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 $\pi _{0}A$ is a commutative 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.

Graded ring structure

Let A be a simplicial commutative ring. Then the ring structure of A gives $\pi _{*}A=\oplus _{i\geq 0}\pi _{i}A$ 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 since the smash product is. It is graded-commutative (i.e., $xy=(-1)^{|x||y|}yx$) since the involution $S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}$ introduces minus sign.

If M is a simplicial module over A (that is, M is a simplicial abelian group with the 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 $\operatorname {Spec} A$.

See also

• En-ring

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
• P. Goerss and K. Schemmerhorn, Model categories and simplicial methods