Fuzzy cold dark matter

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 $π_{0} A$ is a commutative ring and $π_{i} A$ are modules over that ring (in fact, $π_{*} A$ is a graded ring.)

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 $π_{*} A = \oplus_{i \geq 0} π_{i} A$ a structure of graded-commutative graded ring as follows.

By the Dold–Kan correspondence, $π_{*} A$ 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 circle, let $x : (S^{1})^{\land i} \to A, y : (S^{1})^{\land j} \to A$ be two maps. Then the composition

(S^{1})^{\land i} \times (S^{1})^{\land j} \to A \times A \to A

,

the second map the multiplication of A, induces $(S^{1})^{\land i} \land (S^{1})^{\land j} \to A$ . This in turn gives an element in $π_{i + j} A$ . We have thus defined the graded multiplication $π_{i} A \times π_{j} A \to π_{i + j} A$ . It is associative since the smash product is. It is graded-commutative (i.e., $x y = (- 1)^{| x | | y |} y x$ ) since the involution $S^{1} \land S^{1} \to S^{1} \land S^{1}$ introduces minus sign.

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 $Spec A$ .

References

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Fuzzy cold dark matter

Graded ring structure

Spec

References

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