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2014-01-29T01:25:18Z

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In algebra, a '''simplicial commutative ring''' is a commutative [[monoid]] in the category of [[simplicial abelian group]]s, or, equivalently, a [[simplicial object]] in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be shown that <math>\pi_0 A</math> is a commutative [[ring (mathematics)|ring]] and <math>\pi_i A</math> are modules over that ring (in fact, <math>\pi_* A</math> is a graded ring.)

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 <math>\pi_* A = \oplus_{i \ge 0} \pi_i A</math> a structure of graded-commutative graded ring as follows.

By the [[Dold–Kan correspondence]], <math>\pi_* A</math> is the homology of the chain complex corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing <math>S^1</math> for the circle, let <math>x:(S^1)^{\wedge i} \to A, \, \, y:(S^1)^{\wedge j} \to A</math> be two maps. Then the composition
:<math>(S^1)^{\wedge i} \times (S^1)^{\wedge j} \to A \times A \to A</math>,
the second map the multiplication of ''A'', induces <math>(S^1)^{\wedge i} \wedge (S^1)^{\wedge j} \to A</math>. This in turn gives an element in <math>\pi_{i + j} A</math>. We have thus defined the graded multiplication <math>\pi_i A \times \pi_j A \to \pi_{i + j} A</math>. It is associative since the smash product is. It is graded-commutative (i.e., <math>xy = (-1)^{|x||y|} yx</math>) since the involution <math>S^1 \wedge S^1 \to S^1 \wedge S^1</math> introduces minus sign.

== Spec ==
By definition, the category of affine [[derived scheme]]s is the opposite category of the category of simplicial commutative rings; an object corresponding to ''A'' will be denoted by <math>\operatorname{Spec} A</math>.

== References ==
*http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory/
*http://mathoverflow.net/questions/45273/what-facts-in-commutative-algebra-fail-miserably-for-simplicial-commutative-ring
*A. Mathew, [http://people.fas.harvard.edu/~amathew/SCR.pdf Simplicial commutative rings, I].
*B. Toën, [http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf Simplicial presheaves and derived algebraic geometry]

{{algebra-stub}}

[[Category:Commutative algebra]]
[[Category:Ring theory]]
[[Category:Algebraic structures]]

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en>Rpyle731: stub sort