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 ${\displaystyle \pi _{0}A}$ 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.

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

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