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.)
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
- http://mathoverflow.net/questions/94846/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