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 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 since the smash product is. It is graded-commutative (i.e., ) since the involution introduces 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 has the structure of a graded module over . (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 .
- A. Mathew, Simplicial commutative rings, I.
- B. Toën, Simplicial presheaves and derived algebraic geometry
- P. Goerss and K. Schemmerhorn, Model categories and simplicial methods
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