Simplicial commutative ring explained

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 = ⊕ _{i} \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} \to A, y:(S ^{1)} ^{\wedge} \to A be two maps. Then the composition(S ^{1)} ^{\wedge} x (S ^{1)} ^{\wedge} \to A x A \to A ,the second map the multiplication of A, induces (S ^{1)} ^{\wedge} \wedge (S ^{1)} ^{\wedge} \to A . This in turn gives an element in \pi _{i} A . We have thus defined the graded multiplication \pi _{i} A x \pi _{j} A \to \pi _{i} 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 .

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