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 formula_1 is a commutative ring and formula_2 are modules over that ring (in fact, formula_3 is a graded ring over formula_1.)
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 formula_5 the structure of a graded-commutative graded ring as follows.
By the Dold–Kan correspondence, formula_3 is the homology of the chain complex corresponding to "A"; in particular, it is a graded abelian group. Next, to multiply two elements, writing formula_7 for the simplicial circle, let formula_8 be two maps. Then the composition
the second map the multiplication of "A", induces formula_10. This in turn gives an element in formula_11. We have thus defined the graded multiplication formula_12. It is associative since the smash product is. It is graded-commutative (i.e., formula_13) since the involution formula_14 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 formula_15 has the structure of a graded module over formula_3. (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 formula_17.