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 ${\displaystyle \pi _{0}A}${\displaystyle \pi _{0}A} is a ring and ${\displaystyle \pi _{i}A}${\displaystyle \pi _{i}A} are modules over that ring (in fact, ${\displaystyle \pi _{*}A}${\displaystyle \pi _{*}A} is a graded ring over ${\displaystyle \pi _{0}A}${\displaystyle \pi _{0}A}.)

A topology-counterpart of this notion is a commutative ring spectrum.

• The ring of polynomial differential forms on simplexes.

Let A be a simplicial commutative ring. Then the ring structure of A gives ${\displaystyle \pi _{*}A=\oplus _{i\geq 0}\pi _{i}A}${\displaystyle \pi _{*}A=\oplus _{i\geq 0}\pi _{i}A} the structure of a graded-commutative graded ring as follows.

By the Dold–Kan correspondence, ${\displaystyle \pi _{*}A}${\displaystyle \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 ${\displaystyle S^{1}}${\displaystyle S^{1}} for the simplicial circle, let ${\displaystyle x:(S^{1})^{\wedge i}\to A,,円,円y:(S^{1})^{\wedge j}\to A}${\displaystyle x:(S^{1})^{\wedge i}\to A,,円,円y:(S^{1})^{\wedge j}\to A} be two maps. Then the composition

${\displaystyle (S^{1})^{\wedge i}\times (S^{1})^{\wedge j}\to A\times A\to A}${\displaystyle (S^{1})^{\wedge i}\times (S^{1})^{\wedge j}\to A\times A\to A},

the second map the multiplication of A, induces ${\displaystyle (S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A}${\displaystyle (S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A}. This in turn gives an element in ${\displaystyle \pi _{i+j}A}${\displaystyle \pi _{i+j}A}. We have thus defined the graded multiplication ${\displaystyle \pi _{i}A\times \pi _{j}A\to \pi _{i+j}A}${\displaystyle \pi _{i}A\times \pi _{j}A\to \pi _{i+j}A}. It is associative because the smash product is. It is graded-commutative (i.e., ${\displaystyle xy=(-1)^{|x||y|}yx}${\displaystyle xy=(-1)^{|x||y|}yx}) since the involution ${\displaystyle S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}}${\displaystyle S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}} introduces a 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 ${\displaystyle \pi _{*}M}${\displaystyle \pi _{*}M} has the structure of a graded module over ${\displaystyle \pi _{*}A}${\displaystyle \pi _{*}A} (cf. Module spectrum).

By definition, the category of affine derived schemes is the opposite category of the category of simplicial commutative rings; an object corresponding to A will be denoted by ${\displaystyle \operatorname {Spec} A}${\displaystyle \operatorname {Spec} A}.

• E_n-ring

• What is a simplicial commutative ring from the point of view of homotopy theory?
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• Reference request - CDGA vs. sAlg in char. 0
• A. Mathew, Simplicial commutative rings, I.
• B. Toën, Simplicial presheaves and derived algebraic geometry
• P. Goerss and K. Schemmerhorn, Model categories and simplicial methods

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