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 π 0 A {\displaystyle \pi _{0}A} is a ring and π i A {\displaystyle \pi _{i}A} are modules over that ring (in fact, π ∗ A {\displaystyle \pi _{*}A} is a graded ring over π 0 A {\displaystyle \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 π ∗ A = ⊕ i ≥ 0 π 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, π ∗ 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 S 1 {\displaystyle S^{1}} for the simplicial circle, let x : ( S 1 ) ∧ i → A , y : ( S 1 ) ∧ j → A {\displaystyle x:(S^{1})^{\wedge i}\to A,\,\,y:(S^{1})^{\wedge j}\to A} be two maps. Then the composition
( S 1 ) ∧ i × ( S 1 ) ∧ j → A × A → 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 ( S 1 ) ∧ i ∧ ( S 1 ) ∧ j → A {\displaystyle (S^{1})^{\wedge i}\wedge (S^{1})^{\wedge j}\to A} . This in turn gives an element in π i + j A {\displaystyle \pi _{i+j}A} . We have thus defined the graded multiplication π i A × π j A → π 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., x y = ( − 1 ) | x | | y | y x {\displaystyle xy=(-1)^{|x||y|}yx} ) since the involution S 1 ∧ S 1 → S 1 ∧ 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 π ∗ M {\displaystyle \pi _{*}M} has the structure of a graded module over π ∗ A {\displaystyle \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 Spec A {\displaystyle \operatorname {Spec} A} .
See also E_n-ring
References What is a simplicial commutative ring from the point of view of homotopy theory? What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy? 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
