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 ring and are modules over that ring
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 the structure of a 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 simplicial 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 because the smash product is. It is graded-commutative since the involution introduces a minus sign.
If M is a simplicial module over A, then the similar argument shows that has the structure of a graded module over .