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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 Simplicial commutative ring ： ウィキペディア英語版 Simplicial commutative ring In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be shown that $\backslash pi\_0\; A$ is a commutative ring and $\backslash pi\_i\; A$ are modules over that ring (in fact, $\backslash pi\_$ * A is a graded ring over $\backslash pi\_0\; A$.) A topology-counterpart of this notion is a commutative ring spectrum. == Graded ring structure == Let ''A'' be a simplicial commutative ring. Then the ring structure of ''A'' gives $\backslash pi\_$ * A = \oplus_ \pi_i A the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, $\backslash pi\_$ * A is the homology of the chain complex corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing $S^1$ for the simplicial circle, let $x:(S^1)^\; \backslash to\; A,\; \backslash ,\; \backslash ,\; y:(S^1)^\; \backslash to\; A$ be two maps. Then the composition :$(S^1)^\; \backslash times\; (S^1)^\; \backslash to\; A\; \backslash times\; A\; \backslash to\; A$, the second map the multiplication of ''A'', induces $(S^1)^\; \backslash wedge\; (S^1)^\; \backslash to\; A$. This in turn gives an element in $\backslash pi\_\; A$. We have thus defined the graded multiplication $\backslash pi\_i\; A\; \backslash times\; \backslash pi\_j\; A\; \backslash to\; \backslash pi\_\; A$. It is associative since the smash product is. It is graded-commutative (i.e., $xy\; =\; (-1)^\; yx$) since the involution $S^1\; \backslash wedge\; S^1\; \backslash to\; S^1\; \backslash wedge\; S^1$ introduces minus sign. If ''M'' is a simplicial module over ''A'' (that is, ''M'' is a simplicial abelian group with an action of ''A''), then the similar argument shows that $\backslash pi\_$ * M has the structure of a graded module over $\backslash pi\_$ * A. (cf. module spectrum.) 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「Simplicial commutative ring」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.