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Simplicial commutative ring - Wikipedia

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

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

Graded ring structure

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Let A be a simplicial commutative ring. Then the ring structure of A gives {\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} 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}} for the simplicial circle, let {\displaystyle x:(S^{1})^{\wedge i}\to A,\,\,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},

the second map the multiplication of A, induces {\displaystyle (S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A}. This in turn gives an element in {\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}. It is associative because the smash product is. It is graded-commutative (i.e., {\displaystyle xy=(-1)^{|x||y|}yx}) since the involution {\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} has the structure of a graded module over {\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}.