Simplicial commutative ring - Alchetron, the free social encyclopedia

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 over π 0 A .)

Graded ring structure

Let A be a simplicial commutative ring. Then the ring structure of A gives π ∗ A = ⊕ i ≥ 0 π i A the structure of a 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 simplicial circle, let x : ( S 1 ) ∧ i → A , y : ( S 1 ) ∧ j → A be two maps. Then the composition

( S 1 ) ∧ i × ( S 1 ) ∧ j → A × A → A ,

the second map the multiplication of A, induces ( S 1 ) ∧ i ∧ ( S 1 ) ∧ j → A . This in turn gives an element in π i + j A . We have thus defined the graded multiplication π i A × π j A → 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 ∧ S 1 → S 1 ∧ S 1 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 π ∗ M has the structure of a graded module over π ∗ 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 Spec ⁡ A .

References

Simplicial commutative ring Wikipedia

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Contents

Graded ring structure

Spec

References