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 [math]\displaystyle{ \pi_0 A }[/math] is a ring and [math]\displaystyle{ \pi_i A }[/math] are modules over that ring (in fact, [math]\displaystyle{ \pi_* A }[/math] is a graded ring over [math]\displaystyle{ \pi_0 A }[/math].) 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 [math]\displaystyle{ \pi_* A = \oplus_{i \ge 0} \pi_i A }[/math] the structure of a graded-commutative graded ring as follows.
By the Dold–Kan correspondence, [math]\displaystyle{ \pi_* A }[/math] is the homology of the chain complex corresponding to A; in particular, it is a graded abelian group. Next, to multiply two elements, writing [math]\displaystyle{ S^1 }[/math] for the simplicial circle, let [math]\displaystyle{ x:(S^1)^{\wedge i} \to A, \, \, y:(S^1)^{\wedge j} \to A }[/math] be two maps. Then the composition
- [math]\displaystyle{ (S^1)^{\wedge i} \times (S^1)^{\wedge j} \to A \times A \to A }[/math],
the second map the multiplication of A, induces [math]\displaystyle{ (S^1)^{\wedge i} \wedge (S^1)^{\wedge j} \to A }[/math]. This in turn gives an element in [math]\displaystyle{ \pi_{i + j} A }[/math]. We have thus defined the graded multiplication [math]\displaystyle{ \pi_i A \times \pi_j A \to \pi_{i + j} A }[/math]. It is associative because the smash product is. It is graded-commutative (i.e., [math]\displaystyle{ xy = (-1)^{|x||y|} yx }[/math]) since the involution [math]\displaystyle{ S^1 \wedge S^1 \to S^1 \wedge S^1 }[/math] 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 [math]\displaystyle{ \pi_* M }[/math] has the structure of a graded module over [math]\displaystyle{ \pi_* A }[/math] (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 [math]\displaystyle{ \operatorname{Spec} A }[/math].
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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