simplicial commutative ring

{{Short description|Simplicial commutative ring: commutative monoid in simplicial abelian groups}}

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 $\pi_* A = \oplus_{i \ge 0} \pi_i A$ 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 $x:(S^1)^{\wedge i} \to A, \, \, y:(S^1)^{\wedge j} \to A$ be two maps. Then the composition

: $(S^1)^{\wedge i} \times (S^1)^{\wedge j} \to A \times A \to A$ ,

the second map the multiplication of A, induces $(S^1)^{\wedge i} \wedge (S^1)^{\wedge j} \to A$ . This in turn gives an element in $\pi_{i + j} A$ . We have thus defined the graded multiplication $\pi_i A \times \pi_j A \to \pi_{i + j} A$ . It is associative because the smash product is. It is graded-commutative (i.e., $xy = (-1)^$

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 $\operatorname{Spec} A$ .

References

[https://mathoverflow.net/q/118500 What is a simplicial commutative ring from the point of view of homotopy theory?]
[https://mathoverflow.net/q/45273 What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?]
[https://mathoverflow.net/q/94846 Reference request - CDGA vs. sAlg in char. 0]
A. Mathew, [https://math.uchicago.edu/~amathew/SCR.pdf Simplicial commutative rings, I].
B. Toën, [http://wayback.archive-it.org/all/20090625184038/http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf Simplicial presheaves and derived algebraic geometry]
P. Goerss and K. Schemmerhorn, [http://www.math.northwestern.edu/~pgoerss/papers/ucnotes.pdf Model categories and simplicial methods]

Category:Commutative algebra

Category:Ring theory

Category:Algebraic structures

simplicial commutative ring

Examples

Graded ring structure

Spec

See also

References