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 $$\pi_0 A$$ is a ring and $$\pi_i A$$ are modules over that ring (in fact, $$\pi_* A$$ is a graded ring over $$\pi_0 A$$.)

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

Examples

 * The ring of polynomial differential forms on simplexes.

Graded ring structure
Let A be a simplicial commutative ring. Then the ring structure of A gives $$\pi_* A = \oplus_{i \ge 0} \pi_i A$$ the structure of a graded-commutative graded ring as follows.

By the Dold–Kan correspondence, $$\pi_* A$$ 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 $$x:(S^1)^{\wedge i} \to A, \, \, y:(S^1)^{\wedge j} \to A$$ be two maps. Then the composition
 * $$(S^1)^{\wedge i} \times (S^1)^{\wedge j} \to A \times A \to A$$,

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

Spec
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 $$\operatorname{Spec} A$$.