Simplicial commutative ring

Template:Short description Template:Mergeto 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., $xy=(-1{)}^{|x||y|}yx$) 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 $\mathrm{Spec}A$.

See also

• E_n-ring

References

• What is a simplicial commutative ring from the point of view of homotopy theory?
• What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?
• 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

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