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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 Fundamental theorem of algebraic K-theory ： ウィキペディア英語版 Fundamental theorem of algebraic K-theory In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring ''R'' to $R()$ or $R(t^)$. The theorem was first proved by Bass for $K\_0,\; K\_1$ and was later extended to higher K-groups by Quillen. Let $G\_i(R)$ be the algebraic K-theory of the category of finitely generated modules over a noetherian ring ''R''; explicitly, we can take $G\_i(R)\; =\; \backslash pi\_i(B^+\backslash text\_R)$, where $B^+\; =\; \backslash Omega\; BQ$ is given by Quillen's Q-construction. If ''R'' is a regular ring (i.e., has finite global dimension), then $G\_i(R)\; =\; K\_i(R),$ the ''i''-th K-group of ''R''.〔By definition, $K\_i(R)\; =\; \backslash pi\_i(B^+\backslash text\_R),\; \backslash ,\; i\; \backslash ge\; 0$.〕 This is an immediate consequence of the resolution theorem, which compares the K-theories of two different categories (with inclusion relation.) For a noetherian ring ''R'', the fundamental theorem states: *(i) $G\_i(R())\; =\; G\_i(R),\; \backslash ,\; i\; \backslash ge\; 0$. *(ii) $G\_i(R(t^))\; =\; G\_i(R)\; \backslash oplus\; G\_(R),\; \backslash ,\; i\; \backslash ge\; 0,\; \backslash ,\; G\_(R)\; =\; 0$. The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for $K\_i$); this is the version proved in Grayson's paper. == References == *Daniel Grayson, , 1976 * *C. Weibel "(The K-book: An introduction to algebraic K-theory )" 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「Fundamental theorem of algebraic K-theory」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.