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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mitchell's embedding theorem ： ウィキペディア英語版 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 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 $\backslash mathcal\; \backslash subset\; \backslash operatorname(\backslash mathcal,\; Ab)$ be the category of left exact functors from the abelian category $\backslash mathcal$ to the category of abelian groups $Ab$. First we construct a contravariant embedding $H:\backslash mathcal\backslash to\backslash mathcal$ by $H(A)\; =\; h\_A$ for all $A\backslash in\backslash mathcal$, where $h\_A$ is the covariant hom-functor, $h\_A(X)=\backslash operatorname\_\backslash mathcal(A,X)$. 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 $\backslash mathcal$ is an abelian category by using localization theory (also Swan). This is the hard part of the proof. It is easy to check that $\backslash mathcal$ has an injective cogenerator :$I=\backslash prod\_\_\_\backslash to\; R\backslash operatorname.$ The composition $GH:\backslash mathcal\backslash to\; R\backslash operatorname$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mitchell's embedding theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.