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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 Q-construction ： ウィキペディア英語版 Q-construction In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category ''C'', the construction creates a topological space $B^+C$ so that $\backslash pi\_0\; (B^+C)$ is the Grothendieck group of ''C'' and, when ''C'' is the category of finitely generated projective modules over a ring ''R'', for $i\; =\; 0,\; 1,\; 2$, $\backslash pi\_i\; (B^+C)$ is the ''i''-th K-group of ''R'' in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space ''BC''.) One puts :$K\_i(C)\; =\; \backslash pi\_i(B^+C)$ and call it the ''i''-th K-group of ''C''. Similarly, the ''i''-th K-group of ''C'' with coefficients in a group ''G'' is defined as the homotopy group with coefficients: :$K\_i(C;\; G)\; =\; \backslash pi\_i(B^+\; C;\; G)$. The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context. For example, one can define equivariant K-theory as $\backslash pi\_$ * of $B^+$ of the category of equivariant sheaves on a scheme. Waldhausen's S-construction generalizes the Q-construction in a stable sense; in fact, the former, which uses a more general Waldhausen category, produces a spectrum instead of a space. Grayson's binary complex also gives a construction of algebraic K-theory for exact categories.〔Daniel R. Grayson, (Algebraic K-theory via binary complexes )〕 See also module spectrum#K-theory for a K-theory of a ring spectrum. == Operations == Every ring homomorphism $R\; \backslash to\; S$ induces $B^+P(R)\; \backslash to\; B^+P(S)$ and thus $K\_i(P(R))\; =\; K\_i(R)\; \backslash to\; K\_i(S)$ where $P(R)$ is the category of finitely generated projective modules over ''R''. One can easily show this map (called transfer) agrees with one defined in Milnor's ''Introduction to algebraic K-theory''. The construction is also compatible with the suspension of a ring (cf. Grayson). 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「Q-construction」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.