# 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 is a commutative ring and are modules over that ring (in fact, is a graded ring over .)

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 a structure of graded-commutative graded ring as follows.

By the Dold–Kan correspondence, is the homology of the chain complex corresponding to *A*; in particular, it is a graded abelian group. Next, to multiply two elements, writing for the circle, let be two maps. Then the composition

the second map the multiplication of *A*, induces . This in turn gives an element in . We have thus defined the graded multiplication . It is associative since the smash product is. It is graded-commutative (i.e., ) since the involution introduces minus sign.

## 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 .

## See also

## References

- http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory/
- http://mathoverflow.net/questions/45273/what-facts-in-commutative-algebra-fail-miserably-for-simplicial-commutative-ring
- A. Mathew, Simplicial commutative rings, I.
- B. Toën, Simplicial presheaves and derived algebraic geometry