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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 is a commutative ring and are modules over that ring (in fact, is a graded ring over .) 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 the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, is the homology of the chain complex corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing for the simplicial circle, let be two maps. Then the composition :, the second map the multiplication of ''A'', induces . This in turn gives an element in . We have thus defined the graded multiplication . It is associative since the smash product is. It is graded-commutative (i.e., ) since the involution 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 has the structure of a graded module over . (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
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