = Abelian categories
+ | year       = 1964
+ | publisher  = Harper and Row
+}}
+*{{cite book
+ | author     = Barry Mitchell
+ | title      = The full imbedding theorem
+ | year       = 1964
+ | 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
+}}
+{{refend}}
+[[Category:Module theory]]
+[[Category:Additive categories]]
+[[Category:Theorems in algebra]]

Clairaut's equation: Difference between revisions

Latest revision as of 16:23, 30 August 2013

Sketch of the proof

References

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